Feynman-Kac Formula

Feynman-Kac formula connects the solution to a SDE to the solution of a PDE. For example, the Black-Scholes formula is an application of Feynman-Kac.

Let $X(t)$ satisfies the following SDE driven by standard Brownian Motion:

$\textnormal{d}X_u = \beta(u,X_u)\textnormal{d}u + \gamma(u,X_u\textnormal{d}W_u).$

Let $h(y)$ be a Borel measurable function, and $t \in [0,T]$ . Define

$\begin{aligned} g(t,x) := \mathbf{E} [&\int_t^Te^{-\int_t^TV(\tau,X_{\tau})\textnormal{d}\tau}f(r,X_r)\textnormal{d}r \\ &+e^{-\int_t^TV(\tau,X_{\tau})}h(X_T)|X_t = x ] \end{aligned}.$

Then, $g(t,x)$ satisfies PDE

$g_t + \beta g_x + \frac{1}{2}\gamma^2 g_{xx} - Vg + f = 0.$

Proof.

To get an idea of how this formula is proved, we start with a base case. The general case is proved by the same manner with a little more complicated calculation.

Suppose $V=f=0$ .

Let $\mathcal{F}(t)$ be the natural filtration associated with the standard Brownian Motion in the SDE. By the Markov property of the solution to SDE, we have $\forall s \in [0,T]$ ,

$\mathbf{E}[h(X_T)|\mathcal{F}(s)] = \mathbf{E}[h(X_T)|\sigma(X_s)] = g(s,X_s).$

Then, for $0<s<t<T$

$\begin{aligned} \mathbf{E}[g(t,X_t)|\mathcal{F}(s)] & = \mathbf{E}[ \mathbf{E}[h(X_T)| \mathcal{F}(t) ] |\mathcal{F}(s)] \\ &= \mathbf{E}[h(X_T)|\mathcal{F}(s)] = g(s,X_s) \end{aligned}$

Hence, $g(t,X_t)$ is a martingale.

Then, we apply Ito’s formula to $g(t,X_t)$ . Its drift term should be zero because it is a martingale. This leads to the Feynman-Kac PDE.

$\begin{aligned} \textnormal{d}g(t,X_t) & = g_t\textnormal{d}t + g_x \textnormal{d} x + \frac{1}{2}g_{xx} \textnormal{d} X \textnormal{d} X \\ & = [g_t+\beta g_x + \frac{1}{2}\gamma^2 g_{xx}] \textnormal{d} t + \gamma g_x \textnormal{d} W\end{aligned}.$

By setting the $\textnormal{d} t$ term equal to $0$ , we get the PDE:

$g_t+\beta g_x + \frac{1}{2}\gamma^2 g_{xx} = 0.$

For the general case, we only need to factor out the $\int_0^t$ integral and make the integral inside the expectation only contains $\int_0^T$ term. We will get $G(t,X_t)$ is a martingale with $G(t,X_t)$ defined as:

$G(t,x):= e^{-\int_0^tV(\tau,X_{\tau})\textnormal{d}\tau}(g(t,x)+\int_0^t e^{-\int_t^rV (\tau,X_{\tau})\textnormal{d}\tau } f(r,X_r) \textnormal{d}r)$

Similarly, by applying Ito’s formula to above martingale and setting the drift term equal to 0, we will get the result.

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