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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