# Stochastic Differential Equations

## Table of Contents

- Books
- Overview
- Notation
- Definitions
- Theorems
- Ito vs. Stratonovich
- Variations of Brownian motion
- Karhunen-Loève Expansion
- Diffusion Processes
- Solving SDEs
- Numerical SDEs
- Connections between PDEs and SDEs
- TODO Introduction to Stochastic Differential Equations
- Stochastic Partial Differential Equations (rigorous)

## Books

- Handbook of Stochastic Methods
- Øksendal
- Probability with Martingales

## Overview

- A lot from this section comes from the book lototsky2017stochastic

## Notation

Correlation between two points of a

*random field*or a*random process*:to allow the possibility of an infinite number of points. In the discrete case this simply corresponds to the covariance

*matrix*.- denotes the Borel σ-algebra
- Partition
We write

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

the limit is to be understood in the

**mean-square sense**, i.e.

## Definitions

A stochastic process is called a either

**second-order stationary****wide-sense stationary****weakly stationary**

if

- the first moment is
*constant* - covariance function depends only on the difference

That is,

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

for such that , for every .

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

### Martingale continuous processes

Let be a filtration defined on the probability space and let be *adapted* to , i.e. is measurable on , with .

We say is an **martingale** if

### Gaussian process

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

That is, for every finite dimensional vector

for some *symmetric non-negative definite* matrix , for all and .

## Theorems

### Bochner's Theorem

Let be a *continuous positive definite* function.

Then there exists a unique nonnegative measure on such that and

i.e. is the Fourier transform of the function .

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

The measure is called the **spectral measure** of the process .

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

then the Fourier transform of the covariance function is called the **spectral density** of the process:

## 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

- "Expected" since the standard chain rule should work fro

- White noise is regarded as

### Equivalence

Suppose solves the following Stratonovich SDE

then solves the Itô SDE:

Suppose solves the Itô SDE:

then solves the Stratonovich SDE:

That is, letting ,

- Stratonovich → Itô:
- Itô → Stratonovich:

In the multidimensional case,

To show this we consider staisfying the two SDEs

i.e. the satisfying the stochastic integrals

We have

Equivalently, we can write

Assuming the is smooth, we can then Taylor expand about and evaluate at :

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

Using the fact that satisfies the Itô integral, we have

and that

we get

Thus, we have the identity

Matching the coefficients with the Itô integral satisfied by , we get

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

Important to note the following though: here we have assumed that the 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 and still obtain conversion rules between the two formulations of the SDEs.

#### Stratonovich satisfies chain rule proven using conversion

For some function , 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

We will write , where .

The spectral density of this process is

This is called a **Cauchy** or **Lorentz distribution**.

The correlation time is then

A real-valued Gaussian stationary process defiend on 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** , , with Hurst parameter is a centered Gaussian process with continuous sample paths whose covariance is given by

Hence, the *Hurst parameter* controls:

- the correlations between the increments of
**fractional Brownian motion** - the regularity of the paths: they become smoother as increases.

A fractional Brownian motion has the following properties

- When , then becomes standard Brownian motion.
We have

It has stationary increments, and

It has the following self-similarity property:

where the equivalence is

*in law*.

## Karhunen-Loève Expansion

### Notation

- where
- be an orthonormal basis in

### Stuff

Let .

Suppose

We assume are *orthogonal* or *independent*, and

for some positive numbers .

Then

due to orthogonality of and for .

Hence, for the expansion of above, we need to be *valid*!

The above expression for also implies

Hence, we also need the set to be a set of *eigenvalues and eigenvectors* of the *integral operator* whose kernel is the correlation function of $X_{t}, i.e. need to study the operator

which we will now consider as an operator on .

It's easy to see that is self-adjoint and nonnegative in :

Furthermore, it is a *compact* operator, i.e. if is a bounded sequence on , then has a *convergent subsequence*.

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

Furthermore, for every , we can write

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

Let be an process with zero mean and continuous correlation function .

Let be the eigenvalues and eigenfunctions of the operator defined

Then

where

The series converges in to , *uniformly* in !

