Stochastic Differential Equations

Table of Contents

Books

  • Handbook of Stochastic Methods
  • Øksendal
  • Probability with Martingales

Overview

Notation

  • Correlation between two points of a random field or a random process:

    stochastic_differential_equations_b6fefbbfecc163da9eda0b78abeba4a3f1409b78.png

    to allow the possibility of an infinite number of points. In the discrete case this simply corresponds to the covariance matrix.

  • stochastic_differential_equations_4cfeb31220b71ee5c6cf136d1c2f47a3bdd153fd.png denotes the Borel σ-algebra
  • Partition stochastic_differential_equations_adbed33ba6743f55440990abf67d367b2c3cbc58.png
  • We write

    stochastic_differential_equations_ed2b1428da41d5cce17c5a43ae8fac9ffbca3762.png

  • Whenever we write stochastic integrals as Riemann sums, e.g.

    stochastic_differential_equations_3fca1544394ed0c9f753432cbfe77596efc50150.png

    the limit is to be understood in the mean-square sense, i.e.

    stochastic_differential_equations_712c66ed152d6299393d867dacfde4c761121fab.png

Definitions

A stochastic process stochastic_differential_equations_d24d8ff8fdbefc1be713e8fde780aa760b1e4653.png is called a either

  • second-order stationary
  • wide-sense stationary
  • weakly stationary

if

  • the first moment stochastic_differential_equations_901323913af379db3f4c91225077dd1d49832b5a.png is constant
  • covariance function depends only on the difference stochastic_differential_equations_a9240dd66a25078ca2735b0ab4a55fa2c835c56f.png

That is,

stochastic_differential_equations_e4bbac19f1c89ac9aabd01fc2aa0269cbf5c2b13.png

A stochastic process is called (strictly) stationary if all FDDs are invariant under time translation, i.e. for all stochastic_differential_equations_4028f9016d4f7a233195bc931939fa3360d350cd.png, for all times stochastic_differential_equations_9031fd9ea61eddd973e9aaf239a41ab0e39146a1.png, and stochastic_differential_equations_bab97b711c11e32a34dc71cdfc8688b699715f57.png,

stochastic_differential_equations_3879477542882ed82a96d0a6f2d542dffb99cce4.png

for stochastic_differential_equations_9b29b54e0b40c77a57b81caebd04f9c429fd621f.png such that stochastic_differential_equations_757ea720d20cfb357676642dfa913f066ec77993.png, for every stochastic_differential_equations_059da22782fe5de96d484ed620f73a04f028353f.png.

The autocorrelation function of a second-order stationary process enables us to associate a timescale to stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png, the correlation time stochastic_differential_equations_390bdfc6df6f0678c6b797a3e4ff42e6906e4f54.png:

stochastic_differential_equations_16450924bc93ad385541e5bac28810e11998669f.png

Martingale continuous processes

Let stochastic_differential_equations_91195f4c6a97664010c0350a15b1cb62e43a8141.png be a filtration defined on the probability space stochastic_differential_equations_f102c350ebf78bdee7fed177da9ef88677e0bba1.png and let stochastic_differential_equations_269b624bd3f043caca2980f808a87e8acbe430c4.png be adapted to stochastic_differential_equations_2d2ef1bebb51aa5422b07a53d7a29158099d4746.png, i.e. stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png is measurable on stochastic_differential_equations_2d2ef1bebb51aa5422b07a53d7a29158099d4746.png, with stochastic_differential_equations_416fb2d3b913de6dcc978023b371f63af8dc19a7.png.

We say stochastic_differential_equations_3442b2d48281c4f83a46a8c3529cf9bea69732a7.png is an stochastic_differential_equations_2d2ef1bebb51aa5422b07a53d7a29158099d4746.png martingale if

stochastic_differential_equations_da9de33d4969d56f4c0aeed161664cec6b250671.png

Gaussian process

A 1D continuous-time Gaussian process is a stochastic process for which stochastic_differential_equations_f191f14fe87bfb31cb022158faa708a689eae8f2.png, and all finite-dimensional distributions are Gaussians.

That is, for every finite dimensional vector

stochastic_differential_equations_bf9c9c831afed21df4d1ddbb664f37ec0096f661.png

for some symmetric non-negative definite matrix stochastic_differential_equations_3d1eecc5cbbd1a2fe8e079afc3b0aba8b06e675c.png, for all stochastic_differential_equations_4028f9016d4f7a233195bc931939fa3360d350cd.png and stochastic_differential_equations_392eca1c0e8a6c11d08f54c4600872d42c4cb5f6.png.

Theorems

Bochner's Theorem

Let stochastic_differential_equations_a69a3cf8b5bd3a05a68b76e881cb376a1c4ef4ce.png be a continuous positive definite function.

Then there exists a unique nonnegative measure stochastic_differential_equations_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png on stochastic_differential_equations_492d525117d0dcc93d066c8759f46b98cf9980ca.png such that stochastic_differential_equations_9e35e0b9e7edba64d2cd75349c4b39a409fa3e7c.png and

stochastic_differential_equations_f6d159b4c7e095f86fbec8ee4ead2a3a194a21c1.png

i.e. stochastic_differential_equations_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png is the Fourier transform of the function stochastic_differential_equations_a69a3cf8b5bd3a05a68b76e881cb376a1c4ef4ce.png.

Let stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png be a second-order stationary process with autocorrelation function stochastic_differential_equations_a69a3cf8b5bd3a05a68b76e881cb376a1c4ef4ce.png whose Fourier transform is stochastic_differential_equations_0ac13efc8f002a53139de9c9a1b92f54049365c9.png.

The measure stochastic_differential_equations_0ac13efc8f002a53139de9c9a1b92f54049365c9.png is called the spectral measure of the process stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png.

If the spectral measure is absolutely continuous wrt. the Lebesgue measure on stochastic_differential_equations_492d525117d0dcc93d066c8759f46b98cf9980ca.png with density stochastic_differential_equations_9258c6b3a034b8ea36395782a0d44fd02f749f41.png, i.e.

stochastic_differential_equations_b7dcde9b59bf8f158a01f70b37db75aa813a260f.png

then the Fourier transform stochastic_differential_equations_9258c6b3a034b8ea36395782a0d44fd02f749f41.png of the covariance function is called the spectral density of the process:

stochastic_differential_equations_a1bad37ded0e12fc7f66dac58f93421bf2ba92a1.png

Ito vs. Stratonovich

  • Purely matheamtical viewpoint: both Ito and Stratonovich calculi are correct
  • Ito SDE is appropriate when continuous approximation of a discrete system is concerned
  • Stratonovich SDE is appropriate when the idealization of a smooth real noise process is concerned

Benefits of Itô stochastic integral:

Benefits of Stratonovich stochastic integral:

  • Leads to the standard Newton-Leibniz chain rule, in contrast to Itô integral which requires correction
  • SDEs driven by noise with nonzero correlation time converge to the Stratonovich SDE, in the limit as the correlation time tends to 0

Practical considerations

  • Rule of thumb:
    • White noise is regarded as short-correlation approximation of a coloured noise → Stratonovich integral is natural
      • "Expected" since the standard chain rule should work fro smooth noise with finite correlation

Equivalence

Suppose stochastic_differential_equations_0aa7a82dd2200a79757418fb0536194414c8367e.png solves the following Stratonovich SDE

stochastic_differential_equations_97d74a7ca6d6b24f0ec7c4c531fb1a82e23e8053.png

then stochastic_differential_equations_0aa7a82dd2200a79757418fb0536194414c8367e.png solves the Itô SDE:

stochastic_differential_equations_50c7458b26f6d83989afc858829745b632735d3e.png

Suppose stochastic_differential_equations_0aa7a82dd2200a79757418fb0536194414c8367e.png solves the Itô SDE:

stochastic_differential_equations_023f965f2a0ff5b8c5b26c59f0f92771081641b6.png

then stochastic_differential_equations_0aa7a82dd2200a79757418fb0536194414c8367e.png solves the Stratonovich SDE:

stochastic_differential_equations_0ca6fd4eaf5df063b38d3d1e4f0dc19360c21145.png

That is, letting stochastic_differential_equations_70b63aa7d1b5f9535a68a669d097e02b8ffe9e36.png,

  • Stratonovich → Itô: stochastic_differential_equations_3d15729343378bd62a469aa72e1bea426ec700a8.png
  • Itô → Stratonovich: stochastic_differential_equations_3f1e067f6fccb15969bc87bfe81df35f68d76d03.png

In the multidimensional case,

stochastic_differential_equations_ea931816e81cf8e60e41f8b0042284c9dc20b541.png

To show this we consider stochastic_differential_equations_aa07b3a8458adb2855b54064282b1f340d44fbd4.png staisfying the two SDEs

stochastic_differential_equations_071094367ff0bbc633e4fd70803526290254f6f4.png

i.e. the satisfying the stochastic integrals

stochastic_differential_equations_136a9c05e7082f70f75daf629dd090ae2789ab70.png

We have

stochastic_differential_equations_a70f51ab34b6d01bd582c89e78cfb39185495f4d.png

Equivalently, we can write

stochastic_differential_equations_40b540e387cf659721b119b4fbd9e65709abdab7.png

Assuming the stochastic_differential_equations_cbe22aae4a254975871e1a038c5f05012644de2b.png is smooth, we can then Taylor expand about stochastic_differential_equations_2066bde249cf63535bfebda55cc5d0655225f8c9.png and evaluate at stochastic_differential_equations_d6e3dd48ff5cfa99112aeda72a20e39a03c4d382.png:

stochastic_differential_equations_af52ddf5aeabe1f0c539c7a02c36e9b9c726d178.png

Substituting this back into the Riemann series expression for the Stratonovich integral, we have

stochastic_differential_equations_11384e7fdae7abbebb579f43a4eb5e8d9c5d6dd9.png

Using the fact that stochastic_differential_equations_0aa7a82dd2200a79757418fb0536194414c8367e.png satisfies the Itô integral, we have

stochastic_differential_equations_6c782e70c94ac1919fccafcb5ab13e812d63b033.png

and that

stochastic_differential_equations_c1d7dc0dc502013ff60f764df893ca6d7af70727.png

we get

stochastic_differential_equations_69d64522c7a77e4228f1c144cc92297bf000f568.png

Thus, we have the identity

stochastic_differential_equations_d6487bb3fdc4baf718fb630bc948b0eb2fa3f23a.png

Matching the coefficients with the Itô integral satisfied by stochastic_differential_equations_0aa7a82dd2200a79757418fb0536194414c8367e.png, we get

stochastic_differential_equations_c37763402a85a8d7e6d6410a3863f3525a34a6c8.png

which gives us the conversion rules between the Stratonovich and Itô formulation!

Important to note the following though: here we have assumed that the stochastic_differential_equations_ef2c7c7b1ee0ca46af75c3b9f843f7ca173adbb8.png is smooth, i.e. infinitely differentiable and therefore locally Lipschitz. Therefore our proof only holds for this case.

I don't know if one can relax the smoothness constraint of stochastic_differential_equations_de32b094129e4e03f562577e6b8b51603d8ca54e.png and still obtain conversion rules between the two formulations of the SDEs.

Stratonovich satisfies chain rule proven using conversion

For some function stochastic_differential_equations_3b4155ffb83fa2b8083106e9e638e123ccdff388.png, one can show that the Stratonovich formulation satisfies the standard chain rule by considering the Stratonovich SDE, converting to Itô, apply Itô's formula, and then converting back to Stratonovich SDE!

Variations of Brownian motion

Ornstein-Uhlenbeck process

Consider a mean-zero second-order stationary process with correlation function

stochastic_differential_equations_4d1f7d25d1abcedb59f59e6c915aa1000a7d364a.png

We will write stochastic_differential_equations_5bb3b76947643a0f6b8062d195c5eec3635d67e7.png, where stochastic_differential_equations_c765079ce73fda08d248c04a425ccf2d716d6e00.png.

The spectral density of this process is

stochastic_differential_equations_916a2bc7466754699cdbe5edce8696146bdb1c01.png

This stochastic_differential_equations_9258c6b3a034b8ea36395782a0d44fd02f749f41.png is called a Cauchy or Lorentz distribution.

