Boris L. Rozovsky

Stochastic Integration in a Hilbert Space

This chapter is about stochastic calculus for continuous martingales and local martingales in a Hilbert space. The topics include definitions and investigations of martingales, local martingales and a Wiener process in a Hilbert space, construction of stochastic integrals with respect to these processes, and a detailed proof of the Ito formula for the square of a norm of a continuous semimartingale.

Examples and Auxiliary Results

The first of the four sections in this chapter presents examples of linear stochastic evolution systems (LSESs) arising in various applications. The following three sections collect a number of auxiliary results which are used systematically throughout the book. In particular, Sect. 1.5 surveys the theory of stochastic ordinary differential equations.

Hypoellipticity of Itô’s Second Order Parabolic Equations

Smoothness of solutions of deterministic parabolic equations increases as the smoothness assumptions on their coefficients increase. This is a typical feature of parabolic equations. Moreover, under wide assumptions, the smoothness of solutions for t > 0 depends only on the smoothness of coefficients and does not depend on the smoothness of the initial functions. This is important, for example, in the study of the fundamental solution of a parabolic equation, since we can consider this solution as a solution of the corresponding Cauchy problem where the initial function is the Dirac delta function. Hypoellipticity is a particular case of the growth of smoothness property mentioned above.

Chaos Expansion for Linear Stochastic Evolution Systems

Separation of variables is widely used to study evolution equations. For deterministic equations, there are two variables to separate: time and space; the result is often an orthogonal expansion of the solution in the eigenfunctions of the operator in the equation.

Filtering, Interpolation and Extrapolation of Diffusion Processes

Recall that the filtering problem for diffusion processes first appeared in Sect. 1.2.2, where we discussed the motivation and general setting. Accordingly, we now go directly to the mathematical formulation of the problem.

Itô’s Second-Order Parabolic Equations

Let us fix \(T_{0},T\in \mathbb {R}_{+}\) with T0 ≤ T, and \(\mathrm {d},\mathrm {d}_1\in \mathbb {N}\). We also fix the stochastic basis \(\mathbb {F}=(\varOmega , \mathscr {F}, \{\mathscr {F}_{t}\}_{t\in [0,T]}, \mathbb {P})\) with the usual assumptions, and a standard Wiener process w on \(\mathbb {F}\) with values in \(\mathbb {R}^{\mathrm {d}_1}.\)

Itô’s Partial Differential Equations and Diffusion Processes

In this chapter we continue the study of the Cauchy problem for second-order parabolic Ito equations, this time concentrating on qualitative, rather than analytical, aspects of the problem. The main objective is to establish various connections between these equations and diffusion processes.

Stochastic Analysis in Infinite Dimensions

This chapter contains somewhat abstract but necessary, material on functional analysis and stochastic calculus. To save time, one can move on to the following chapters and come back as necessary.

Linear Equations: Square-Integrable Solutions

There are many standard references on SODEs and even more standard references on deterministic PDEs. Here are a few of each, listed in a non-decreasing order of difficulty:

The Polynomial Chaos Method

Separation of variables is a powerful idea in the study of partial differential equations, and the polynomial chaos method is a particular implementation of this idea for stochastic equations. While the elementary outcome ω is typically never mentioned explicitly in the notation of random objects, it is a variable that can potentially be separated from other variables, and the objective of this chapter is to outline a systematic approach to doing just that. Along the way, it quickly becomes clear that many ideas are closely connected to another modern branch of stochastic analysis, namely, Malliavin Calculus, and we explore these connections throughout.