I Have to solve the following PDE: \begin{equation} \begin{cases} \dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{\partial^2 F}{\partial x\partial y}=0\\ F(t,x,y)=xy^2 \end{cases} \end{equation} My problem is the presence of the mixed derivative term and I don't have any Idea on how to manage it. If the mixed derivative term is not present I know that I can introduce two independent brownian motion under the probability measure $\mathbb{P}$ and describe the dynamics of the process $X$ and $Y$ as: \begin{equation} dX_t=dW_t^1\\ dY_t=dW_t^2 \end{equation} and then apply the formula: \begin{equation} F(t,X_t,Y_t)=E^\mathbb{P}_t[X_TY_T^2]=E_t^\mathbb{P}[X_T]E_t^\mathbb{P}[Y^2_T] \end{equation} If the mixed derivative is present I know that the two process can be correlated in some ways but I'm not able to write explicit caluclations. Can someone help me?
2026-04-01 19:12:28.1775070748
Feymann Kac formula to solve Pde
67 Views Asked by user117449 https://math.techqa.club/user/user117449/detail AtRelated Questions in STOCHASTIC-CALCULUS
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