The correlation time is then

stochastic_differential_equations_e76861174beb60d4904369294762132ff909d9b0.png

A real-valued Gaussian stationary process defiend on stochastic_differential_equations_492d525117d0dcc93d066c8759f46b98cf9980ca.png with correlation function as given above is called a Ornstein-Uhlenbeck process.

Look here to see the derivation of the Ornstein-Uhlenbeck process from it's Markov semigroup generator.

Fractional Brownian Motion

A (normalized) fractional Brownian motion stochastic_differential_equations_08e439d48d7aec2de8642a5f2b78e6812be456bb.png, stochastic_differential_equations_809038084ca90dd90f9bde810a570e42ccd59b74.png, with Hurst parameter stochastic_differential_equations_657b8d01e112525b363b6569e524490fc0eeaaa4.png is a centered Gaussian process with continuous sample paths whose covariance is given by

stochastic_differential_equations_3388bc0027857fe34032d1d99f963573b910d160.png

Hence, the Hurst parameter controls:

  • the correlations between the increments of fractional Brownian motion
  • the regularity of the paths: they become smoother as stochastic_differential_equations_14a40f189f2ce698341c03cd5c099a337431c49f.png increases.

A fractional Brownian motion has the following properties

  1. When stochastic_differential_equations_d0d553244a02ac76b18e8e0ceb3f6fe80fae99a2.png, then stochastic_differential_equations_f75cc71491cc5f6410acbfd5619db759899dd845.png becomes standard Brownian motion.
  2. We have

    stochastic_differential_equations_abe55f08f25dc0b7a6084dcbfb8efe6132da54d8.png

  3. It has stationary increments, and

    stochastic_differential_equations_843aba59deb2a7a544e19f91a36533e5db95629c.png

  4. It has the following self-similarity property:

    stochastic_differential_equations_6eb609494e646c820fc0013277d1de003421217f.png

    where the equivalence is in law.

Karhunen-Loève Expansion

Notation

  • stochastic_differential_equations_14bbd3042cee20afa2c679d631174506f4f3dcd9.png where stochastic_differential_equations_4118f3a6603f379f7f04731bd16a998a80af1255.png
  • stochastic_differential_equations_ee4411dc59e01200286fcebd478705a471aa6c88.png be an orthonormal basis in stochastic_differential_equations_224aae5907c7f36cef5ef997eba42b139b2130c3.png
  • stochastic_differential_equations_03f94ddd05578749c78f22cd4df4a843d54e1206.png
  • stochastic_differential_equations_719de29b20d6dc683c764c700bba774afc1bc487.png

Stuff

Let stochastic_differential_equations_1e7c4cf43681a9741fbd9d0d93a6e48f24ca20f2.png.

Suppose

stochastic_differential_equations_85d0b4dee2711458acb6c38fcaeea74d7d622fc9.png

We assume stochastic_differential_equations_ed68bec31eed1be9a13d0aa43405da1a7953201b.png are orthogonal or independent, and

stochastic_differential_equations_c1c9aac077a20793be5654cfc656289b1de1e620.png

for some positive numbers stochastic_differential_equations_d43f2a09e7a35aea0bc954aa52add718bc987125.png.

Then

stochastic_differential_equations_9cae6eff77372fbeebaea110172e8b6b03815048.png

due to orthogonality of stochastic_differential_equations_2ce73aaf3979a944b1f3cada28e0297e8973cba7.png and stochastic_differential_equations_34dd97d52e71f76366362427597380b22c23536a.png for stochastic_differential_equations_7709f5c4ce81a164f8eaaeb01ef632eb76b85dc9.png.

Hence, for the expansion of stochastic_differential_equations_981e568ee4a5054e25521bdd79fe25c3c2830f61.png above, we need stochastic_differential_equations_636b319745e2a2d8eaa9b44d72201ddd548e4a53.png to be valid!

The above expression for stochastic_differential_equations_07dba20997ebf113f96e739384a740efd8bc5997.png also implies

stochastic_differential_equations_75ee8bc19a94b089b607aee1d30ea4c1427d9bd5.png

Hence, we also need the set stochastic_differential_equations_e9a46b333b6d614906e8fd14da4589d37ec470e6.png to be a set of eigenvalues and eigenvectors of the integral operator whose kernel is the correlation function of $Xt, i.e. need to study the operator

stochastic_differential_equations_03c09e3dcf3cbc13249f9e470a2d6ed846b73d0d.png

which we will now consider as an operator on stochastic_differential_equations_fbf1b6a894caca47a3d73428873e64f9763e3200.png.

It's easy to see that stochastic_differential_equations_980ba78e56bdd74e2c28da31957ba01c25234a8e.png is self-adjoint and nonnegative in stochastic_differential_equations_b3da908bd310e68de6607fba0fec2bcfdfdc6cde.png:

stochastic_differential_equations_9acb3c51041eacb0e6ab948ee6fa78b1a6ded83d.png

Furthermore, it is a compact operator, i.e. if stochastic_differential_equations_11cf69a8247ff31f4e0d5d986b000d83bbd5dfa5.png is a bounded sequence on stochastic_differential_equations_b3da908bd310e68de6607fba0fec2bcfdfdc6cde.png, then stochastic_differential_equations_bf678e8dea65c40d20a02af2058160969ac4417e.png has a convergent subsequence.

Spectral theorem for compact self-adjoint operators can be used to deduce that stochastic_differential_equations_980ba78e56bdd74e2c28da31957ba01c25234a8e.png has a countable sequence of eigenvalues tending to stochastic_differential_equations_96f53e8f2667720f54bd85623f46cbf545733989.png.

Furthermore, for every stochastic_differential_equations_e2a0179e90d44fd5864d9b7201b7253a4d8ccdf7.png, we can write

stochastic_differential_equations_756b6ea03bec8d98440706372836ae66e4a52d68.png

where stochastic_differential_equations_9f0f9eae834a0bead4cfd065018940165f4ecc85.png and stochastic_differential_equations_8e7f6e53ab111b368863053e550aa0f4932bcbf4.png are the eigenfunctions of the operator stochastic_differential_equations_980ba78e56bdd74e2c28da31957ba01c25234a8e.png corresponding to the non-zero eigenvalues and where the convergence is in stochastic_differential_equations_5d229ea7e71b918baa91f8bbda3a25e08b64ceb5.png, i.e. we can "project" stochastic_differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png onto the subspace spanned by eigenfunctions of stochastic_differential_equations_980ba78e56bdd74e2c28da31957ba01c25234a8e.png.

Let stochastic_differential_equations_48f66a3557a7ababc1429fa822e43bbbc2e2ac7a.png be an stochastic_differential_equations_5d229ea7e71b918baa91f8bbda3a25e08b64ceb5.png process with zero mean and continuous correlation function stochastic_differential_equations_07dba20997ebf113f96e739384a740efd8bc5997.png.

Let stochastic_differential_equations_e42511b2d8a8583b45d545e83c2c9c732f7ed7a1.png be the eigenvalues and eigenfunctions of the operator stochastic_differential_equations_980ba78e56bdd74e2c28da31957ba01c25234a8e.png defined

stochastic_differential_equations_919aa02ec92d62b7cccddd102d771d0a49356547.png

Then

stochastic_differential_equations_079426387974f6d5bbcb2af109c1f470d405466b.png

where

stochastic_differential_equations_d2685f3aae75d80ed1e42cdc07c39ec6350f8e82.png

The series converges in stochastic_differential_equations_5d229ea7e71b918baa91f8bbda3a25e08b64ceb5.png to stochastic_differential_equations_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, uniformly in stochastic_differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png!

Karhunen-Loève expansion of Brownian motion

  • Correlation function of Brownian motion is stochastic_differential_equations_249c6cf3141f721e563cd330b99ed54510e6ebce.png.
  • Eigenvalue problem stochastic_differential_equations_b0a97d061a7ed47fa8cddef45032b66350fd9a91.png becomes

    stochastic_differential_equations_01584b7a0dccb3eb38bbf3daefe469b22deb79cf.png

  • Assume stochastic_differential_equations_ac25eeebb16c84a3e9de9146c2882bcb20d35aef.png (since stochastic_differential_equations_8441b3a29b827eb54f5941a74cd06a22158873f3.png would imply ψn(t) = 0$)
  • Consider intial condition stochastic_differential_equations_9b50c3c5095e3def3ea0a38afe0602245b66e068.png which gives stochastic_differential_equations_aed879b4cf9cad2102a5f2b0361aada345b8fc4f.png
  • Can rewrite eigenvalue problem

    stochastic_differential_equations_40b17ba986f647b46a42e35fe72aafbe786a4649.png

  • Differentiatiate once

    stochastic_differential_equations_e8399757fcca0ed2b53b93f616c8ca1325178aff.png

    using FTC, we have

    stochastic_differential_equations_4f5b4b4485d7f77041efd0e80b5ca0893bf29768.png

    hence

    stochastic_differential_equations_a09ad8fbfa1542e3521aa22e2fb14a9d24667dbb.png

  • Obtain second BC by observing stochastic_differential_equations_16eec9fb556df079e3496bb47ae9a947a7e6badb.png (since LHS in the above is clearly 0)
  • Second differentiation

    stochastic_differential_equations_a86ba02f1a61e2637e34217181110f13ed2dbfa1.png

  • Thus, the eigenvalues and eigenfunctions of the integral operator whose kernel is the covariance function of Brownian motion can be obtained as solutions to the Sturm-Lioville problem

    stochastic_differential_equations_9228bd7b146641c4e86c2895b7b95e499b7f394b.png

  • Eigenvalues and (normalized) eigenfunctions are then given by

    stochastic_differential_equations_c08b41e4142c3a57a13a3b8a96f16d03e78b39b7.png

  • Karhunen-Loève expansion of Brownian motion on stochastic_differential_equations_5cb819dbdaa11557a460fe04a5ef95eede596daa.png is then

    stochastic_differential_equations_7438162195a6ff2f600416731bbcaac253aa6aba.png

Diffusion Processes

Notation

  • stochastic_differential_equations_12b2a1240e0f4c77cdd07ea0cd6ddc2b4e98482a.png denotes a Borel set
  • stochastic_differential_equations_f102c350ebf78bdee7fed177da9ef88677e0bba1.png denotes a probability space
  • stochastic_differential_equations_803119c4a5913283ae6cd2510fd0756df89b84d3.png denotes a stochastic processes with stochastic_differential_equations_2013b8f2d322da7273b190ad0053ed0d0bf196d5.png and state space stochastic_differential_equations_fd1cd9131432292b0427c985bc4643a3f1a23228.png
  • stochastic_differential_equations_b3813d7a7ac6722041a1a450ba4af1dc6e81f340.png denotes the σ-algebra generated by stochastic_differential_equations_04a5544f47ccf44be619d1f127d8bf32894a663f.png, which is the smallest σ-algebra s.t. stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png is a measurable function (random variable) wrt. it.

Markov Processes and the Chapman-Kolmogorov Equation

We define the σ-algebra generated by stochastic_differential_equations_2b7751b19b96dc7c47e3ca69a4c2d85ce59c0140.png, denoted stochastic_differential_equations_739f27061f9b4c4709797cd178016bf1d2b1264e.png, to be the smallest σ-algebra s.t. the family of mappings stochastic_differential_equations_04a5544f47ccf44be619d1f127d8bf32894a663f.png is a stochastic process with

  • sample space stochastic_differential_equations_0a8d11f2eea0051b25ff9023c04a02f1327c0c98.png
  • state space stochastic_differential_equations_fd1cd9131432292b0427c985bc4643a3f1a23228.png
  • Idea: encode all past information about a stochastic process into an appropriate collection of σ-algebras

Let stochastic_differential_equations_f102c350ebf78bdee7fed177da9ef88677e0bba1.png denote a probability space.

Consider stochastic process stochastic_differential_equations_803119c4a5913283ae6cd2510fd0756df89b84d3.png with stochastic_differential_equations_5aea7acb24e183e2e5894435aeceb51ac678e06f.png and state space stochastic_differential_equations_fd1cd9131432292b0427c985bc4643a3f1a23228.png.

