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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 ?

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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.

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