Two-Variable Feynman–Kac Problem

Article Quant Q&A · Author: Femman

Summary

The document considers a terminal value problem whose differential operator has second-derivative coefficients corresponding to a two-variable covariance matrix. The terminal condition is a polynomial in the state variables, and the proposed Feynman–Kac representation discounts its expected terminal value over the remaining time. The central issue is choosing Brownian increments whose variances and covariance match the operator.

The answer gives a construction in which the first state is driven by one Brownian motion, while the second shares that motion and also receives an independent Brownian component scaled by the square root of two. This produces the stated covariance matrix and supplies the appropriate correlated diffusion. The response addresses the modeling step that caused confusion, but does not work through the expectation to obtain the full solution. Readers should verify the generator and discount convention when applying the method to a different equation.

Key ideas

The diffusion covariance matrix is determined by the second-derivative terms in the differential operator.
Correlated state variables can be represented using shared and independent Brownian components.
The suggested diffusion produces variances of one and three with covariance one.
The Feynman–Kac representation evaluates discounted expected terminal value under the matching diffusion.
The answer specifies the Brownian drivers but does not derive the final expectation explicitly.

Tags

Statistics Derivatives pricing

Full text

# Feynman Kac Terminal value problem two variables


# Feynman Kac Terminal value problem two variables












So, I need some help to move forward with this problem.

$$ \begin{cases} \frac{\partial F(t,x,y)}{\partial t}+\frac{1}{2}\frac{\partial^2 F(t,x,y)}{\partial x^2}+\frac{9}{2}\frac{\partial^2 F(t,x,y)}{\partial y^2}+\frac{\partial^2 F(t,x,y)}{\partial x \partial y}-F(t,x,y)=0 \ \ \ (t,x) \in[0,T) x R^2\\ F(T,x,y)=x^2y\\ \end{cases} $$

I have concluded that the C matrix defined as $\sigma*\sigma^T=\begin{matrix} 1 & 1\\ 1 & 3 \end{matrix}$

and that $\mu_{1}=\mu_{2}=0$

and that I should solve $$F(t,x,y)=E_{t,x,y}[e^{-(T-t)}x^2y]$$

but my problem is to figure out how $x$ and $y$ should look like. My attempt was $$dX(s)=dW_{1}(s)+dW_{2}(s)$$ $$dY(s)=3dW_{2}(s)+dW_{1}(s)$$ since they should be correlated given the problem, but I'm unsure if I have understood how to define $dX(s)$ and $dY(s)$ properly.

Well, if i preceed with $dX(s)$ and $dY(s)$ as defined above I get

$$F(t,x,y)=E_{t,x,y}[e^{-(T-t)}x^2y]=\\e^{-(T-t)}E_{t,x,y}\bigg[\bigg(x+\big(W_{1}(T)-W_{1}(t)\big)+\big(W_{2}(T)-W_{2}(t)\big)\bigg)^2\bigg(y+3\big(W_{2}(T)-W_{2}(t)\big)+\big(W_{1}(T)-W_{1}(t)\big)\bigg)\bigg]$$

which does not lead me to the correct answer. Anyone with some guiding for me?

## Answer by Raskolnikov (score 0, accepted)

https://quant.stackexchange.com/a/37088

Try with

\begin{align} dX(s)&=dW_{1}(s) \\ dY(s)&=dW_{1}(s)+\sqrt{2}dW_{2}(s) \end{align}

The covariance matrix of these differentials is $$\left(\begin{array}{cc} 1 & 1 \\ 1 & 3\end{array}\right) \; .$$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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

Solving a Two-Variable Feynman–Kac Terminal Value Problem

Summary

Key ideas

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