#### Karhunen-Loève expansion of Brownian motion

- Correlation function of Brownian motion is .
Eigenvalue problem becomes

- Assume (since would imply ψ
_{n}(t) = 0$) - Consider intial condition which gives
Can rewrite eigenvalue problem

Differentiatiate once

using FTC, we have

hence

- Obtain second BC by observing (since LHS in the above is clearly 0)
Second differentiation

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

Eigenvalues and (normalized) eigenfunctions are then given by

Karhunen-Loève expansion of Brownian motion on is then

## Diffusion Processes

### Notation

### Markov Processes and the Chapman-Kolmogorov Equation

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

*sample space**state space*

- Idea: encode all past information about a stochastic process into an appropriate collection of σ-algebras

Let denote a probability space.

Consider stochastic process with and state space .

A **filtration** on is a *nondecreasing* family of sub-σ-algebras of :

We set

The **filtration generated by** our stochastic process is

A filtration is generated by events of the form

with and .

Let be a stochastic process defined on a probability space with values in , and let be the filtration generated by .

Then is a **Markov process** if

for all with , and .

Equivalently, it's a **Markov process** if

for and with $Γ ∈ (E).

### Chapman-Kolmogorov Equation

The transition function for fixed is a probability measure on with

It is measurable in , for fixed , and satisfies the **Chapman-Kolmogorov equation**

for all

- with

Assuming that , we can write

since .

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

- System moves from to at some
*intermediate*step - Moves from to at time

### 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

Let be the *transition* function of a homogenous Markov process and let , and define the operator

Linear operator with

which means that , and

i.e. .

We can study properties of time-homogenous Markov process by studying properties of the **Markov semigroup** .

This is an example of a strongly continuous semigroup.

Let be set of all such that the limit

exists.

The operator is called the **(infinitesimal) generator of the operator semigroup **

- Also referred to as the
**generator of the Markov process**

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

Furthermore, consider function

Compute time-derivative

And we also have

Consequently, satisfies the IVP

which defines the **backward Kolmogorov equation**.

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

#### Example: Brownian motion 1D

*Transition function*for Brownian motion is given by the fundamental solution to the heat equation in 1DCorresponding Markov semigroup is the

*heat semigroup*- Generator of the 1D Brownian motion is then the 1D Laplacian
The backward Kolmorogov equation is then the heat equation

#### Adjoint semigroup

Let be a Markov semigroup, which then acts on .

The **adjoint semigroup** acts on *probability measures*:

The image of a probability measure under is again a probability measure.

The operators and are *adjoint* in the sense:

We can write

where is the adjoint of the generator of the Markov process:

Let be a Markov process with generator with , and let denote the adjoint Markov semigroup.

We define

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

Assuming that the *initial distribution* and the law of the process each have a density wrt. Lebesgue measure, denoted and , respectively, the law becomes

which defines the **forward Kolmorogov equation**.

##### "Simple" forward Kolmogorov equation

Consider SDE

Since is Markovian, its evoluation can be characterised by a transition probability :

Consider , then

### 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 , i.e. a fixed point of the semigroup :

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

We say the process is **ergodic wrt. the measure **.

Furthermore, if we consider a Markov process in with generator and Markov semigroup , we say that is **ergodic** provided that is a *simple eigenvalue* of , i.e.

has *only constant solutions*.

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

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

Assume that has a density wrt. the Lebesgue measure. Then

by definition of the generator of the adjoint semigroup.

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

#### 1D Ornstein-Uhlenbeck process and its generator

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

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

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

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

Thus,

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

which is just a *Gaussian measure*!

*Observe that in the above expression, the stuff on LHS before corresponds to in the equation we solved for .*

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

and spectral density

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 .

### Diffusion Processes

#### Notation

- means depends on terms dominated by
*linearity*in

#### 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 in with transition function is called a **diffusion process** if the following conditions are satisfied:

*(Continuity)*For every and ,uniformly over

(

**Drift coefficient**) There exists a function s.t. for every and every ,uniformly over .

(

**Diffusion coefficient**) There exists a function s.t. for every and every ,uniformly over .

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

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

and **diffusion coefficient**:

#### Backward Kolmogorov Equation

Let , and let

with fixed .

