Feynman–Kac formula

The Feynman–Kac formula represents solutions of certain linear parabolic partial differential equations as expectations over stochastic processes, linking diffusion and probabilistic descriptions.

Itô's lemma: A stochastic calculus rule for differentiating functions of stochastic processes. Applying it to a function along a diffusion produces the differential terms in the formula.
Parabolic partial differential equation: A partial differential equation describing time evolution with diffusion-like smoothing. The classical formula applies to a class of linear equations of this type.
Black–Scholes equation: A parabolic partial differential equation for the value of a European option under the Black–Scholes model. Its solution can be expressed as a discounted expectation over risk-neutral asset paths.
Richard Feynman: An American theoretical physicist known for contributions to quantum mechanics and quantum electrodynamics. His path-integral ideas helped establish the formula's connection between trajectories and evolution equations.
Brownian motion: A continuous-time stochastic process with independent, normally distributed increments and continuous paths. It supplies the random paths whose weighted expectations solve diffusion equations.
Markov process: A stochastic process whose future, conditional on its present, is independent of its past. The process in the expectation is typically Markov, allowing its generator to define the equation.
Monte Carlo method: A computational method that estimates quantities using repeated random sampling. Path sampling can approximate the expectations that represent equation solutions.
Mark Kac: A Polish-American mathematician whose work connected probability theory with analysis and mathematical physics. His probabilistic treatment of the heat equation gave the result its mathematical foundation.
Infinitesimal generator: An operator describing the instantaneous evolution of a Markov process on functions. The generator becomes the spatial differential operator in the associated equation.
Initial value problem: A differential equation problem specified by the solution's value at an initial time. Its prescribed data appear as the terminal payoff in the probabilistic representation.

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