>[!quote] In a Nutshell >Theorem that links [[Stochastic Processes|stochastic processes]] to [[PDEs - Partial Differential Equations|parabolic partial differential equations]]. This can be used to ... >- solve certain PDEs by simulating stochastic processes, see e.g. [[MPPI - Model Predictive Path Integral Control Framework]] >- [[Expectations|expectations]] of certain random processes can be computed by deterministic methods --- We want to find a solution $u:\mathbb{R} \times [0,T] \to \mathbb{R}$ to the partial differential equation $ \frac{\partial}{\partial t}u(x,t) + \mu(x,t) \frac{\partial}{\partial x}u(x,t) + \tfrac{1}{2} \sigma^2(x,t) \frac{\partial^2}{\partial x^2}u(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 [[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|functions]]. >[!success] Theorem >Then the Feynman–Kac formula expresses $u(x,t)$ as a [[Expectations|conditional expectation]] under the [[Measure|probability measure]] $Q$ via $ u(x,t) = \mathbb{E}_{x \sim Q} \left[g_\tau(t,T) \, \psi(X_T) + \int_t^T g_s(t,\tau) \, f(X_\tau,\tau) \, d\tau \,\Bigg|\, X_t=x \right] $where $X$ is an [[Itô Calculus and Integral|Itô process]] satisfying $ dX_t = \mu(X_t,t) \, dt + \sigma(X_t,t) \, dW^Q_t.$$g_\tau,g_s$ are functions defined as $g_\kappa(a, b) = \exp\left(-\int_a^b V(X_\kappa, \kappa) \, d\kappa\right),\quad \kappa \in \{\tau,s\}$and $W_t^{Q}$ a [[Wiener Process or Brownian Motion|Wiener process]] under $Q$. >[!brainwaves] Intuition >In order to evaluate the solution, we can e.g. use [[Monte Carlo Methods|Monte Carlo]] methods to evaluate the [[Expectations|expectation]]. To generate samples for that, we use the given [[Itô Diffusion|Itô process]]. > >The "under $Qquot; simply means that the Gaussian used to sample $dW_t$ is always truly zero mean (and [[Covariance and Variance|variance]] $d t$). It is a technicality required when e.g. transforming the measure.