Assume, furthermore, that the functions and are *smooth* in both and .

Then solves the *final* value problem

for .

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*:

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

Let , and introduce . Then,

Further, we can let , therefore

where

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 and :

Then the transition probability density is the solution to the IVP

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"

- =
- for all
If multivariate case,

*typically*one will assume white noise to be indep.:Suppose we have some SDE for , i.e. some expression for .

*Change of variables*then from Itô's lemma we havewhere is computed by straight forward substitution by expression for and using the "properties" of

Letting and in Itô's lemma we get

Letting and , then Itô's lemma gets us

Hence satisfies the geometric Brownian motion SDE.

### Numerically

## Numerical SDEs

### Notation

We willl consider an SDE with the exact solution

### Convergence

The **strong error** is defined

We say a method **converges strongly** if

We say the method has **strong order** if

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

#### Error on individual path level

Euler-Maryuama has

Markov inequality says

for any .

Let gives

i.e.

- That is,
*along any path the error is small with high probability*

### Stochastic Taylor Expansion

#### Deterministic ODEs

Consider

For some function, we then write

where

and the 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 :

where

- 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
- For an example of this being used in a "standard" way; see how we proved Euler's method for ODEs

- 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 SDEsConsider

Satisfies the Itô integral

- Can do the same as before for each of the integral terms:
:

but in this stochastic case, we have to use Itô's formula

so

:

Then we can substitute these expressions into our original expression for :

Observe that we can bring the terms and out of the integrals:

- 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

### Forward Kolmogorov / Fokker-Plank equation

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

#### Derivation of Forward Kolmogorov

Suppose we have an SDE of the form

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

which then satisfies the Itô integral

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

and RHS

Taking the derivative wrt. , LHS becomes

and RHS

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 rather than derivatives of ", 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

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

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

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 :*as*Absorbing boundary conditions:*where we assume that for (the boundary of the domain )*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:

where

Here we've used defined as

rather than

as defined before.

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

Furthermore, if and , i.e. time-independent (*autonomous* system), then

in which case the Forward Kolmogorov equation becomes

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

### Notation of generator of the Markov process and its adjoint

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

## TODO Introduction to Stochastic Differential Equations

### Notation

- indicates the dependence on the initial condition .
Stopping time

### Motivation

We will consider stochastic differential equations of the form

or, equivalently, componentwise,

which is

*really*just notation forNeed to define

*stochastic integral*for sufficiently large class of functions.

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

- Since Brownian motion is

### Itô and Stratonovich Stochastic Integrals

Let

where is a Brownian motion and , such that

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

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 ** as the ( is the underlying probability space) limit of the Riemann sum approximation

where

- where is s.t.
increments

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

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

Let be a stochastic integral.

The **Stratonovich stochastic integral** is when , i.e.

We will often use the notation

to denote the **Stratonovich stochastic integral**.

Observe that does *not* satisfy the Martingale property, since we are taking the *midpoint* in , therefore is *correlated* with !

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

Suppose that there exist s.t.

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

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

with being white noise.

The solution then satisfies the integral equation

which motivates the notation

since, in a way,

(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 be a Itô stochastic integral, then

From which it follows that for any square-integrable functions

but we also know that

Comparing with the equation above, we see that

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

##### Martingale

For Itô stochastic integral we have

and

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

The quadratic variation of this martingale is

### Solutions of SDEs

Consider SDEs of the form

where

A process with continuous paths defined on the probability space is called a **strong solution** to the SDE if:

- is a.s. continuous and adapted to the filtration
- and a.s.
For every , the stochastic integral equation

Let

satisfy the following conditions

There exists positive constant s.t. for all and

For all and ,

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

Then the SDE

has a **unique strong solution** with

where by *unique* we mean

for all possible solutions and .