A filtration on stochastic_differential_equations_76a6b32a6ae7c805ba0c86839c9d8034d38269ec.png is a nondecreasing family stochastic_differential_equations_b13636dcea0c803fe5587d55092d5cf942fa8bda.png of sub-σ-algebras of stochastic_differential_equations_c9dac3d8c7645fbcf8461e182d255264c7484369.png:

stochastic_differential_equations_69c1ec697c0fd735c6a6689461c4b00a725f02c8.png

We set

stochastic_differential_equations_afd2d4180c3f6cd5b8057ebf0aa48b6bb84af357.png

The filtration generated by our stochastic process stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png is

stochastic_differential_equations_92baeff5b641f887ccb01bc3547d5cfa92a6f60d.png

A filtration stochastic_differential_equations_a1a55f76a82938898b7538ef7199ff921982f5d7.png is generated by events of the form

stochastic_differential_equations_b82bedf500350bc2c8fe4d5a96ca4c9970c8e1ae.png

with stochastic_differential_equations_7e0548363691b8f4e260d4c425e2d27079ecbb5e.png and stochastic_differential_equations_790d14f488eb46d06254da85f071983f52ac5534.png.

Let stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png be a stochastic process defined on a probability space stochastic_differential_equations_f102c350ebf78bdee7fed177da9ef88677e0bba1.png with values in stochastic_differential_equations_abf17ab8d81a8a9c9a3b8d37b4e26b6e2983cdc7.png, and let stochastic_differential_equations_a1a55f76a82938898b7538ef7199ff921982f5d7.png be the filtration generated by stochastic_differential_equations_205b671c937a14feac020f3cca438c8c5a197907.png.

Then stochastic_differential_equations_ba57ac4b5cc8e92752096b46a10dbabbbfc7e2f9.png is a Markov process if

stochastic_differential_equations_25db1937e106fbf0f39e82cb414bd337347e4e24.png

for all stochastic_differential_equations_fe0a516c3b15752ca60b3151fe4fc8ebc6aa30c5.png with stochastic_differential_equations_1e7cccf6308d069c26d963ec51048089768a8982.png, and stochastic_differential_equations_478ce666e884cdc0273c01c7f270e50b6e959a16.png.

Equivalently, it's a Markov process if

stochastic_differential_equations_4b0deb1cc77459ac93f6168f3e037b9b1594d01c.png

for stochastic_differential_equations_ddadf38e9b6f2913a5b64670ddde97d2cddaf0d0.png and stochastic_differential_equations_7e0548363691b8f4e260d4c425e2d27079ecbb5e.png with $Γ ∈ (E).

Chapman-Kolmogorov Equation

The transition function stochastic_differential_equations_fe7601cfed79dc62479379bef4f498cddf3d9c16.png for fixed stochastic_differential_equations_71a8b29228c2f55b15f7bed0aa4e358d00c84a23.png is a probability measure on stochastic_differential_equations_abf17ab8d81a8a9c9a3b8d37b4e26b6e2983cdc7.png with

stochastic_differential_equations_3814e736002bdefe34205a5a285f2c18ec86bfce.png

It is stochastic_differential_equations_92dcfc70ed5a1cfdb7ed761e2fda06960cd99850.png measurable in stochastic_differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png, for fixed stochastic_differential_equations_676a81b7f1ab4f3b61a5d3c14c8efe29c3ddd870.png, and satisfies the Chapman-Kolmogorov equation

stochastic_differential_equations_8175674fccb9b03487b21d23f2863c5a3f092ebc.png

for all

  • stochastic_differential_equations_b50d3f9b65d6a48f7ccf964f392ac87b16b3c3e6.png
  • stochastic_differential_equations_478ce666e884cdc0273c01c7f270e50b6e959a16.png
  • stochastic_differential_equations_13fca4efa0e3d8a20e413d059a4b9f71bc800be9.png with stochastic_differential_equations_d3601cace05cca464756267163e23b493eef6fce.png

Assuming that stochastic_differential_equations_f58fe9f2460438bf91c94b19e93e21918220e43b.png, we can write

stochastic_differential_equations_2d5baaa5c1bbf9c77698276c1deb67471e2a82e1.png

since stochastic_differential_equations_56a6bacec8d0078cf2612a1acc7942ecda682911.png.

In words, the Chapman-Kolmogorov equation tells us that for a Markov process, the transition from stochastic_differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png at time stochastic_differential_equations_aa1698cb8ee1665238ec3e91824191643c62ee93.png to the set stochastic_differential_equations_12b2a1240e0f4c77cdd07ea0cd6ddc2b4e98482a.png at time stochastic_differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png can be done in two steps:

  1. System moves from stochastic_differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png to stochastic_differential_equations_eb83d466c7d035356e9f39998f357cee73da1e26.png at some intermediate step stochastic_differential_equations_8b0e7a5ea4d56f5afdac3c2658ca87bd50b0ad6f.png
  2. Moves from stochastic_differential_equations_eb83d466c7d035356e9f39998f357cee73da1e26.png to stochastic_differential_equations_12b2a1240e0f4c77cdd07ea0cd6ddc2b4e98482a.png at time stochastic_differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png

Generator of a Markov Process

  • Chapman-Kolmogorov equation suggests that a time-homogenous Markov process can be described through a semigroup of operators, i.e. a one-parameter family of linear operators with the properties

    stochastic_differential_equations_cd4c4bba724602801877879133f3b183e19f8922.png

Let stochastic_differential_equations_f98a47c50b5509394314555ed66dcc302691f62a.png be the transition function of a homogenous Markov process and let stochastic_differential_equations_17d30f4b4993004b41c46520e85a311a135da1a5.png, and define the operator

stochastic_differential_equations_1181425b5ab3ae7543742b69c8906c8462540c04.png

Linear operator with

stochastic_differential_equations_837972bf2e4e065fa528d3e089a60de1fc89f349.png

which means that stochastic_differential_equations_a9d922cdb048520616ea984ae7224b701c027866.png, and

stochastic_differential_equations_ab2823337d5daf351f6dcdc631da313745ba184f.png

i.e. stochastic_differential_equations_c940740799059316b1f4bbed34743aac3d4f56df.png.

We can study properties of time-homogenous Markov process stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png by studying properties of the Markov semigroup stochastic_differential_equations_29f9bc93a72f4a53ea800c87ee69c7038e63449b.png.

This is an example of a strongly continuous semigroup.

Let stochastic_differential_equations_509bf5694a9c030beb9706e35c8757438777e090.png be set of all stochastic_differential_equations_9fa427e550077da0b171a2d0f95238fb970c065b.png such that the limit

stochastic_differential_equations_24bb6e693a14208a155cfc1e2536ed9bed9f855a.png

exists.

The operator stochastic_differential_equations_65bc8b31fcd6491c190267a8568860bc64601981.png is called the (infinitesimal) generator of the operator semigroup stochastic_differential_equations_29f9bc93a72f4a53ea800c87ee69c7038e63449b.png

  • Also referred to as the generator of the Markov process stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png

This is an example of a (infinitesimal) generator of a strongly continuous semigroup.

The semigroup property of the generator of the Markov process implies that we can write

stochastic_differential_equations_284066b2fc9a15800902628b58e4ad7c50de91ef.png

Furthermore, consider function

stochastic_differential_equations_d427b999de2476e45d3714a81260529bfa643bf6.png

Compute time-derivative

stochastic_differential_equations_35df84f6bd6803f8552e43fedb1a3bb639c3dd4a.png

And we also have

stochastic_differential_equations_b2a7530c9039dea1b7ae88b802309681d4efc7e3.png

Consequently, stochastic_differential_equations_42683648b06176ce7644a8b4be60ac09f9660b5c.png satisfies the IVP

stochastic_differential_equations_26090d6dc7a49b3e7a2163d88a3de97e12623633.png

which defines the backward Kolmogorov equation.

This equation governs the evolution of the expectation of an observalbe stochastic_differential_equations_17d30f4b4993004b41c46520e85a311a135da1a5.png.

Example: Brownian motion 1D

  • Transition function for Brownian motion is given by the fundamental solution to the heat equation in 1D
  • Corresponding Markov semigroup is the heat semigroup

    stochastic_differential_equations_9204cffb4f489f46c9ddda706895a61c3476acb6.png

  • Generator of the 1D Brownian motion is then the 1D Laplacian stochastic_differential_equations_e1aaaf561f9678348cc5b2931711799340da6219.png
  • The backward Kolmorogov equation is then the heat equation

    stochastic_differential_equations_c9e423f7f65704d71cadc19e02f79982ab482a17.png

Adjoint semigroup

Let stochastic_differential_equations_29f9bc93a72f4a53ea800c87ee69c7038e63449b.png be a Markov semigroup, which then acts on stochastic_differential_equations_bdd92f5c64d99f50f83b5643cdeacf814cfd4e78.png.

The adjoint semigroup stochastic_differential_equations_d8309cd7b9d9cdb36c3d0582e2a8d59cbf29cd4d.png acts on probability measures:

stochastic_differential_equations_9169f6c7ef36ea15aff4da483755258e8cfa2c0c.png

The image of a probability measure stochastic_differential_equations_acb0106122b7f90d9bd5639367a141a7e53d8327.png under stochastic_differential_equations_d8309cd7b9d9cdb36c3d0582e2a8d59cbf29cd4d.png is again a probability measure.

The operators stochastic_differential_equations_29f9bc93a72f4a53ea800c87ee69c7038e63449b.png and stochastic_differential_equations_d8309cd7b9d9cdb36c3d0582e2a8d59cbf29cd4d.png are adjoint in the stochastic_differential_equations_5d229ea7e71b918baa91f8bbda3a25e08b64ceb5.png sense:

stochastic_differential_equations_f0429c0dbd8757d6f328fc487d45c769130d008e.png

We can write

stochastic_differential_equations_5c66cf82ffc07ef98c66a81e5de17571187ba416.png

where stochastic_differential_equations_e6c1741051caf3b86f62f32817d3d17d70f9d35b.png is the stochastic_differential_equations_5d229ea7e71b918baa91f8bbda3a25e08b64ceb5.png adjoint of the generator of the Markov process:

stochastic_differential_equations_b8912cad24e46042d16cbde8fb2aaa56eac7771e.png

Let stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png be a Markov process with generator stochastic_differential_equations_29f9bc93a72f4a53ea800c87ee69c7038e63449b.png with stochastic_differential_equations_4f06f584023dc72b4b56a85b63566b497ee76b88.png, and let stochastic_differential_equations_d8309cd7b9d9cdb36c3d0582e2a8d59cbf29cd4d.png denote the adjoint Markov semigroup.

We define

stochastic_differential_equations_53240dcc71f7ba8e16dff86dc64bbc4bf0b35b28.png

This is the law of the Markov process. This follows the equation

stochastic_differential_equations_2f1032a7ec386158190803a71444674b6a457124.png

Assuming that the initial distribution stochastic_differential_equations_acb0106122b7f90d9bd5639367a141a7e53d8327.png and the law of the process stochastic_differential_equations_d9ab696cadf928e277d7deea7318f056ec21aede.png each have a density wrt. Lebesgue measure, denoted stochastic_differential_equations_ff90d651779a50f195d486fe0a14a2724b82845d.png and stochastic_differential_equations_c9435c3cafb1beaa655ecbc35d913df137df5db5.png, respectively, the law becomes

stochastic_differential_equations_dee4dd8027b027caa2c9ed56594c79c684664469.png

which defines the forward Kolmorogov equation.

"Simple" forward Kolmogorov equation
  • Consider SDE

    stochastic_differential_equations_f6fa7f305632d4ccfa2a944829f31e2f88728517.png

  • Since stochastic_differential_equations_0aa7a82dd2200a79757418fb0536194414c8367e.png is Markovian, its evoluation can be characterised by a transition probability stochastic_differential_equations_ff5b6dbbed9bab6c708994144de676c990c004a7.png:

    stochastic_differential_equations_206874abdc369524003d7adde44fbd3a51344025.png

  • Consider stochastic_differential_equations_7f3b46a7b7ccd5e0315a37bb58d5dbcf7a7d79aa.png, then

    stochastic_differential_equations_bc48b7ea07505a77dca4cc159d87dff08afaa6b3.png

Ergodic Markov Processes

Using the adjoint Markov semigroup, we can define the invariant measure as a probability measure that is invariant under time evolution of stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png, i.e. a fixed point of the semigroup stochastic_differential_equations_d8309cd7b9d9cdb36c3d0582e2a8d59cbf29cd4d.png:

stochastic_differential_equations_1492db47b71ecf869cdc0145d35cc04c89b0359f.png

A Markov process is said to be ergodic if and only if there exists a unique invariant measure stochastic_differential_equations_acb0106122b7f90d9bd5639367a141a7e53d8327.png.

