Feynman-Kac Theorem Theorem7.2Feynman-Kac formulaThe solution to βuβt+Luβru+f=0\frac{\partial u}{\partial t} + Lu - ru + f = 0βtβuβ+Luβru+f=0 with terminal condition u(T,x)=g(x)u(T,x) = g(x)u(T,x)=g(x), where LLL is the generator of the SDE dXt=b(Xt)dt+Ο(Xt)dBtdX_t = b(X_t) dt + \sigma(X_t) dB_tdXtβ=b(Xtβ)dt+Ο(Xtβ)dBtβ, is:u(t,x)=E[eββ«tTr(Xs)dsg(XT)+β«tTeββ«tsr(Xu)duf(Xs)dsβ£Xt=x].u(t,x) = \mathbb{E}\left[e^{-\int_t^T r(X_s) ds} g(X_T) + \int_t^T e^{-\int_t^s r(X_u) du} f(X_s) ds \mid X_t = x\right].u(t,x)=E[eββ«tTβr(Xsβ)dsg(XTβ)+β«tTβeββ«tsβr(Xuβ)duf(Xsβ)dsβ£Xtβ=x]. This formula is fundamental to mathematical finance (Black-Scholes pricing) and quantum mechanics (Feynman path integrals).
Feynman-Kac Theorem Theorem7.2Feynman-Kac formulaThe solution to βuβt+Luβru+f=0\frac{\partial u}{\partial t} + Lu - ru + f = 0βtβuβ+Luβru+f=0 with terminal condition u(T,x)=g(x)u(T,x) = g(x)u(T,x)=g(x), where LLL is the generator of the SDE dXt=b(Xt)dt+Ο(Xt)dBtdX_t = b(X_t) dt + \sigma(X_t) dB_tdXtβ=b(Xtβ)dt+Ο(Xtβ)dBtβ, is:u(t,x)=E[eββ«tTr(Xs)dsg(XT)+β«tTeββ«tsr(Xu)duf(Xs)dsβ£Xt=x].u(t,x) = \mathbb{E}\left[e^{-\int_t^T r(X_s) ds} g(X_T) + \int_t^T e^{-\int_t^s r(X_u) du} f(X_s) ds \mid X_t = x\right].u(t,x)=E[eββ«tTβr(Xsβ)dsg(XTβ)+β«tTβeββ«tsβr(Xuβ)duf(Xsβ)dsβ£Xtβ=x]. This formula is fundamental to mathematical finance (Black-Scholes pricing) and quantum mechanics (Feynman path integrals).