Lecture 10: Forward and Backward equations for SDEs

Lecture 10: Forward and Backward equations for SDEs

Readings

Recommended:

? Pavliotis [2014] 2.2-2.6, 3.4, 4.1-4.2 ? Gardiner [2009] 5.1-5.3

Other sections are recommended too ? this is a great book to read (and own as a reference), and it is strongly suggested to start looking through it.

Optional:

? Oksendal [2005] 7.3, 8.1, ? Koralov and Sinai [2010] 21.3, 21.4

Consider the SDE

dXt = b(Xt ,t)dt + (Xt ,t)dWt , Xt ,Wt Rd.

(1)

We assume throughout this lecture that b, satisfy a global Lipschitz condition and linear growth condition. We work with a vector-valued process here, since it will be no more complicated than a scalar one. We have studied how to solve for the actual solution trajectories themselves. For the next few lectures we will consider another approach to study properties of the solutions, via partial differential equations.

We start with an important theorem: solutions to (1) are Markov processes. We won't prove this here,

but rather point out that it is a natural consequence of the fact that the increments Xt - Xs =

t s

b(Xu)d

u

+

(Xu)dWu can be calculated using only (Xu)su ................
................

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