We say the process is ergodic wrt. the measure stochastic_differential_equations_acb0106122b7f90d9bd5639367a141a7e53d8327.png.

Furthermore, if we consider a Markov process stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png in stochastic_differential_equations_abf17ab8d81a8a9c9a3b8d37b4e26b6e2983cdc7.png with generator stochastic_differential_equations_63148cdf8e9010dfbe97cd2a2a64b740b0c652df.png and Markov semigroup stochastic_differential_equations_29f9bc93a72f4a53ea800c87ee69c7038e63449b.png, we say that stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png is ergodic provided that stochastic_differential_equations_96f53e8f2667720f54bd85623f46cbf545733989.png is a simple eigenvalue of stochastic_differential_equations_63148cdf8e9010dfbe97cd2a2a64b740b0c652df.png, i.e.

stochastic_differential_equations_fc57124ccedc3ae6251027c6ce56076ef5690627.png

has only constant solutions.

Thus, we can study the ergodic properties of a Markov process stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png by studying the null space of the its generator.

We can then obtain an equation for the invariant measure in terms of the adjoint stochastic_differential_equations_e6c1741051caf3b86f62f32817d3d17d70f9d35b.png of the generator.

Assume that stochastic_differential_equations_acb0106122b7f90d9bd5639367a141a7e53d8327.png has a density stochastic_differential_equations_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png wrt. the Lebesgue measure. Then

stochastic_differential_equations_4a4126085a9432a4698875075b5ea300cef5e8bc.png

by definition of the generator of the adjoint semigroup.

Furthermore, the long-time average of an observable stochastic_differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png converges to the equilibrium expectation wrt. the invariant measure

stochastic_differential_equations_745cfe5c9d8c9c2f66ba485a6e3db36c60a66acb.png

1D Ornstein-Uhlenbeck process and its generator

The 1D Ornstein-Uhlenbeck process is an ergodic Markov process with generator

stochastic_differential_equations_89012310ffb18225d3dfe88009fd511d43bfa1a1.png

The null-space of stochastic_differential_equations_63148cdf8e9010dfbe97cd2a2a64b740b0c652df.png comprises of constants in stochastic_differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png, hence it is an ergodic Markov process.

In order to find the invariant measure, we need to solve the stationary Fokker-Planck equation:

stochastic_differential_equations_033e3a6b1587a28a82fde3e79944b85df9641ce0.png

Which clearly require that we have an expression for stochastic_differential_equations_e6c1741051caf3b86f62f32817d3d17d70f9d35b.png. We have stochastic_differential_equations_63148cdf8e9010dfbe97cd2a2a64b740b0c652df.png, so

stochastic_differential_equations_de4f3b0123c606a3f794a98833c4f9db3ab28f55.png

Thus,

stochastic_differential_equations_e4935ce0bfeed82e2fcb7cf002b20f74f95397db.png

Substituting this expression for stochastic_differential_equations_e6c1741051caf3b86f62f32817d3d17d70f9d35b.png back into the equation above, we get

stochastic_differential_equations_beb6e6cf6aa16ff70ea6e1bf78c03d96ad7e8315.png

which is just a Gaussian measure!

Observe that in the above expression, the stuff on LHS before stochastic_differential_equations_95013ade91f0db312518e0219d7e7e0935643ca0.png corresponds to stochastic_differential_equations_139197f0b8b1630105d2171b8c97a89dcd9e947b.png in the equation we solved for stochastic_differential_equations_e6c1741051caf3b86f62f32817d3d17d70f9d35b.png.

If stochastic_differential_equations_989fc7b6561d6e9def44bdc66c96e0435529c6b0.png (i.e. distributed according to the invariant measure stochastic_differential_equations_acb0106122b7f90d9bd5639367a141a7e53d8327.png derived above), then stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png is a mean-zero Gaussian second-order stationary process on stochastic_differential_equations_dfba454ddab9caf4e291e5dbf30de4a218c340d6.png with correlation function

stochastic_differential_equations_0ac4f0d974fe48471e142aa1d5f9a4e2f8a24fa3.png

and spectral density

stochastic_differential_equations_b94943cb17dac9f0908fe79087f19c2b398ced8c.png

as seen before!

Furthermore, Ornstein-Uhlenbeck process is the only real-valued mean-zero Gaussian second-order stationary Markov process with continuous paths defined on stochastic_differential_equations_492d525117d0dcc93d066c8759f46b98cf9980ca.png.

Diffusion Processes

Notation

  • stochastic_differential_equations_2d14b6c346953049a11ddb7803d43d2b9de32373.png means depends on terms dominated by linearity in stochastic_differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png

Stuff

A Markov process consists of three parts:

  • a drift
  • a random part
  • a jump process

A diffusion process is a Markov process with no jumps.

A Markov process stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png in stochastic_differential_equations_492d525117d0dcc93d066c8759f46b98cf9980ca.png with transition function stochastic_differential_equations_fe7601cfed79dc62479379bef4f498cddf3d9c16.png is called a diffusion process if the following conditions are satisfied:

  1. (Continuity) For every stochastic_differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png and stochastic_differential_equations_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png,

    stochastic_differential_equations_5b87a4a7d0247afcdebb4180517976d8fdfda1f5.png

    uniformly over stochastic_differential_equations_4eae80be90b44cb22695ffbdd83655bb2f84815b.png

  2. ( Drift coefficient ) There exists a function stochastic_differential_equations_2c025c4674df2a5d20647204720d76f198c93d02.png s.t. for every stochastic_differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png and every stochastic_differential_equations_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png,

    stochastic_differential_equations_a6d8f0361f25a37cad2091c3203f77346b9e19ff.png

    uniformly over stochastic_differential_equations_4eae80be90b44cb22695ffbdd83655bb2f84815b.png.

  3. ( Diffusion coefficient ) There exists a function stochastic_differential_equations_3255a4d5447d9116546f79dcbbe7fee4f370c372.png s.t. for every stochastic_differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png and every stochastic_differential_equations_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png,

    stochastic_differential_equations_bcd2bf161e9840dac51a1a653496efc32882940e.png

    uniformly over stochastic_differential_equations_4eae80be90b44cb22695ffbdd83655bb2f84815b.png.

Important: above we've truncated the domain of integration, since we do not know whether the first and second moments of stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png are finite. If we assume that there exists stochastic_differential_equations_0809b466cd9de765ec30e362bd8646103cb497eb.png such that

stochastic_differential_equations_00a40ebe30699ce0fb3bbe28b816ddb5fa86d72d.png

then we can extend integration over all of stochastic_differential_equations_492d525117d0dcc93d066c8759f46b98cf9980ca.png and use expectations in the definition of the drift and the diffusion coefficient , i.e. the drift:

stochastic_differential_equations_6b73573cd0f9669329ca37150fcab8201fbe9896.png

and diffusion coefficient:

stochastic_differential_equations_274d7b34e280ff6dfcfd7a8dc376f81ac229a79c.png

Backward Kolmogorov Equation

Let stochastic_differential_equations_2d5828f056c439872207a4d68d6fbffb3319100f.png, and let

stochastic_differential_equations_78a7940bda88abcb4100aac430e3722cc5693d14.png

with fixed stochastic_differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png.

Assume, furthermore, that the functions stochastic_differential_equations_2c025c4674df2a5d20647204720d76f198c93d02.png and stochastic_differential_equations_3255a4d5447d9116546f79dcbbe7fee4f370c372.png are smooth in both stochastic_differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png and stochastic_differential_equations_aa1698cb8ee1665238ec3e91824191643c62ee93.png.

Then stochastic_differential_equations_511de4f49f40103b9312f81ba00f0b7a9eaff0c1.png solves the final value problem

stochastic_differential_equations_d2f364b7391b46f13f670062fc7b192d25b73e88.png

for stochastic_differential_equations_099d49a981404f86a20aa78570cff068c262a2f2.png.

For a proof, see Thm 2.1 in pavliotis2014stochastic. It's clever usage of the Chapman-Kolmogorov equation and Taylor's theorem.

For a time-homogenous diffusion process, where the drift and the diffusion coefficents are independent of time:

stochastic_differential_equations_18d93dab3b7e18d2297aa125eb6d0aff0fa4f9d6.png

we can rewrite the final value problem defined by the backward Kolmogorov equation as an initial value problem.

Let stochastic_differential_equations_4a6b33831ec3c40f590ea34595b4259ebc6a3bf9.png, and introduce stochastic_differential_equations_496415488886d4560da4d971655540d5c4a7f6fa.png. Then,

stochastic_differential_equations_fd0538fd462fbdf4ec5c307e7d48827b1e2a2aab.png

Further, we can let stochastic_differential_equations_6895ac723a44d1e88cb199e38db32983bcf92d72.png, therefore

stochastic_differential_equations_f6aec3adbd1222cf99ed8c0be79960b4f864793c.png

where

stochastic_differential_equations_98344da1ad4f3821412e17b514156290c498a5e5.png

is the solution to the IVP.

Forward Kolmorogov Equation

Assume that the conditions of a diffusion process are satisfied, and that the following are smooth functions of stochastic_differential_equations_eb83d466c7d035356e9f39998f357cee73da1e26.png and stochastic_differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png:

  • stochastic_differential_equations_b88d3fa03ce3a156e7022dfdada2aee63fe11f13.png
  • stochastic_differential_equations_18e42b27b9555e75f55b6e60e8203dd81df6fa5e.png
  • stochastic_differential_equations_49d9b323f93687a367b5d95dafd21340636f787a.png

Then the transition probability density is the solution to the IVP

stochastic_differential_equations_9026a622423d9c311d9e93ebd9fac7ff0518656a.png

For proof, see Thm 2.2 in pavliotis2014stochastic. It's clever usage of Chapman-Kolmogorov equation.

Solving SDEs

This section is meant to give an overview over the calculus of SDEs and tips & tricks for solving them, both analytically and numerically. Therefore this section might be echoing other sections quite a bit, but in a more compact manner most useful for performing actual computations.

Notation

Analytically

"Differential calculus"

  • stochastic_differential_equations_99d14ba6122686edc84fe7a46904a0ff92661039.png = stochastic_differential_equations_bc18e257c96aae58fddbfd75baff1f836550f7f6.png
  • stochastic_differential_equations_3ff08c793bf862ab22e3536be43bd83e0e2a42c5.png for all stochastic_differential_equations_d327b6ec5911a32fd49b0007db4b1b3e0cf2c3fc.png
  • If multivariate case, typically one will assume white noise to be indep.:

    stochastic_differential_equations_53124fe8c38f2908ef78d922dcfc9faf90185245.png

  • Suppose we have some SDE for stochastic_differential_equations_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, i.e. some expression for stochastic_differential_equations_00c22577ff0b61e3523f55277fe72f9d488ea505.png. Change of variables stochastic_differential_equations_410afa506a604effa22569ebf16cf047150e7988.png then from Itô's lemma we have

    stochastic_differential_equations_b452ed536d374347d531c2f91134fb5f972f4a8c.png

    where stochastic_differential_equations_8ad6fc5d048202c816537d1807fb41ee9fe8da7b.png is computed by straight forward substitution by expression for stochastic_differential_equations_00c22577ff0b61e3523f55277fe72f9d488ea505.png and using the "properties" of stochastic_differential_equations_3baef031291e31e2065321996f2508cba0f5a65d.png

  • Letting stochastic_differential_equations_662ea491dc68d6a64de6402238c98389b1b0def1.png and stochastic_differential_equations_3e6e49b3b95ec8c6ffa0f9c9caf4be9b97c02caf.png in Itô's lemma we get

    stochastic_differential_equations_5314b7853f0a659630caad7f3b395ba475ebab7a.png

  • Letting stochastic_differential_equations_662ea491dc68d6a64de6402238c98389b1b0def1.png and stochastic_differential_equations_4bb987ed8ab0922bcedab55899073eaad8931fc0.png, then Itô's lemma gets us

    stochastic_differential_equations_7e2753eb8737a3e6006095d4e1f025ce7127d3f6.png

    Hence stochastic_differential_equations_07367af3bd9ed9a87c2fc26a39044a014117effe.png satisfies the geometric Brownian motion SDE.

