0
votes
0answers
29 views

Stochastic PDE representation

I am trying to find a pde which $u$ satisfies when $u(x) = E^{x}[\cos(X_1)]$ where $dX_t = \sin(nX_t)\,dt + dW_t$ and $X_0 = x$. I have tried using Feynman-Kac but I can't seem to get it into the ...
2
votes
0answers
62 views

Initial Conditions for Finite Difference of PDE

I am having trouble with figuring out what my initial conditions should be for a simple finite difference algorithm I wrote in Matlab. Specifically, I'm trying to show that the regular 1-Dimensional ...
2
votes
2answers
99 views

Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$

Is there a way to solve $$ \partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0? $$ This appeared as a condition for $$ X_t=u(t,B_t)e^{\int_0^tv(B_s)ds} $$ to be a martingale. With $B$ a ...
4
votes
0answers
239 views

Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?

The following is the definition of infinitesimal generator from Oksendal. Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
1
vote
2answers
188 views

Random diffusion coefficient in the Fourier equation

I'm stuck on the following simple problem: It's given the Fourier equation: $$\partial_t{u(x,t)}=\partial_x[k(t)\partial_xu(x,t)]$$where the diffusion coefficient $k(t)$ is a random variable with a ...