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Solving a PDE with a Mixed Derivative Using Perfectly Correlated Brownian Motion

Article Quant Q&A · Author: Pefok
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Summary

The document works through a Feynman–Kac representation for a two-variable partial differential equation containing a mixed second derivative. It connects the diffusion coefficients in the equation to the covariance of the underlying Brownian motions. With unit variance in each coordinate and a mixed-derivative coefficient corresponding to unit covariance, the proposed processes are driven by the same Brownian motion, equivalent to perfect correlation.

For the terminal payoff given as the product of the first coordinate and the square of the second, the derivation evaluates its expectation after a shared Gaussian increment. The resulting expression adds time-dependent terms from the increment’s second moment, and the author substitutes the expression back into the PDE as a sanity check. This illustrates how covariance enters the generator and why independence would not represent the stated equation. The post does not work through the integrability conditions it mentions, and its argument is limited to this particular payoff and coefficient structure.

Key ideas

  • A mixed second derivative in the diffusion equation corresponds to covariance between the driving processes.
  • Unit variance in both coordinates and unit covariance imply perfectly correlated Brownian increments.
  • Feynman–Kac evaluates the terminal payoff using the joint distribution of those increments.
  • The proposed solution is checked by substitution into the partial differential equation.
  • The post leaves the stated integrability verification unfinished.

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# Feymann Kac for multidimensional pde


# Feymann Kac for multidimensional pde












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 processes can be correlated in some ways but I'm not able to write explicitely the dynamic of the two processes calculations. Can someone help me?

EDIT: I've tried to reasoning like this: we can assume that the two processes evolves according the following dynamics with $W_1$ and $W_2$ independent Brownian Motions: \begin{equation} dX=\sigma_1dW_1\\ dY=\sigma_2(\rho dW_1+\sqrt{1-\rho^2}dW_2) \end{equation} Now lookind at the coefficients of the PDE we have $\sigma_1=1$, $\sigma_2=1$ and since the quadratic covariation should be still 1 it must be $\sigma_2\rho=1\Rightarrow\rho=1$. Hence in principal the two processes follows the same dynamic: \begin{equation} dX=dW_1\\ dY=dW_1 \end{equation} Now this implies that the solution of the two SDEs are: \begin{equation} X_T=X_t+W_T^1-W_t^1\\ Y_T=Y_t+W_T^1-W_t^1 \end{equation} Applying now Feymann-Kac we find: \begin{equation} F(t,X_t,Y_t)=E_t(X_TY_T^2)=E((X_t+\sqrt{T-t}Z_1)(Y_t^2+2Y_t\sqrt{T-t}Z_1+(T-t) Z_1^2))\\ =X_tY_t^2+X_t(T-t)+2Y_t(T-t) \end{equation} Where I used that $Z_1\sim N(0,1)$ and $E(Z_1^3)=0$ by the properties of gaussian density. Now removing the stochasticity in the processes we find the solution: \begin{equation} F(t,x,y)=xy^2+(x+2y)(T-t) \end{equation} As sanity check I plug it into the Pde to find: \begin{equation} -x-2y+\frac{1}{2}2x+2y=0 \end{equation} To conclude I have now to check that $\dfrac{\partial F}{\partial X}$ and $\dfrac{\partial F}{\partial X}$ are in $M^2(0,T)$ but I avoid this computation.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.