# Stochastic Differential Equations

• Zeev Schuss
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 170)

## Abstract

Dynamics driven by white noise, often written as
$$d{\bf {\it x}} = {\bf {\it a}}({\bf {\it x}}, t) dt + {\bf {\it B}}(x, t) d{\bf {\it w}},\ \ \ {\bf {\it x}}(0) = {\bf {\it x}}0,$$
(4.1)
or
$$\dot{x} = {\bf {\it a}}({\bf {\it x}}, t) + {\bf {\it B}}(x, t) \dot{w},\quad {\bf {\it x}}(0) = {\bf {\it x}}0,$$
is usually understood as the integral equation
$${\bf {\it x}}(t) = {\bf {\it x}}(0) + \int_0^t {\bf {\it a}}({\bf {\it x}}(s), s) ds + \int^t_0 {\bf {\it B}}({\bf {\it x}}(s), s) d{\bf {\it w}}(s),$$
(4.2)
where $$a(x, t)$$ and $$B(x, t)$$ are random coefficients, which can be interpreted in several different ways, depending on the interpretation of the stochastic integral in (4.2) as Itô, Stratonovich, backward, or otherwise. Different interpretations lead to very different solutions and to qualitative differences in the behavior of the solution. For example, a noisy dynamical system of the form (4.1) may be stable if the Itô integral is used in (4.2), but unstable if the Stratonovich or the backward integral is used instead. Different interpretations lead to different numerical schemes for the computer simulation of the equation. A different approach, based on path integrals, is given in Chapter 5.

## Keywords

Brownian Motion Stochastic Differential Equation Conditional Expectation Uniqueness Theorem Planck Equation
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