Numerically

Numerical SDEs

Notation

  • stochastic_differential_equations_2d615c313187e80ce659fd2cd62e9c3e71e29f11.png
  • We willl consider an SDE with the exact solution

    stochastic_differential_equations_7a53dc6c78e5ef5d8558bfb30b14c48fc33666f8.png

Convergence

The strong error is defined

stochastic_differential_equations_f775d92685d5ac25f0869270858d609ef24a4f37.png

We say a method converges strongly if

stochastic_differential_equations_adf98ccabe9e9d1ec93713bc842ad89430cc5ce3.png

We say the method has strong order stochastic_differential_equations_7225b076f6e6326f1636b11d1aad8de58bcc4761.png if

stochastic_differential_equations_4e1a74acf66cf718e3b423cf08b9f62347ad583c.png

Basically, this notion of convergence talks about how "accurately" paths are followed.

Error on individual path level

Methods

Euler-Maruyama

stochastic_differential_equations_96cf3f4a845e45755724118ef0cb374e2b34f6e3.png

Milstein method

Stochastic Taylor Expansion

Deterministic ODEs

  • Consider

    stochastic_differential_equations_a67d77adb747d07f6b70f368a967af1778c0bf56.png

  • For some function, we then write

    stochastic_differential_equations_d8981631245c25474d2a7b37a5fb550053ae946c.png

    where

    stochastic_differential_equations_a380b5fe4b79085eeb3a25f89de8a6cdc7494a37.png

    and the stochastic_differential_equations_12ef45689d21247a45a5a2f0f99f2a0b31f30375.png is due to the above

  • This looks quite a bit like the numerical quadrature rule, dunnit kid?!
    • Because it is!
  • We can then do the same for stochastic_differential_equations_f55d570f640e0f779ca9761c1efa0405ccfe8e09.png:

    stochastic_differential_equations_8778aad4edb60a863209bd0a825f0ba33f6e2776.png

    where

    stochastic_differential_equations_fff6f063e371c318013f8f2ec5d06a353539466a.png

  • We can then substitute this back into the original integral, and so on, to obtain higher and higher order
  • This is basically just performing a Taylor expansion around a point wrt. the stepsize stochastic_differential_equations_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png
  • Reason for using integrals rather than the the "standard" Taylor expansion to motivate the stochastic way of doing this, where we cannot properly talk about taking derivatives

Stochastic

  • Idea: extend the "Taylor expansion method" of obtaining higher order numerical methods for ODEs to SDEs
  • Consider

    stochastic_differential_equations_ece32d2772f0275f4ecbc8c4a96f65fbb64c4f2c.png

  • Satisfies the Itô integral

    stochastic_differential_equations_7758b7b7569cb1ebac7f9ee2df2b8e1971a24b07.png

  • Can do the same as before for each of the integral terms:
  • Then we can substitute these expressions into our original expression for stochastic_differential_equations_ab409669b13c6fe1aa2b19db9a1cfd511eb97466.png:

    stochastic_differential_equations_19f58d6f94f345004d8251a957f18d8872c62149.png

  • Observe that we can bring the terms stochastic_differential_equations_9b8771ac15ba3b1110797de4f19e651dd42da78b.png and stochastic_differential_equations_f377a4399d4cd6014216a3e9420abda2ffe0b6a0.png out of the integrals:

    stochastic_differential_equations_cd6a875f593b07fe27b0145d77042d0c67d2785e.png

  • Observe that the two terms that we just brought out of the integrals define the Euler-Maruyama method!!
    • Bloody dope, ain't it?

Connections between PDEs and SDEs

Suppose we have an SDE of the form

stochastic_differential_equations_487d47b32b660722f9ad57084cdc38c7b33f4941.png

Forward Kolmogorov / Fokker-Plank equation

Then the Fokker-Planck / Forward Kolmogorov equation is given by the following ODE:

stochastic_differential_equations_8e166f0b19f08960adf348716fe60fb33e4ef2d3.png

Derivation of Forward Kolmogorov

Suppose we have an SDE of the form

stochastic_differential_equations_487d47b32b660722f9ad57084cdc38c7b33f4941.png

Consider twice differentiable function stochastic_differential_equations_1b28688a2f600a83d425c4c444313a873bb8c2ae.png, then Itô's formula gives us

stochastic_differential_equations_76de34ba9696f050489a83eb71defac5737a5325.png

which then satisfies the Itô integral

stochastic_differential_equations_78b1a9b43c41734ae3fdda94d76e2bc826b680e5.png

Taking the (conditional) expectation, the last term vanish, so LHS becomes

stochastic_differential_equations_eabd0263ddf82b59d6a0284db57dd6fddb5d3cfb.png

and RHS

stochastic_differential_equations_5ef6b6b46ae6b1ce25c6bdf5343d219ef0ddbb11.png

Taking the derivative wrt. stochastic_differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png, LHS becomes

stochastic_differential_equations_a1ce59e91f793708e24fb32a9a4c69a79c212818.png

and RHS

stochastic_differential_equations_435d68e1f583c3aabe9b619823b9b632f791dc30.png

The following is actually quite similar to what we do in variational calculus for our variations!

Here we first use the standard "integration by parts to get stochastic_differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png rather than derivatives of stochastic_differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png", and then we make use of the Fundamental Lemma of the Calculus of Variations!

Using integration by parts in the above equation for RHS, we have

stochastic_differential_equations_10f2a9b064f53a1eb46f34f1e68a37f70d698ffa.png

where we have assumed that we do not pick up any extra terms (i.e. the functions vanish at the boundaries). Hence,

stochastic_differential_equations_e52d89c0d6bd3a8f8beeacd09f7efbc5cf179c85.png

Since this holds for all stochastic_differential_equations_e972cf8c5cbd1bffa497d8f1d8aaee1258df83bd.png, by the Fundamental Lemma of Calculus of Variations, we need

stochastic_differential_equations_2c416dc558feaae8ecab2a31bc93aad691fb9cdd.png

which is the Fokker-Planck equation, as wanted!

Assumptions

In the derivation above we assumed that the non-integral terms vanished when we performed integration by parts. This is really assuming one of the following:

  • Boundary conditions at stochastic_differential_equations_f6f02b10995dc1353548c5ff27838dfc549df734.png: stochastic_differential_equations_52c5ab69df198bf7c9bae8098d834a28fbccdf91.png as stochastic_differential_equations_dbe3a34d3caca16ec28db5da88a4b548d7edbf29.png
  • Absorbing boundary conditions: where we assume that stochastic_differential_equations_00c73ccec32459d1b8ae999878f8ffeabc78d448.png for stochastic_differential_equations_ae89f6a9a584f173dbdb75698c45427f27af0ac0.png (the boundary of the domain stochastic_differential_equations_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png)
  • Reflecting boundary conditions: this really refers to the fact that multidimensional variations vanish has vanishing divergence at the boundary. See multidimensional Euler-Lagrange and the surrounding subjects.

Backward equation

And the Backward Kolmogorov equation:

stochastic_differential_equations_032ba7fdd14f99d7eb2d7a9a6a7ab31eb6b0ba19.png

where

stochastic_differential_equations_11b04569b71871847ceeebff5110b696c1b167e4.png

Here we've used stochastic_differential_equations_3b641f1a2bcf446797177eefc8e6b8a5fbf45480.png defined as

stochastic_differential_equations_462c778c30201aeeddbd000f4f075e30049cff07.png

rather than

stochastic_differential_equations_7890c792d3c356bbba8982407e21cef36e18aa85.png

as defined before.

I do this to stay somewhat consistent with notes in the course (though there the operator stochastic_differential_equations_29f9bc93a72f4a53ea800c87ee69c7038e63449b.png is not mentioned explicitly), but I think using stochastic_differential_equations_6895ac723a44d1e88cb199e38db32983bcf92d72.png would simplify things without loss of generality (could just define the equations using stochastic_differential_equations_700b09d2c929f317cb9e52dd1279bf74f04d1355.png and stochastic_differential_equations_7f97a0795f7095629e43223b98495888186cf4f3.png, and substitute back when finished).

Furthermore, if stochastic_differential_equations_609c747e8ed8329e7f491aab412ea412aae106c6.png and stochastic_differential_equations_cd161f523ca33a0dbfb86515acda591bb3f3b031.png, i.e. time-independent (autonomous system), then

stochastic_differential_equations_8dc6fa11a5f6e7581f6b28c3c63103c914db6ec7.png

in which case the Forward Kolmogorov equation becomes

stochastic_differential_equations_561a61e65586b6cf2e48623995c1c640ce219011.png

where we've used the fact that in this case stochastic_differential_equations_7d1e7e21ff199c66d9bb974fec8dbf715e1e5a9d.png.

Notation of generator of the Markov process and its adjoint

In the notation of the generator of the Markov process we have

stochastic_differential_equations_704368718fb50b811d3a689146a18cd02218d5d3.png

TODO Introduction to Stochastic Differential Equations

Notation

  • stochastic_differential_equations_3a2084903965fa6859187e02fa40f108c34f2d3c.png indicates the dependence on the initial condition stochastic_differential_equations_f9b57a73f6b1f70643a841e2abcae5208910d403.png.
  • Stopping time

    stochastic_differential_equations_e080ca4de3827977722f493798ebac83deb0bca3.png

Motivation

  • We will consider stochastic differential equations of the form

    stochastic_differential_equations_ee7908bd86152e68de577521cca3f681d6ea66c8.png

    or, equivalently, componentwise,

    stochastic_differential_equations_c924439897f1b4d1e2d7463c7503820f32d78832.png

    which is really just notation for

    stochastic_differential_equations_a5d06db5f4c70e3cdb8041d3281552d3858f7888.png

  • Need to define stochastic integral

    stochastic_differential_equations_a98124854a4330501360ffc1a657adbb487e2aca.png

    for sufficiently large class of functions.

    • Since Brownian motion is not of bounded variation → Riemann-Stieltjes integral cannot be defined in a unique way

Itô and Stratonovich Stochastic Integrals

Let

stochastic_differential_equations_8b686faf5faf9ce8cd6fed4bdd01dcf025fe5509.png

where stochastic_differential_equations_af8078928800b128476208139c7201ed7a609c6d.png is a Brownian motion and stochastic_differential_equations_9297e420aa186538945a7ec2305c79ca3df02d61.png, such that

stochastic_differential_equations_0bcd8e28d6de6481ca8b85ace51e2d1158f17abc.png

Integrand is a stochastic process whose randomness depends on stochastic_differential_equations_af8078928800b128476208139c7201ed7a609c6d.png, and in particular, that is adapted to the filtration stochastic_differential_equations_2d2ef1bebb51aa5422b07a53d7a29158099d4746.png generated by the Brownian motion stochastic_differential_equations_af8078928800b128476208139c7201ed7a609c6d.png, i.e. that is an stochastic_differential_equations_55b83e3fb2211751b65dfef8b631b86b24f9cac8.png function for all stochastic_differential_equations_9297e420aa186538945a7ec2305c79ca3df02d61.png.

Basically means that the integrand depends only on the past history of the Brownian motion wrt. which we are integrating.

We then define the stochastic integral stochastic_differential_equations_2656f386e1e724be72bb89369646112ab2a2b447.png as the stochastic_differential_equations_1e0a1c83c0d528bcfeed5053a52cefebb52a9528.png (stochastic_differential_equations_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png is the underlying probability space) limit of the Riemann sum approximation

stochastic_differential_equations_47e2f6e5288a7030a069dfcb53e3710819cb2d32.png

where

  • stochastic_differential_equations_858f17f2c637d2771ea374b0b834febebe5974ff.png where stochastic_differential_equations_e2031b3655575802069e137605b86dd499ee42e4.png is s.t. stochastic_differential_equations_e8afd82db748a55d06cc5ca6708769c56d27f46d.png
  • stochastic_differential_equations_44a15174f3675c7b929a408a60552c136c1681ef.png
  • increments

    stochastic_differential_equations_fd788b3c4b72b0446344c686334e2534691353e9.png

What we are really saying here is that stochastic_differential_equations_2656f386e1e724be72bb89369646112ab2a2b447.png (which itself is a random variable, i.e. a measurable function on the probability space stochastic_differential_equations_35888a61b906c65212fa0adbff6551b976615c68.png) is the limit of the Riemann sum in a mean-square sense, i.e. converges in stochastic_differential_equations_5d229ea7e71b918baa91f8bbda3a25e08b64ceb5.png!

stochastic_differential_equations_c39292da93c24f1724c1be582c080b361c4524e4.png

I found this Stackexchange answer to be very informative regarding the use of Riemann sums to define the stochastic integral.

