A Burgers-KPZ Type Parabolic Equation \par\noindent for the Path-Independence of the Density of the Girsanov Transformation

Abstract

Let $X_t$ solve the multidimensional Itô's stochastic differential equations on ${\mathbb{R}}^d$ $$d{X}_t=b(t,X_t)dt+\sigma (t,X_t)d{B}_t$$ where $b:[0,\infty )\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ is smooth in its two arguments, $\sigma :[0,\infty )\times {\mathbb{R}}^d\to {\mathbb{R}}^d\otimes {\mathbb{R}}^d$ is smooth with $\sigma (t,x)$ being invertible for all $(t,x)\in [0,\infty )\times {\mathbb{R}}^d$, $B_t$ is $d$-dimensional Brownian motion. It is shown that, associated to a Girsanov transformation, the stochastic process $${\int }_0^t⟨({\sigma }^{-1}b)(s,X_s),d{B}_t⟩+\frac{1}{2}{\int }_0^t|{\sigma }^{-1}b{|}^2(s,X_s)ds$$ is a function of the arguments $t$ and $X_t$ (i.e., path-independent) if and only if $b=\sigma {\sigma }^{\ast }\nabla v$ for some scalar function $v:[0,\infty )\times {\mathbb{R}}^d\to \mathbb{R}$ satisfying the time-reversed KPZ type equation $$\frac{\partial }{\partial t}v(t,x)=-\frac{1}{2}\left[\left(Tr(\sigma {\sigma }^{\ast }{\nabla }^{2}v)\right)(t,x)+|{\sigma }^{\ast }\nabla v{|}^2(t,x)\right].$$ The assertion also holds on a connected complete differential manifold.

19 pages