Solving a Stochastic PDE with two variables in time

Question

I am trying to work on exercise 5.13 in the book Arbitrage Theory in Continuous time by Thomas Bjork. The equation to solve is; \begin{eqnarray*} \frac{\partial F}{\partial t} (t,x,y) + \frac{1}{2} \sigma^2 \frac{\partial^2 F}{\partial x^2} (t,x,y) + \frac{1}{2} \delta^2 \frac{\partial^2 F}{\partial y^2} (t,x,y) &= 0\\ F(T,x,y) &= xy\end{eqnarray*} I seem not to see how I can start off, especially due to the fact that I am used to solving when the equation has only one variable in time. How can I go about this when they are now two variables in time? Thanks a lot.

Answer 1

Just looking at the boundary condition, you can guess the solution.

Or you can :

We define the two SDEs $$dX_t=\sigma dW_t$$ $$dY_t=\delta dZ_t$$ where $W_t$ and $Z_t$ are independent Brownian motions.

By applying the Feynman-Kac formula, the solution of your PDE can be written as

$$u(t,x,y)=E\left[X_TY_T|X_t=x,Y_t=y\right]$$

We can write $$X_T=X_t+\sigma\left(W_T-W_t\right)$$ $$Y_T=Y_t+\delta\left(Z_T-Z_t\right)$$

Using the independence property of the Brownian motions increments

$$u(t,x,y)=E\left[\left(x+\sigma\left(W_T-W_t\right)\right)\left(y+\delta\left(Z_T-Z_t\right)\right)\right]=xy$$