Let stochastic_differential_equations_2656f386e1e724be72bb89369646112ab2a2b447.png be a stochastic integral.

The Itô stochastic integral is when stochastic_differential_equations_8441b3a29b827eb54f5941a74cd06a22158873f3.png, i.e.

stochastic_differential_equations_e33e7af189d6cb6763748af13307bc1968ebde68.png

Let stochastic_differential_equations_2656f386e1e724be72bb89369646112ab2a2b447.png be a stochastic integral.

The Stratonovich stochastic integral is when stochastic_differential_equations_881dc8d425b9dd0dfea68a97acdeaddbcbd3007f.png, i.e.

stochastic_differential_equations_9e5833d8b91f9f766c761a6091c0cbbc97a64951.png

We will often use the notation

stochastic_differential_equations_3de1c4d2841be92ea81078b5a336eb431591317e.png

to denote the Stratonovich stochastic integral.

Observe that stochastic_differential_equations_f5dc80d64c062c7dded8cd9236a6634d73ed715a.png does not satisfy the Martingale property, since we are taking the midpoint in stochastic_differential_equations_5280249fd7bb1260b18cc0a62473a916d732b85e.png, therefore stochastic_differential_equations_77f6a5a212ab15cbc624680da6dd581c8d180cc0.png is correlated with stochastic_differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png!

Itô instead uses the Martingale property by evaluating stochastic_differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png at the start point of the integral, and thus there is no auto correlation between the stochastic_differential_equations_667e9e095b2365541881ae2836e81f3a3483e28f.png and stochastic_differential_equations_e12921e4a4770eec9069228e10bcbd7adc5cbf69.png, making it much easier to work with.

Suppose that there exist stochastic_differential_equations_d4ee61cbec16b658d6d5cca4c2e769e309f92a96.png s.t.

stochastic_differential_equations_28bb369eb064f0c60694ac79e36d947c9e2ba409.png

Then the Riemann sum approximation for the stochastic integral converges in stochastic_differential_equations_574a256a5591356a713c80a21e42d745ea725b3e.png to the same value for all stochastic_differential_equations_44a15174f3675c7b929a408a60552c136c1681ef.png.

From the definition of a stochastic integral, we can make sense of a "noise differential equation", or a stochastic differential equation

stochastic_differential_equations_d86e34ffa3de68cd94ffbc3444cdec84d1a9aef8.png

with stochastic_differential_equations_31608205cc1442fed386863d5856a2596ae3918e.png being white noise.

The solution stochastic_differential_equations_0aa7a82dd2200a79757418fb0536194414c8367e.png then satisfies the integral equation

stochastic_differential_equations_21903c8cfb6ff891ea2fa07455d6a497ee159143.png

which motivates the notation

stochastic_differential_equations_0cafb5779674ef85e5709fa9768978490c2db574.png

since, in a way,

stochastic_differential_equations_2df2e2376acdec38c5a9321875be4e49a7abfdd8.png

(sometimes you will actually see the Brownian motion (this definition for example) defined in this manner, e.g. lototsky2017stochastic)

Properties of Itô stochastic integral

Itô isometry

Let stochastic_differential_equations_2656f386e1e724be72bb89369646112ab2a2b447.png be a Itô stochastic integral, then

stochastic_differential_equations_111f01323143a1cf7886f8f1c874f11bcc7ae4c5.png

From which it follows that for any square-integrable functions stochastic_differential_equations_1ef38efeac3ecbd4ca5ee34c5260dedc9797f765.png

stochastic_differential_equations_5cc5e6637a7e5910bc92cd7bb3fbaf3c1b3c93e5.png

stochastic_differential_equations_f7daf4025221d0b04fed35fc476e9450823a2aa7.png

but we also know that

stochastic_differential_equations_fd3395c91a1c02a37905bb5622c6c2fdea7c8b25.png

Comparing with the equation above, we see that

stochastic_differential_equations_4fb00db40dafce48cc8b4f3ed5a4d449b9bc33f9.png

WHAT HAPPENED TO THE ABSOLUTE VALUE MATE?! Well, in the case where we are working with real functions, the Itô isometry is satisfied for stochastic_differential_equations_8fe33cb7f9fd17eee4f84ee75dc613338949185b.png since it's equal to stochastic_differential_equations_e4a19f037ef9eec5296dd27c6048b25d9ee890f2.png, which would give us the above expression.

Martingale

For Itô stochastic integral we have

stochastic_differential_equations_c087ac9ae98cfadae8abab7c40300483e9bcc5e0.png

and

stochastic_differential_equations_79841410a56956a69ca77d99aa8c39b9fcb8471f.png

where stochastic_differential_equations_b96f2cd4e7c172af731a037a7e1395466d695470.png denotes the filtration generated by stochastic_differential_equations_7428dc41de7274d2f95425c4aec187eb59d7d612.png, hence the Itô integral is martingale.

The quadratic variation of this martingale is

stochastic_differential_equations_b1dcd7ac1dc49e1e6e91015a2c98624d0ba68aca.png

Solutions of SDEs

  • Consider SDEs of the form

    stochastic_differential_equations_b5900b1f9a1afd6d8d2d11a54bb9fda458ddc079.png

    where

    • stochastic_differential_equations_fac17b9bc509d20e411f43a313c6443cadb1053b.png
    • stochastic_differential_equations_7079f17ca5e1c4bc4cfa78e27ec1260f3d1543b6.png

A process stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png with continuous paths defined on the probability space stochastic_differential_equations_5812bdb1d6a84feb0094e2f9aa8a2f11c19aa8b1.png is called a strong solution to the SDE if:

  1. stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png is a.s. continuous and adapted to the filtration stochastic_differential_equations_2d2ef1bebb51aa5422b07a53d7a29158099d4746.png
  2. stochastic_differential_equations_10e93f7248bae64d626e44dc60aed87276eaa73f.png and stochastic_differential_equations_afb7b5fbd53e24d3ff2e1ba58a66ba7efebf2831.png a.s.
  3. For every stochastic_differential_equations_809038084ca90dd90f9bde810a570e42ccd59b74.png, the stochastic integral equation

    stochastic_differential_equations_6295629625a273e74e6e835a4471f175311b518d.png

Let

  • stochastic_differential_equations_fac17b9bc509d20e411f43a313c6443cadb1053b.png
  • stochastic_differential_equations_7079f17ca5e1c4bc4cfa78e27ec1260f3d1543b6.png

satisfy the following conditions

  1. There exists positive constant stochastic_differential_equations_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png s.t. for all stochastic_differential_equations_b50d3f9b65d6a48f7ccf964f392ac87b16b3c3e6.png and stochastic_differential_equations_9297e420aa186538945a7ec2305c79ca3df02d61.png

    stochastic_differential_equations_7803a016fa87e2d1c409970cf3418acfca0ce5af.png

  2. For all stochastic_differential_equations_4fa1097acc8fafc1ffd719f1e5dc0304f596c073.png and stochastic_differential_equations_9297e420aa186538945a7ec2305c79ca3df02d61.png,

    stochastic_differential_equations_7c5092650aa977b71ddd5857f626224a31f927b5.png

Furthermore, suppose that the initial condition stochastic_differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png is a random variable independent of the Brownian motion stochastic_differential_equations_478362a95ee639558cb989749b83ad3c85f95868.png with

stochastic_differential_equations_1e1e0fb10d5404a74c93b01699b83a803fd3842e.png

Then the SDE

stochastic_differential_equations_b5900b1f9a1afd6d8d2d11a54bb9fda458ddc079.png

has a unique strong solution stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png with

stochastic_differential_equations_f16cbdb39d3187d65291fcd27d13719e0cae2bb7.png

where by unique we mean

stochastic_differential_equations_7f54c4b42059dfe14e6b60a4dcc0c32648cb2901.png

for all possible solutions stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png and stochastic_differential_equations_f4f964dc00744218eacf10746ab373b9e4bb5cea.png.

Itô's Formula

Notation

Stuff

Assume that the conditions used in thm:unique-strong-solution hold.

Let stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png be the solution of

stochastic_differential_equations_b24b4ac2e0c4328629ec0a6f970a24dbb136c232.png

and let stochastic_differential_equations_cf12fa5b226af3eef6993073f34915a4a492c85b.png. Then the process stochastic_differential_equations_e00c22407c5818a8749de8eec9fc72224fc0edfe.png satisfies

stochastic_differential_equations_9c54cfeed085cccc7319a48ead55d246dd780903.png

where the generator stochastic_differential_equations_63148cdf8e9010dfbe97cd2a2a64b740b0c652df.png is defined

stochastic_differential_equations_03cebc38cead7aa533cc9e8fc91214caa7c090b3.png

If we further assume that noise in different components are independent, i.e.

stochastic_differential_equations_9ec039634465cbbffbf5bfdcf5475d4ba7c59f85.png

then stochastic_differential_equations_e6b54f65aae289decdb4bd3a7e70d5e82c4b080e.png simplifies to

stochastic_differential_equations_4378d245308b6465a08d8bf272dafbf15c2bf51e.png

Finally, this can then be written in "differential form":

stochastic_differential_equations_601fcff78e9ee570576dad3902ad9237e63e3078.png

In pavliotis2014stochastic it is stated that the proof is very similar to the proof of the validity of the proof of the validity of the backward Kolmogorov equation for diffusion processes.

Feynman-Kac formula

  • Itô's formula can be used to obtain a probabilistic description of solutions ot more general PDEs of parabolic type

Itô's formula can be used to obtain a probabilistic description of solutions of more general PDEs of parabolic type.

Let stochastic_differential_equations_3a2084903965fa6859187e02fa40f108c34f2d3c.png be a diffusion process with

  • drift stochastic_differential_equations_f8ba0bcc022c6d477cbfde325f315454468205bb.png
  • diffusion stochastic_differential_equations_9415c02b15000bd744e4bf67b2e9b48acc384758.png
  • generator stochastic_differential_equations_63148cdf8e9010dfbe97cd2a2a64b740b0c652df.png with stochastic_differential_equations_ca732dd4f2d717b31733f2441ceaf2cc45063842.png

and let

  • stochastic_differential_equations_e765d059d65c0a1d9c2ddcaba39d31c385031170.png
  • stochastic_differential_equations_61e752f508433b4fe3f470efa7e24f82c4f111fc.png

be bounded from below.

Then the function

stochastic_differential_equations_bfa2a3baaece7d34a7e82c39ef050ec3de81caa8.png

is the solution to the IVP

stochastic_differential_equations_fecc5e1a58f069d1ef53b8b3db25c654a0e9a150.png

The representation stochastic_differential_equations_42683648b06176ce7644a8b4be60ac09f9660b5c.png is then called the Feynman-Kac formula of the solution to the IVP.

This is useful for theoretical analysis of IVP for parabolic PDEs of the form above, and for their numerical solution using Monte Carlo approaches.

Examples of solvable SDEs

Ornstein-Uhlenbeck process

Properties
  • Mean-reverting
  • Additive noise
Stuff

stochastic_differential_equations_4df6f974686d85956b6a27f39f2229cf56eea3ad.png

We observe that using Itô's formula

stochastic_differential_equations_9b8c8b780b55ebc7743828bca25d46ce98299961.png

which from the Ornstein-Uhlenbeck equation we see that

stochastic_differential_equations_89a51c682869056478e0c1fc4fc7b94743703b33.png

i.e.

stochastic_differential_equations_22a23ca0d682e5a020a83788d46ff1bb7a50aeed.png

which gives us

stochastic_differential_equations_5c006fff09e9d17ba2482d90d31fe31c34cbbfb7.png

and thus

stochastic_differential_equations_6142de66f2d5fd06c2a23a2cca6bd420b39412bf.png

with stochastic_differential_equations_9eede1cc5c0a0b7c233825627b7c473f22adf6be.png assumed to be non-random. This is the solution to the Ornstein-Uhlenbeck process.

