Applying Itō’s lemma to the unknown function <math>\ f</math> one gets

<math>df=\left(\mu(x,t)\frac{\partial f}{\partial x}+\frac{\partial f}{\partial t}+\frac{1}{2}\sigma^2(x,t)\frac{\partial^2 f}{\partial x^2}\right)\,dt+\sigma(x,t)\frac{\partial f}{\partial x}\,dW.</math>

The first term in parentheses is the above PDE and is zero by hypothesis. Integrating both sides one gets

<math>\int_t^T df=f(X_T,T)-f(x,t)=\int_t^T\sigma(x,t)\frac{\partial f}{\partial x}\,dW.</math>

Reorganising and taking the expectation of both sides:

<math>f(x,t)=\textrm{E}\left[f(X_T,T)\right]-\textrm{E}\left[\int_t^T\sigma(x,t)\frac{\partial f}{\partial x}\,dW\right].</math>

Since the expectation of an Itō integral with respect to a Wiener process <math>\ W</math> is zero, one gets the desired result:

<math>f(x,t)=\textrm{E}\left[f(X_T,T)\right]=\textrm{E}\left[\psi(X_T)\right]=\textrm{E}\left[\psi(X_T)|X_t=x\right].</math>

When originally published by Kac in 1949^[1], the Feynman-Kac formula was presented as a formula for determining the distribution of certain Wiener functionals. Suppose we wish to find the expected value of the function

<math> e^{-\int_0^t V(x(\tau))\, d\tau} </math>

in the case where <math>\ x(\tau)</math> is some realization of a diffusion process starting at <math>\ x(0) = 0</math>. The Feynman-Kac formula says that this expectation is equivalent to the integral of a solution to a diffusion equation. Specifically, under the conditions that <math>\ u V(x) \geq 0</math>,

<math> E\left( e^{- u \int_0^t V(x(\tau))\, d\tau} \right) = \int_{-\infty}^{\infty} w(x,t)\, dx </math>

where <math>\ w(x,0) = \delta(x)</math> and

<math>

\frac{\partial w}{\partial t} = \frac{1}{2} \frac{\partial^2 w}{\partial x^2} – u V(x) w. </math>

The Feynman-Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form. If

<math> I = \int f(x(0)) e^{-u\int_0^t V(x(t))\, dt} g(x(t))\, Dx </math>

where the integral is taken over all random walks, then

<math> I = \int w(x,t) g(x)\, dx </math>

where <math>\ w(x,t)</math> is a solution to the parabolic partial differential equation

<math> \frac{\partial w}{\partial t} = \frac{1}{2} \frac{\partial^2 w}{\partial x^2} – u V(x) w </math>

with initial condition <math>\ w(x,0) = f(x)</math>.

• Itō’s lemma
• Kunita-Watanabe theorem
• Girsanov theorem
• Kolmogorov forward equation (also known as Fokker-Planck equation)

• Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press.

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