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Path Integral Methods for Parabolic Partial Differential Equations with Examples from Computational Finance

2. Some Background: Solutions to Second-order Parabolic Partial Differential Equations as Path Integrals

This section outlines some well-known principles and relations on which the method developed in this paper is based. A complete, detailed, and rigorous description of these principles can be found in the ground-breaking work of Stroock and Varadhan [2] and also in [3, 4]. The computational aspects of the methods developed in these works have not been explored fully, and to a large extent the present work is a modest attempt to fill this gap.

The Feynman integral of a real function defined on the space , which consists of continuous sample paths with the property , can be expressed as the following formal expression

which is understood as the limit of the following sequence of multiple integrals

where k1, and is the result of the evaluation of at the piecewise linear sample path from the space , whose values at the points , , are given by the vector . It would not be an exaggeration to say that the convergence of the sequence in (2.2)--and therefore the fact that the integral in (2.1) is meaningful--is a miracle, without which very little of what we currently know about the universe would have taken place. Indeed, "miracle" is the only term that can explain the fact that, as was pointed out in [4], the infinities that arise under the exponent in (2.1) not only offset the infinities that arise under the product sign, but also offset them in such a way that the result is a perfectly well defined and highly nontrivial probability measure on the space , known as the Wiener measure. Not only is the integral in (2.1) the perfectly meaningful "expected value of ," but when is given by , , for some fixed and some reasonably behaved function , then, as a function of and , this expected value coincides with the solution of the standard heat equation

with boundary condition . In fact, this statement admits a generalization: if is replaced by a general elliptic differential operator of the form

assuming that the coefficients and are reasonably behaved, the solution of the boundary value problem (2.3) can still be expressed as a path integral similar to the one in (2.1). The only difference is that in this case the path integral is understood as the limit of the sequence

where is the transition probability density of the Markov process , , determined by the stochastic equation

in which , , is a given Brownian motion process. In other words, is characterized by the fact that for any test function one has

We will say that the process , , is a sample path realization of the Markovian semigroup , , being the differential operator from (2.4), and will refer to the transition density as the integral kernel of the semigroup , . This integral kernel is also known as the fundamental solution of the equation , that is, a solution which satisfies the initial condition , where stands for the usual Dirac delta function with mass concentrated at the point . This means that, formally, can be expressed as .

A thorough exploration of this generalization of the Feynman integral can be found in the seminal work of Stroock and Varadhan [2]. The terminology is somewhat different from the one used here, that is, instead of "path integrals," it comes down to a study of probability distributions on the pathspace associated with equations similar to the one in (2.6). Such a framework is much more rigorous and powerful. However, our goal is to develop a numerical procedure that computes the solution of the PDE in (2.3) as an expected value, and in this respect Feynman's formalism proves to be quite useful.