### Itô's Formula

#### Notation

#### Stuff

Consider Itô SDE

is a diffusion process with drift and

**diffusion matrix**The generator is then defined as

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

Let be the solution of

and let . Then the process satisfies

where the generator is defined

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

then simplifies to

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

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 be a diffusion process with

- drift
- diffusion
- generator with

and let

be bounded from below.

Then the function

is the solution to the IVP

The representation 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

We observe that using Itô's formula

which from the Ornstein-Uhlenbeck equation we see that

i.e.

which gives us

and thus

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

Further, we observe the following:

since this is a Gaussian process. The covariance ew see

Assuming , these are independent! Therefore

Using Itô isometry, the first factor becomes

since

#### Langevin equation

##### Notation

- position of particle
- velocity of particle

##### Definition

##### Solution

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

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

Subsituting into our expression for :

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

Hence the solution is given by

#### Geometric Brownian motion

The **Geometric Brownian motion** equation is given by

#### Brownian bridge

Consider the process which satisfies the SDE

for .

This is the definiting SDE for a **Brownian bridge**.

##### Solution

#### A random oscillator

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

A *stochastic* version might be

where is constant and the frequency is a white noise with .

##### Solution

We can solve this by letting

so

Then,

Hence,

and thus

## 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

- denotes the space of continuous mappings from metric space to metric space
- If then we write

- is the collection of functions with continuous derivatives
- for and is the collection of functions with continuous derivatives s.t. derivatives of order are Hölder continuous of order .
- is the collection of infinitely differentiable functions with compact support
- denotes the Schwartz space and denotes the dual space (i.e. space of linear operators on )
- denotes the partial derivatives of every order
Partial derivatives:

and

- Laplace operator is denoted by
means

and if we will write

means

for all sufficiently larger .

- means that is a Gaussian rv. with mean and variance
- or are equations driven by
*Wiener process* -
- is the
*sample space*(i.e. underlying space of the measure space) - is the sigma-algebra ( denotes the power set of )
- is an
*increasing*family of sub-algebras and is*right-continuous*(often called a**filtration**) - contains all neglible sets, i.e. contains every subset of that is a subset of an element from with measure zero (i.e. is complete)

- is the
- is an indep. std. Gaussian random variable

### Definitions

A **filtered probability space** is given by

where the sigma-algebra represents the information available up until time , and a process is **adapted** if is .

#### Martingale

A **square-integrable Martingale** on is a process with values in such that

and

A **quadratic variation** of a martingale is the *continuous non-decreasing* real-valued process such that

is a *martingale*.

A **stopping (or Markov) time** on is a *non-negative* random variable such that

### Introduction

If is an orthonormal basis in , then

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

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

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

Consider

- a collection of indep. std. Brownian motions,
orthonormal basis in the space with

a d-dimensional hyper-cube.

For define

Then the process

is *Gaussian*,

We call this process the **Brownian sheet**.

From Ex. 1.1.3 b) we have

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

Writing

where is an orthonormal basis in and is an open set.

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

Sometimes, an alternative notation is used for :

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

#### Integrating over

Often see something like

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

Consider the integral above, which we then define as

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

Now, observe that in the case of 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 !**

#### Alternative description of Gaussian white noise

Zero-mean Gaussian process such that

where is the Dirac delta function.

Similarily, we have

To construct noise that is

*white in time*and*coloured in space*, take a sequence of non-negative numbers and definewhere is an orthonormal basis in .

We say this noise is

**finite-dimensional**if

#### Useful Equalities

If is a smooth function and is a standard Brownian motion, then

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

The **Fourier transform** is defined

which is defined on the generalized functions from by

for and . And the **inverse Fourier transform** is

If is an orthonormal basis in a Hilbert space and , then

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

then

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

#### Useful inequalities

#### Exercises

##### 1.1.1

Observe that

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

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

since are all *standard* normal variables, hence .

I'm not entirely sure about that hint though? Are we not then assuming that the 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 and .

##### 1.1.2

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

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

### Basic Ideas

#### Notation

- is a
*probability space* - and are two measurable spaces
- is a
*random function* **Random process**refers to the case when**Random field**corresponds to when and .