# Where can I find a modern general proof of the Feynman-Kac formula?

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 ?

## 1 Answer

Wikipedia's proof seems quite complete:

https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula

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