Further, we observe the following:

stochastic_differential_equations_04f9c070c840bc797e311bc5e0e0644269472535.png

since this is a Gaussian process. The covariance ew see

stochastic_differential_equations_a3798181d05791a66abcc8b04b3343cd8eeafbd6.png

Assuming stochastic_differential_equations_4721bcaed90482f941978fdbc7b08437489bfc25.png, these are independent! Therefore

stochastic_differential_equations_a8be53afe724ba6c0e8f22381c91564798c5aff5.png

Using Itô isometry, the first factor becomes

stochastic_differential_equations_2cf1caa40399b4dc47c6730ec86c2b0e73461de2.png

since

stochastic_differential_equations_d9b49e81658fc6ee15e3cce98bfbab2a55ccb4fd.png

Langevin equation

Notation
  • stochastic_differential_equations_aa07b3a8458adb2855b54064282b1f340d44fbd4.png position of particle
  • stochastic_differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png velocity of particle
Definition

stochastic_differential_equations_a8717d4112ebed9b36b0328548db9467cb4745ed.png

Solution

Observe that the Langevin equation looks very similar to the Ornstein-Uhlenbeck process, but with stochastic_differential_equations_a155081155e64fe75a7f8b853760eee523803d82.png instead of stochastic_differential_equations_aa07b3a8458adb2855b54064282b1f340d44fbd4.png. We can write this as a system of two SDEs

stochastic_differential_equations_b6e0301661e480a51bd258ce8426be6951115d96.png

The expression for $d V $ is simply a OU process, and since this does not depend on stochastic_differential_equations_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, we simply integrate as we did for the OU process giving us

stochastic_differential_equations_3f1015b615ede6b548f69f8cedd0eae78c9590f1.png

Subsituting into our expression for stochastic_differential_equations_00c22577ff0b61e3523f55277fe72f9d488ea505.png:

stochastic_differential_equations_a94ff5505808e8b05d8df802b72f089897e7987e.png

We notice that the integral here is what you call a "triangular" integral; we're integrating from stochastic_differential_equations_3e2e21e59b05ae931b9586c1db54fbb38b91df9a.png and integrating stochastic_differential_equations_0393b00a5504e029d16630da5125e149c1f2f2b7.png. We can therefore interchange the order of integration by integrating stochastic_differential_equations_aa1698cb8ee1665238ec3e91824191643c62ee93.png from stochastic_differential_equations_fed5cf1c6d1bfc1c034e57a760129fab0ff4f297.png and the integrating stochastic_differential_equations_6eb95865ccf7f3fc048eb8bcc06c72901a0a7724.png from stochastic_differential_equations_1314b1cefa81206d2bbd58619f9198a0c3ed53cd.png! In doing so we get:

stochastic_differential_equations_63611aab48e21649cca0bef1d6a8780782c5e50f.png

Hence the solution is given by

stochastic_differential_equations_1683f964fc3ba47f26c6f34cd286756fa493f56a.png

Geometric Brownian motion

The Geometric Brownian motion equation is given by

stochastic_differential_equations_69604ad54f5a124ada0d3f949bd83f027a0b5eec.png

Brownian bridge

Consider the process stochastic_differential_equations_41e0ac735664c2aff709212f686f4d8074396c22.png which satisfies the SDE

stochastic_differential_equations_2b83eb689986783acdb75bae6565ca684cb440d8.png

for stochastic_differential_equations_7b4b018a75a8515d13e4c484de0beb46e83b04fd.png.

This is the definiting SDE for a Brownian bridge.

Solution

Let

stochastic_differential_equations_bbb838f91e2d060867ccd3178290827f7a93dcb3.png

Ito's formula gives

stochastic_differential_equations_3731d08e7ac7cf6b4cc8485a581ac297eaf252c6.png

which satisfies

stochastic_differential_equations_e9dc650380c3b90e92300625fe6eb585efbd46de.png

Hence,

stochastic_differential_equations_1ea90c96b3a7e47a0b32eff1011a174d74e843a6.png

Imposing the BCs stochastic_differential_equations_16d5e8b9a61a989eb5ccdf09a61bb1a5b106265a.png, we have

stochastic_differential_equations_b61505590fd0d628ee7360f14458d23872f7a46d.png

A random oscillator

The harmonic oscillator ODE stochastic_differential_equations_d1c63aa03cf1a0d3ee94aee58c758be2aab21cbf.png can be written as a system of ODEs:

stochastic_differential_equations_4d05b3f3de1d9b3c6d221dc5080fc5d1798153ff.png

A stochastic version might be

stochastic_differential_equations_ec07671cb98a24f4c48207e3ae5a41ff5bf6aa9f.png

where stochastic_differential_equations_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png is constant and the frequency is a white noise stochastic_differential_equations_8944849d0fc21f85cef3aebba3a8fe1f1f2056f9.png with stochastic_differential_equations_099ec7d796eb834460fb0c7feff95cfab4e2685c.png.

Solution

We can solve this by letting

stochastic_differential_equations_f57bcda31c0e3702e61592f2fd2a3c5314c74740.png

so

stochastic_differential_equations_35e0c0a4bfcdf676f3365cc5bdfa3e549b556c10.png

Then,

stochastic_differential_equations_239ef6a5eff4bbd6cf79c8b6977b60b683ccaed7.png

Hence,

stochastic_differential_equations_6d30bbf72bfc77dffc2aed9e211c965ef160bda1.png

and thus

stochastic_differential_equations_be6904b650154fe276fdc0023e2b0e247c51ac85.png

Stochastic Partial Differential Equations (rigorous)

Overview

This subsection are notes taken mostly from lototsky2017stochastic. This is quite a rigorous book which I find can be quite a useful supplement to the more "applied" descriptions of SDEs you'll find in most places.

I have found some of these more "formal" definitions to be provide insight:

  • Brownian motion expressed as a sum over basis-elements multiplied with white noise in each term
  • Definition of a "Gaussian process" as it is usually called, instead as a "Guassian field", such that each finite collection of random variables form a Gaussian vector.

Notation

  • stochastic_differential_equations_2dbb71c1dd623d22aa8874974f2e3aa4731f8516.png
  • stochastic_differential_equations_3ef982691e494a30041f3515870808b763796769.png
  • stochastic_differential_equations_64db2e0ff3ffa63f81e6211408d3e9cd18072421.png denotes the space of continuous mappings from metric space stochastic_differential_equations_e46729bc781c25bbc7120ee2892cc1c0215af7da.png to metric space stochastic_differential_equations_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png
    • If stochastic_differential_equations_79f2f3575d655ecfcd0d54a3f70ec47c086e8e08.png then we write stochastic_differential_equations_ec2a3b98bb92ad55d143fd17ec8e2a5193cc375d.png
  • stochastic_differential_equations_a28c9639d193434ecb4aae8453875131cf77db18.png is the collection of functions with stochastic_differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png continuous derivatives
  • stochastic_differential_equations_0a22bb43e9b32dee025837e958b682763598da61.png for stochastic_differential_equations_f022b6efaffaed1ec41eab447c6c445c328ffc56.png and stochastic_differential_equations_84feeb83da8919c3f995550cdfb7e928ee8b390d.png is the collection of functions with stochastic_differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png continuous derivatives s.t. derivatives of order stochastic_differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png are Hölder continuous of order stochastic_differential_equations_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png.
  • stochastic_differential_equations_3eaf6b6e39e8985693e4a66a7c1c73c41f1d6e4f.png is the collection of infinitely differentiable functions with compact support
  • stochastic_differential_equations_9b1eebba4179003c9fc7d6173cb300bf65c85d2f.png denotes the Schwartz space and stochastic_differential_equations_676285db441b6a041a679a15c3bb97a31ea885ed.png denotes the dual space (i.e. space of linear operators on stochastic_differential_equations_9b1eebba4179003c9fc7d6173cb300bf65c85d2f.png)
  • stochastic_differential_equations_26849e00c9f9d483034c9d6051324360de853ba9.png denotes the partial derivatives of every order stochastic_differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png
  • Partial derivatives:

    stochastic_differential_equations_e86438326973a96fe10acd9599e896e9180cbf62.png

    and

    stochastic_differential_equations_2316b4aa78165595334728716656715513faa593.png

  • Laplace operator is denoted by stochastic_differential_equations_700ca6709a441581193e12974e482b5a8bf3d377.png
  • stochastic_differential_equations_bab1a43ab3d6b8aad6b3bd6f2b4fde19a07a2e6b.png means

    stochastic_differential_equations_2b658be275075e175a76e0d6097b1eb4b51ae387.png

    and if stochastic_differential_equations_445b00488193f6629e33498f0b83415781b0335d.png we will write stochastic_differential_equations_d1f6aad076334012e463bdb7ad53da1ba54ff362.png

  • stochastic_differential_equations_96adaa83c4c41cd931d2a667b03fb94c7b94aa52.png means

    stochastic_differential_equations_d24fd73176be32aad7fdfbcca9bc01bdffc10512.png

    for all sufficiently larger stochastic_differential_equations_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png.

  • stochastic_differential_equations_d466977ff52a075032415c381a65f38a0b2be1ad.png means that stochastic_differential_equations_733ca772b7d4d03369548c9cbb9921033b7182db.png is a Gaussian rv. with mean stochastic_differential_equations_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png and variance stochastic_differential_equations_68668609fc72c879ba0fabd0c25275964a5e4af8.png
  • stochastic_differential_equations_d2810c911a64bcdc20af0cb0e245a2e8601bd775.png or stochastic_differential_equations_35588526528557bbd60fd2b4366ee86c08cd9f70.png are equations driven by Wiener process
  • stochastic_differential_equations_b9f7ef0364d76f9a2fcb4a933784175692d348f0.png
    • stochastic_differential_equations_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png is the sample space (i.e. underlying space of the measure space)
    • stochastic_differential_equations_86015b70085708a33641b21fd2391631d08aab67.png is the sigma-algebra (stochastic_differential_equations_2cd5dd49c6b9996ac2fa0efe93da1337ed8b5bed.png denotes the power set of stochastic_differential_equations_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png)
    • stochastic_differential_equations_41423890e6dcd0ee9ea1c0ea8bb41dbf2f5c9e8d.png is an increasing family of sub-algebras and stochastic_differential_equations_b077fa0d496e54c22bd22e87c35ca61609d6fdc4.png is right-continuous (often called a filtration)
    • stochastic_differential_equations_b85866eea6317e54095c782ffb17f2b023c424a2.png contains all stochastic_differential_equations_7a5cfa790bd948bb1802f0252f319189fc8f3c72.png neglible sets, i.e. stochastic_differential_equations_b85866eea6317e54095c782ffb17f2b023c424a2.png contains every subset of stochastic_differential_equations_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png that is a subset of an element from stochastic_differential_equations_92757546978c1aad3468f8a69e8d334af2bfadac.png with stochastic_differential_equations_7a5cfa790bd948bb1802f0252f319189fc8f3c72.png measure zero (i.e. stochastic_differential_equations_b85866eea6317e54095c782ffb17f2b023c424a2.png is complete)
  • stochastic_differential_equations_4b624e509a9218ea6c3943b3d6b29a8fd2e4c445.png is an indep. std. Gaussian random variable
  • stochastic_differential_equations_bd50271118b2ccc61cf67e9a291c9931fd986b2f.png

Definitions

A filtered probability space is given by

stochastic_differential_equations_5f24efc41c2303dfd5037021f75f659edda4a3e9.png

where the sigma-algebra stochastic_differential_equations_b85866eea6317e54095c782ffb17f2b023c424a2.png represents the information available up until time stochastic_differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png, and a process stochastic_differential_equations_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is adapted if stochastic_differential_equations_1e93bc9be5682a9bca0c04cca91fbeebc2c17db7.png is stochastic_differential_equations_7fbb93c7663dce7d40551574c9c92f4a0b8f3c90.png.

Martingale

A square-integrable Martingale on stochastic_differential_equations_9360329fce6f3b05ed0cad8714426e714a6dcf1e.png is a process stochastic_differential_equations_ffef3d821c020733c6f2fad718280bd941621c92.png with values in stochastic_differential_equations_abf17ab8d81a8a9c9a3b8d37b4e26b6e2983cdc7.png such that

stochastic_differential_equations_4233e4eec679f538146151d808e18dbdb14ae8df.png

and

stochastic_differential_equations_c4c31e63225b99e7b1bb4de87591f16545579d7b.png

A quadratic variation of a martingale stochastic_differential_equations_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is the continuous non-decreasing real-valued process stochastic_differential_equations_adabd1e2660b596a91adbcea6f872e2d8b35a895.png such that

stochastic_differential_equations_0a524ec5c8a0bde799c6cba9a97a6c14ea39622b.png

is a martingale.

