English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $X_t$ solve the multidimensional It\^o's stochastic differentialequations on $\R^d$ $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dB_t$$ where$b:[0,\infty)\times\R^d\to\R^d$ is smooth in its two arguments,$\sigma:[0,\infty)\times\R^d\to\R^d\otimes\R^d$ is smooth with $\sigma(t,x)$being invertible for all $(t,x)\in[0,\infty)\times\R^d$, $B_t$ is$d$-dimensional Brownian motion. It is shown that, associated to a Girsanovtransformation, the stochastic process$$\int^t_0\langle(\sigma^{-1}b)(s,X_s),dB_t\rangle+\frac{1}{2}\int^t_0|\sigma^{-1}b|^2(s,X_s)ds$$is a function of the arguments $t$ and $X_t$ (i.e., path-independent) if andonly if $b=\sigma\sigma^\ast\nabla v$ for some scalar function$v:[0,\infty)\times\R^d\to\R$ satisfying the time-reversed KPZ type equation$$\frac{\partial}{\partialt}v(t,x)=-\frac{1}{2}\left[\left(Tr(\sigma\sigma^\ast\nabla^2v)\right)(t,x)+|\sigma^\ast\nabla v|^2(t,x)\right].$$ The assertion also holds on a connectedcomplete differential manifold.

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