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Kolmogorov forward equation

This is also called the Fokker-Plank equation.
Let $X$ be a solution to the SDE \begin{eqnarray*} dX(u)&=&\mu(u,X(u))dt+\sigma(u,X(u))dW(u),~u>t\\ X(t)&=&x \end{eqnarray*} Then the density at time $T$, $f(T,y,t,x)$, satisifies the PDE: \[ \frac{\partial}{\partial T} f(T,y,t,x)-\frac{\partial}{\partial y}\left(\mu(T,y) f(T,y,t,x)\right) +\frac{\partial^2}{\partial y^2}\left(\frac{1}{2}\sigma^2(T,y) f(T,y,t,x)\right)=0\] Viewed as a function of $t$ and $x$, $f$ satisfies the Kolmogorov backward equation with $g(X(T))=\delta(X(T)-y)$.

 

Questions: Magnus Wiktorsson
Last update: 2012 Sep 02 18:20:42. Validate: HTML CSS

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