A stopping (or Markov) time on stochastic_differential_equations_9360329fce6f3b05ed0cad8714426e714a6dcf1e.png is a non-negative random variable stochastic_differential_equations_6eb95865ccf7f3fc048eb8bcc06c72901a0a7724.png such that

stochastic_differential_equations_95b91b26dbbc46d7d89759352f4043eb09e8d584.png

Introduction

If stochastic_differential_equations_116786d6c80ea195d311727d233a7e936f307acd.png is an orthonormal basis in stochastic_differential_equations_d639211dcf33fe0a27b27639496e1b5f25f45477.png, then

stochastic_differential_equations_2cf8779c1effaf43f2c9dbb5bc5946283cf9ecd6.png

is a standard Brownian motion ; a Gaussian process with zero mean and covariance given by

stochastic_differential_equations_be3d331886c5d2ffc5df726c4bdb244e65981de6.png

This definition of a standard Brownian motion does make a fair bit of sense.

It basically says that that a Brownian motion can be written as a sum of elements in the basis of the space, with each term being multiplied by some white noise.

The derivative of Brownian motion (though does not exist in the usual sense) is then defined

stochastic_differential_equations_e929f4355420b091a8cd17127e5185647114fe7b.png

While the series certainly diverges, id oes define a random generalized function on stochastic_differential_equations_d5f626a57d9732e418552b8e97fddb4a51f1f4b6.png according to the rule

stochastic_differential_equations_1a1055600f2eba4cbd2c548cdf2493e61fdd3ba8.png

Consider

  • a collection stochastic_differential_equations_2e32bcdd3f2c0c956340088b086179d7b260d27b.png of indep. std. Brownian motions, stochastic_differential_equations_9297e420aa186538945a7ec2305c79ca3df02d61.png
  • orthonormal basis stochastic_differential_equations_a55bfd575dac1605dcbb3915be192cc6010de70e.png in the space stochastic_differential_equations_b19acb320bc8a2ea4109a04d488fa8197b2c2719.png with

    stochastic_differential_equations_76dc3e7fe0b9324f9ae6a6f4523190ace0dfe93b.png

    a d-dimensional hyper-cube.

  • For stochastic_differential_equations_1246a65c51771f0045a75bcd8fc3f887c288e433.png define

    stochastic_differential_equations_db1761cf9b5a4dc91630d418f79ddb20a57ff75c.png

Then the process

stochastic_differential_equations_2b659eda175b49cc0b03db447bfe06799ef4ddda.png

is Gaussian,

stochastic_differential_equations_3e3536ca1657cbcb72ff773f3d3e2e374e38aecd.png

We call this process stochastic_differential_equations_d2514a932c136dc832469d74ea295bafc44a8a96.png the Brownian sheet.

From Ex. 1.1.3 b) we have

stochastic_differential_equations_5879ee2ce950a7f68bfbb5963011e77336b6d179.png

are i.i.d. std. normal. From this we can define

stochastic_differential_equations_a34262b57d7e1a430c6b8f12b2f87baebdc49794.png

Writing

stochastic_differential_equations_c225617c9ea0d8ee881a82ccf9347e05781a905e.png

where stochastic_differential_equations_4dea8b56a6c7502fb4489be310f8c2a7679fd7ed.png is an orthonormal basis in stochastic_differential_equations_b19acb320bc8a2ea4109a04d488fa8197b2c2719.png and stochastic_differential_equations_38382d88f647859111c56a24f04f516a11b5bf09.png is an open set.

We call the process stochastic_differential_equations_cfc6df3bd5df1643c5e545302badee916f9f09d5.png the (Gaussian) space-time white noise. It is a random generalized function stochastic_differential_equations_4251ea8a78f49c0136f6382c64bb134716dd22b0.png:

stochastic_differential_equations_e88fa2ece53f2704e466b9b7324bdeb1b1f2fccc.png

Sometimes, an alternative notation is used for stochastic_differential_equations_6d454988682b116a1335e532b4041414cb49ed11.png:

stochastic_differential_equations_ac10cec74c5382346ffcdc2957b088e250a0198e.png

Unlike the Brownian sheet, space-time white noise stochastic_differential_equations_cfc6df3bd5df1643c5e545302badee916f9f09d5.png is defined on every domain stochastic_differential_equations_38382d88f647859111c56a24f04f516a11b5bf09.png and not just on hyper-cuves stochastic_differential_equations_2ff85917829338ee43f4d991d69b9ef15248d353.png, as log as we can find an orthonormal basis stochastic_differential_equations_4dea8b56a6c7502fb4489be310f8c2a7679fd7ed.png in stochastic_differential_equations_b19acb320bc8a2ea4109a04d488fa8197b2c2719.png.

Integrating over stochastic_differential_equations_c90030b5158c9f6b20bc216bd69891bb54b7ae85.png

Often see something like

stochastic_differential_equations_a748673642aefc22b4f9ce5231c05ce254fbfca2.png

e.g. in Ito's lemma, but what does this even mean?

Consider the integral above, which we then define as

stochastic_differential_equations_ddb54d49056469b2f2c26b3b5fa7b05362655b37.png

as we would do normally in the case of a Riemann-Stieltjes integral.

Now, observe that in the case of stochastic_differential_equations_5a8ee25134a129577635fbafa532a9658d09da80.png being Brownian motion, each of these increments are well-defined!

Furthermore, considering a partial sum of the RHS in the above equation, we find that the partial sum converges to the infinite sum in the mean-squared sense.

This then means that the stochastic integral is a random variable, the samples of which depend on the individual realizations of the paths stochastic_differential_equations_e1b66fb673d9750510e320d48ffa2d0624c56d40.png!

Alternative description of Gaussian white noise

  • Zero-mean Gaussian process stochastic_differential_equations_8ad5e145651278fef5666eeb727531ac6184a274.png such that

    stochastic_differential_equations_219a50e9c2b1bf883bec032d1f65b851be77f760.png

    where stochastic_differential_equations_18f46e29ae5df6d58883632ac9a73c6a3ea7e102.png is the Dirac delta function.

  • Similarily, we have

    stochastic_differential_equations_35b4574da0beac7ca30839937704ca890bb63752.png

  • To construct noise that is white in time and coloured in space, take a sequence of non-negative numbers stochastic_differential_equations_36a09c4115e0acb252d0fd5f6abcd679d6f3310a.png and define

    stochastic_differential_equations_804723427d9e616c42113a80ad01a04af58e69e3.png

    where stochastic_differential_equations_4dea8b56a6c7502fb4489be310f8c2a7679fd7ed.png is an orthonormal basis in stochastic_differential_equations_b19acb320bc8a2ea4109a04d488fa8197b2c2719.png.

  • We say this noise is finite-dimensional if

    stochastic_differential_equations_5969f3c05d21c26d72a13896dfe09610db0d175e.png

Useful Equalities

If stochastic_differential_equations_6a96cc5c9b586790f2129e1d4baac5d3d502416c.png is a smooth function and stochastic_differential_equations_40c82f3764cb554e9f08d1fe3dd382597371ab14.png is a standard Brownian motion, then

stochastic_differential_equations_fb2266a757766fa29bcdec8e67a7adfd84297ce6.png

If stochastic_differential_equations_c5cd2e111c21decdec253f54b39b5789e0778151.png is a std. Brownian motion and stochastic_differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png, an adapted process, then

stochastic_differential_equations_766e5d68d6218ef121f949495d7a80e9e037a4cf.png

The Fourier transform is defined

stochastic_differential_equations_09419f809ba868baa135ca826e5225d55e2f116a.png

which is defined on the generalized functions from stochastic_differential_equations_676285db441b6a041a679a15c3bb97a31ea885ed.png by

stochastic_differential_equations_3d6d22a13aa0d5bfca86da3b2d7ecd8eaa11caf3.png

for stochastic_differential_equations_ef537ab429e1909827d397593e4c319fbbb58ab9.png and stochastic_differential_equations_28929a008f1233a0189f5abc568ec9180b6140c4.png. And the inverse Fourier transform is

stochastic_differential_equations_9b7a8a628a8456344a05a039776e959d509db055.png

stochastic_differential_equations_b993c4d763b25a9b7f6bb8d77e07a8454b2bb2ac.png

If stochastic_differential_equations_4dea8b56a6c7502fb4489be310f8c2a7679fd7ed.png is an orthonormal basis in a Hilbert space stochastic_differential_equations_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and stochastic_differential_equations_8552cce38ec3f5f29f30d80b077098935e64f7af.png, then

stochastic_differential_equations_4d2c5906688db3b9624f2122936495151eb668ca.png

Plancherel's identity (or isometry of the Fourier transform) says that if stochastic_differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png is a smooth function with compact support in stochastic_differential_equations_abf17ab8d81a8a9c9a3b8d37b4e26b6e2983cdc7.png and

stochastic_differential_equations_52139f6d316b0efd689e2e93a76d04649d302404.png

then

stochastic_differential_equations_c09f24572cdcc0d52c972d9cea12183e7890056f.png

This result is essentially a continuum version of Parseval's identity.

Useful inequalities

Exercises

1.1.1

stochastic_differential_equations_6a6800f0051352cf5f1b847e7d8b1eb9fdbfcd76.png

Observe that

stochastic_differential_equations_129203979babd24391413391038a36e00068f389.png

since we're taking the product (see notation). Then, from the hint that stochastic_differential_equations_11fd356946ba1ba22e0b6b2550242ac7870c0c77.png is the Fourier coefficient of the indicator function of the interval stochastic_differential_equations_920128663550910f4a538ceeab54e1e1ade1f443.png, we make use of the Parseval identity:

stochastic_differential_equations_dba647ec888b3c1c1c1fe55b206d85e4f28beb8b.png

where stochastic_differential_equations_e134cf73270051adabbc994f8dae575f60cebfff.png denotes the k-th Fourier coefficient of the indicator function. Then

stochastic_differential_equations_de71c4aebd87945aeb08f39988be0924a661b23b.png

since stochastic_differential_equations_4b624e509a9218ea6c3943b3d6b29a8fd2e4c445.png are all standard normal variables, hence stochastic_differential_equations_1369048560b19e5e9aa4544ec58b95a7dea1c3c0.png.

I'm not entirely sure about that hint though? Are we not then assuming that the stochastic_differential_equations_6b437d3f827db0ad545eaad91184d01b1a3407d6.png is the basis? I mean, sure that's fine, but not specified anywhere as far as I can tell?

Hooold, is inner-product same in any given basis? It is! (well, at least for finite bases, which this is not, but aight) Then it doesn't matter which basis we're working in, so we might as well use the basis of stochastic_differential_equations_0692a571c05746595311f3f150eb3fb71bdcbfe5.png and stochastic_differential_equations_b646f9882ad10e5902c2b81de57b9190de67b3fe.png.

1.1.2

Same procedure as in 1.1.1., but observing that we can separate stochastic_differential_equations_927f053fb51c7300368813a8bf9e754fbaa667ff.png into a time- and position-dependent part, and again using the fact that

stochastic_differential_equations_7024615593d2ce8ec98ad35e2297ab88d6c3ec4e.png

is just the Fourier coefficient of the indicator function on the range stochastic_differential_equations_26c935433532efcde1d9ba65f2ce954cf3078590.png in the i-th dimension.

Basic Ideas

Notation

  • stochastic_differential_equations_67bc6eb59958ebb4af4816afd9546348615fd563.png is a probability space
  • stochastic_differential_equations_84a86600a7a2604c4b83237e6c4166663b92bcab.png and stochastic_differential_equations_5bb447d9c543b2fbdf43bc14d66f4d164bdb7890.png are two measurable spaces
  • stochastic_differential_equations_27dfced334eb8c2b91e16311e57417ecf88d0bf1.png is a random function
  • Random process refers to the case when stochastic_differential_equations_72a907baf0f5471a395428e84ec6bf1c7efa37e2.png
  • Random field corresponds to when stochastic_differential_equations_35938a998c3d4ec360f701c7d06f6c1ae7c3b00a.png and stochastic_differential_equations_79f2f3575d655ecfcd0d54a3f70ec47c086e8e08.png.