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Kolmogorov backward 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 $f(t,x)=E[g(X(T))|X(t)=x]$ satisifies the PDE: \[ \frac{\partial}{\partial t} f(t,x)+\mu(t,x)\frac{\partial}{\partial x} f(t,x) +\frac{1}{2}\sigma^2(t,x)\frac{\partial^2}{\partial x^2} f(t,x)=0\]

 

Questions: Magnus Wiktorsson
Last update: 2012 Sep 02 16:21:49. Validate: HTML CSS

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