Consider the partial differential equation $$ \frac{\partial u}{\partial t}(x, t)+\mu(x, t) \frac{\partial u}{\partial x}(x, t)+\frac{1}{2} \sigma^{2}(x, t) \frac{\partial^{2} u}{\partial x^{2}}(x, t)-V(x, t) u(x, t)+f(x, t)=0 $$ defined for all $x \in \mathbb{R}$ and $t \in[0, T]$, subject to the terminal condition $$ u(x, T)=\psi(x) $$ where $\mu, \sigma, \psi, V, f$ are known functions, $T$ is a parameter and $u: \mathbb{R} \times[0, T] \rightarrow \mathbb{R}$ is the unknown. Then the Feynman-Kac formula tells us that the solution can be written as a conditional expectation $$ u(x, t)=E^{Q}\left[\int_{t}^{T} e^{-\int_{t}^{r} V\left(X_{\tau}, \tau\right) d \tau} f\left(X_{r}, r\right) d r+e^{-\int_{t}^{T} V\left(X_{\tau}, \tau\right) d \tau} \psi\left(X_{T}\right) \mid X_{t}=x\right] $$ under the probability measure $Q$ such that $X$ is an Itô process driven by the equation $d X=\mu(X, t) d t+\sigma(X, t) d W^{Q}$ where $W^{Q}(t)$ is a Wiener process (also called Brownian motion) under $Q$, and the initial condition for $X(t)$ is $X(t)=x$.

Where can I find a complete (no steps left for the reader...) and modern (notation-wise) proof of that result ?


Wikipedia's proof seems quite complete:


Alternatively, there is a very good proof in Oksendal's book "Stochastic Differential Equation," theorem 8.2.1 at page 145.


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.