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Feynman–Kac formula
The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and 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
\( \frac{\partial u}{\partial t} + \mu(x,t) \frac{\partial u}{\partial x} + \frac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2} -V(x,t) u + f(x,t) = 0 \)
defined for all real x and t in the interval [0, T] , subject to the terminal condition
\( u(x,T)=\psi(x),
where \( \mu,\ \sigma,\ \psi, V \) are known functions, \( \ T \) is a parameter and \(u:\mathbb{R}\times[0,T]\to\mathbb{R} is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as an conditional expectation:
\(u(x,t) = E\left[ \int_t^T e^{- \int_t^s V(X_\tau)\, d\tau}f(X_s,s)ds + e^{- \int_t^T V(X_\tau)\, d\tau}\psi(X_T) | X_t=x \right] \)
where X is an Itō process driven by the equation
\(dX = \mu(X,t)\,dt + \sigma(X,t)\,dW, \)
with W(t) is a Wiener process (also called Brownian motion) and the initial condition for X(t) is X(0) = x.
Proof
NOTE: The proof presented below is essentially that of [1] where it is assumed that f(x, t)=0 .
Let u(x, t) be the solution to above PDE. Applying Itō's lemma to the process \(Y(s) = e^{- \int_t^s V(X_\tau)\, d\tau} u(X_s,s) \) one gets
\( dY = de^{- \int_t^s V(X_\tau)\, d\tau} u(X_s,s) + e^{- \int_t^s V(X_\tau)\, d\tau}\,du(X_s,s) +de^{- \int_t^s V(X_\tau)\, d\tau}du(X_s,s) \)
Since \(de^{- \int_t^s V(X_\tau)\, d\tau} =-V(X_s) e^{- \int_t^s V(X_\tau)\, d\tau} \,ds \) , the third term is o(dtdu) and can be dropped. Applying Itō's lemma once again to du(X_s,s), it follows that
\( dY=e^{- \int_t^s V(X_\tau)\, d\tau}\,\left(-V(X_s) u(X_s,s) +\mu(X_s,s)\frac{\partial u}{\partial X}+\frac{\partial u}{\partial s}+\frac{1}{2}\sigma^2(X_s,s)\frac{\partial^2 u}{\partial X^2}\right)\,ds
\;+e^{- \int_t^s V(X_\tau)\, d\tau}\sigma(X,s)\frac{\partial u}{\partial X}\,dW. \)
The first term contains, in parentheses, the above PDE and is therefore zero. What remains is
\( dY=e^{- \int_t^s V(X_\tau)\, d\tau}\sigma(X,s)\frac{\partial u}{\partial X}\,dW. \)
Integrating this equation from t to T , one concludes that
\( Y(T) - Y(t) = \int_t^T e^{- \int_t^s V(X_\tau)\, d\tau}\sigma(X,s)\frac{\partial u}{\partial X}\,dW. \)
Upon taking expectations, conditioned on \( X_t = x , and observing that the right side is an Itō integral, which has expectation zero, it follows that \( E[Y(T)| X_t=x] = E[Y(t)| X_t=x] = u(x,t). \) The desired result is obtained by observing that
\( E[Y(T)| X_t=x] = E[e^{- \int_t^T V(X_\tau)\, d\tau} u(X_T,T)| X_t=x] = E[e^{- \int_t^T V(X_\tau)\, d\tau} \psi(X_T))| X_t=x] \)
Remarks
The expectation formula above is also valid for N-dimensional Itô diffusions. The corresponding PDE for \( u:\mathbb{R}^N\times[0,T]\to\mathbb{R} \) becomes (see H. Pham book below):
\( \frac{\partial u}{\partial t} + \sum_{i=1}^N \mu_i(x,t)\frac{\partial u}{\partial x_i} + \frac{1}{2} \sum_{i=1}^N\sum_{j=1}^N\gamma_{ij}(x,t) \frac{\partial^2 u}{\partial x_i x_j} -r(x,t) u = f(x,t), \)
where,
\( \gamma_{ij}(x,t) = \sum_{k=1}^N\sigma_{ik}(x,t)\sigma_{jk}(x,t), \)
i.e. \(\gamma =\sigma\,\sigma^\prime \) , where \( \sigma^\prime \) denotes the transpose matrix of \sigma).
This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.
When originally published by Kac in 1949,[2] 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
\( e^{-\int_0^t V(x(\tau))\, d\tau} \)
in the case where \( \ x(\tau) is some realization of a diffusion process starting at \( \ x(0) = 0 \) . 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 \(\ u V(x) \geq 0, \)
\( E\left( e^{- u \int_0^t V(x(\tau))\, d\tau} \right) = \int_{-\infty}^{\infty} w(x,t)\, dx \)
where \(\ w(x,0) = \delta(x) \) and
\( \frac{\partial w}{\partial t} = \frac{1}{2} \frac{\partial^2 w}{\partial x^2} - u V(x) w. \)
The Feynman–Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form. If
\( I = \int f(x(0)) e^{-u\int_0^t V(x(t))\, dt} g(x(t))\, Dx \)
where the integral is taken over all random walks, then
\( I = \int w(x,t) g(x)\, dx \)
where \(\ w(x,t) \) is a solution to the parabolic partial differential equation
\( \frac{\partial w}{\partial t} = \frac{1}{2} \frac{\partial^2 w}{\partial x^2} - u V(x) w \)
with initial condition \(\ w(x,0) = f(x). \)
See also
Itō's lemma
Kunita–Watanabe theorem
Girsanov theorem
Kolmogorov forward equation (also known as Fokker–Planck equation)
References
Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press.
Pham, Huyên (2009). Continuous-time stochastic control and optimisation with financial applications. Springer-Verlag.
^ http://www.math.nyu.edu/faculty/kohn/pde_finance.html
^ Kac, Mark (1949). "On Distributions of Certain Wiener Functionals". Transactions of the American Mathematical Society 65 (1): 1–13. doi:10.2307/1990512. JSTOR 1990512.
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