I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the following terminal value problem:

\begin{align} & F_t+\frac{1}{2}σ^2x^2F_{xx}=1\\ & F(x,T)=(\ln(x))^4,\ x>0 \end{align}

How would I compute $F(x,t)$ in closed form, given the closed form of the right hand side $(\ln(x))^4$ using Feynman-Kac?

Thanks in advance,



The idea is to choose a stochastic process of the form $$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t $$ and consider the process $Y_t=F(t,X_t)$. Applying Ito to $Y_t$ gives $$ dY_t=\left(F_t+\mu(t,X_t)F_x(t,X_t)+\frac12\sigma(t,X_t)^2F_{xx}(t,X_t)\right)dt+\sigma(t,X_t)F_x(t,X_t)dWt $$

And then to choose the coefficients $\mu$ and $\sigma$ in such a way to have $Y_t$ martingale. In our case if we choose $\mu(t,x)=0$ and $\sigma(t,x)=\sigma x$ we will have $$dY_t=\sigma(t,X_t)F_x(t,X_t)dWt$$ under some additional hypothesis of polynomial growth of $F$ we can conclude that $Y_t$ is a martingale and then that $Y_t = E_t[Y_T]$ which we can rewrite $$F(t,x)=E[F(T,X_T)|X_t=x)$$ now solving the SDE $dX_t=\sigma X_t dW_t$ gives $$X_T=X_t \exp(\sigma(W_T-W_t)-\frac12 \sigma^2(T-t))$$

and hence in law we have $$X_T|X_t \sim x \exp(\sigma\sqrt{T-T}Z-\frac12 \sigma^2(T-t))$$ where $Z$ is a standard Gaussian, hence $$F(t,x)= E\left[ \left(\ln(x)+\sigma\sqrt{T-T}Z-\frac12 \sigma^2(T-t)\right)^4\right] $$

this amounts of calculating the four first moments of a standard Gaussian.

Hope this helps


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.