Working with Wiener processes (mathematical Brownian motion)

Example calculations:

Another example:

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Another example:

What kind of random variable is:

not true for the original form in terms of the square of an integral

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Continuing our calculations:

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Finally, Z(t) is for each t a Gaussian random variable because the Riemann sum approximations are sums of Gaussian random variables, and one can show that the Riemann sum approximations converge in a "mean-square sense" and the "mean-square" limits of Gaussian random variables are Gaussian random variables. (Oksendal)

We conclude that for each t, Z(t) is a Gaussian random variable with mean 0 and standard deviation

Beyond studying the Wiener process in its own right it plays an ubiquitous role in the theory of stochastic processes in continuous state space. Reason: Levy-Khintchine theorem says that any noise that has independent increments and influences the state variables in a continuous manner, then the noise process must be governed by the Wiener process suitably rescaled (locally).

Simple example of Langevin description of physical Brownian motion

particle undergoing Brownian motion at time t

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Fluctuation-dissipation theorem (nonequilibrium statistical mechanics)

Physicist language: is mean zero Gaussian random function with "correlations" (don't you mean covariance?-- confusing language)

has a "white noise" character: a random function that is independent at each moment of time (and statistically stationary in time and some other conditions…)

One can work on a formal level with these white noise processes (often done in theoretical physics, OK if careful).

But how can we write a meaningful description for the random thermal force in terms of a clean differential equation.

White noise can be expressed formally as

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the derivative of the Wiener process. Intuition: Wiener process has independent increments, so derivative should be uncorrelated at different moments of time and also has Gaussian properties (since discrete difference approximations are just linear combinations of the Gaussian Wiener process).

This is the point for why one needs a separate calculus for white noise: it is not just a random function, it doesn’t exist as a normal random function (delta functions). So how to make sense of the differential equation for the Brownianparticle is not obvious.

At least three ways to proceed mathematically: Work in the sense of generalized random functions (Hida, Gel'fand, etc do this but this is not emphasized in the applied literature.)

Shut up and calculate with the delta functions.

Mathematical SDE point of view:

helpful to work in terms of first order systems

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This is how Stochastic differential equations arewritten in a mathematically proper way (so noreferences to derivatives of Wiener process)

Interpret?

We will say that V(t) satisfies the stochastic differential equation if it satisfies its integrated version. That means we just have to give meaning to integrating against increments of the Wiener process (dW) --stochastic integrals.

Generalization: Remember that noise with independent increments entering a system continuously must look (locally) like a rescaled Wiener process. So a broad class of continuous-time stochastic processes (drift-diffusion processes -- see Kloeden Platen 1.7, 2.4)are actually described by stochastic differential equations of the followng form:

and this SDE is interpreted so that a solution X(t) must satisfy

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psolution X(t) must satisfy

Defining the stochastic integral is subtle because it is not simply a random extension of any traditional integration theory:

Riemann integrals:

Riemann-Stieltjes: you have to be very careful; the standard theory doesn't work because W does not have "bounded variation" So if one blithely tries to use this integration theory one gets different answers for different choices of discretization.

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