1 Feynman-Kac Formula
References: Wikipedia.
The Feynman-Kac Formula establishes a link between [[parabolic-partial-differential-equations]] and [[knowledge-mathematics-analysis-probability-theory-stochastic-processes-stochastic-processes|Stochastic Processes]]. It offers a method of solving certain PDEs by simulating random paths of a stochastic process. Conversely, an important class of expectations of random processes can be computed by deterministic methods.
Consider the PDE \[u_{t}(x,t)+\mu(x,t)u_{x}(x,t)+\frac{1}{2}\sigma^2(x,t)u_{xx}(x,t)-V(x,t)u(x,t)+f(x,t)=0,\]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 \(u:\mathbb{R}\times[0,T]\to \mathbb{R}\) is the unknown. The the Feyman-Kac formula expresses \(u(x,t)\) as a conditional expectation under probability measure \(\mathbb{Q}\) \[u(x,t)=\mathbb{E}^\mathbb{Q}\left[ g_{\tau}(t,T)\psi(X_{T})+\int_{t}^{T}{g_{s}(t,\tau)f(X_{\tau, \tau})}~d{\tau}\middle| X_{t}=x \right], \]where \(X\) is an Itô Process satisfying \[dX_{t}=\mu(X_{t},t)dt+\sigma(X_{t},t)dW_{t}^\mathbb{Q},\] \(g_{\tau}.g_{s}\) are functions defined by \[g_{\kappa}(a,b)=\exp \left( -\int_{a}^{b}{V(X_{\kappa}, \kappa)}~d{\kappa} \right) \] for \(\kappa=\tau,s\) and \(W_{t}^\mathbb{Q}\) is Brownian Motion under \(\mathbb{Q}\).