Volume 9, Issue 2
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Path Integral Methods for Parabolic Partial Differential Equations with Examples from Computational Finance
5. Approximation of the Fundamental Solution by Way of a Discrete Dynamic Programming Procedure
In general, there are many ways in which the solution of a given SDE can be approximated by a discrete time series (see  for a thorough exploration and a comprehensive list of references on this topic). Unfortunately, none of the available approximation schemes was developed for a purpose similar to ours, so we will develop one which actually is. Another extremely interesting approximation method, whose computational aspects are yet to be explored, can be found in .
To begin with, notice that the stochastic equation in (2.6) can be written in integral form as
One possible approximation of this equation can be obtained by replacing the coefficients and by constants, which gives the following SDE
This approximation, which we will call the -order approximation, is so crude that the SDE it leads to admits a trivial solution:
Intuitively, it should be clear that, when is sufficiently small, the distribution of the random variable (r.v.) is not very far from that of the r.v. . If we were to implement the plan outlined in Section 3 with this level of approximation of the Markovian semigroup associated with the equation , the recursive procedure in (3.1) would come down to
Assuming that is the result of polynomial interpolation, the above integral is certainly computable for every choice of : in fact, it only has to be computed for , where stands for the set of interpolation nodes. One can show that as and as the number of interpolation nodes increases to , converges to the actual solution uniformly in the interpolation region. However, this convergence is known to be rather slow. The -order approximation of SDEs is as old as the theory of SDEs itself and can be traced back to the original works of Itô--in fact, this approximation is crucial for his construction of the stochastic integral. One may say that the -order approximation is "optimal" if all that one needs to do is show the existence of solutions to SDEs and derive various properties of the associated probability distributions, in which case the speed of the convergence is not an issue. In our case, however, it is and one can do considerably better by using the -order Taylor approximation of the coefficients and . Our plan, then, is to replace the stochastic equation in (5.1) with the following SDE
which admits this explicit solution
In spite of the "explicit" form of this solution, the density of the r.v. is still difficult to compute. Therefore, we need to develop yet another level of approximation. In order to approximate the random variable in the right side of (5.2) with something computable, notice that if the integrand in the first and the third integral is replaced by the constant 1 this will produce an error of order (a small of ). The approximation of the second integral is somewhat more involved. A straightforward application of the Itô formula yields
which shows that
Thus, (5.2) can be rewritten as
Of course, this identity holds only if , but this is the only truly interesting case, since implies that
The stochastic integral in this expression is a Gaussian r.v. of variance . Thus, we will approximate expressions of the form with
In particular, the recursive procedure in (3.1) will be replaced with
in the case of a terminal value problem, and with
in the case of a free boundary value problem. In Section 6 we will show that the computation in Mathematica of the last two integrals for a fixed is completely straightforward when the object is the result of polynomial interpolation.
It is not hard to check that in the special case where and one has
which means that, with this choice for and , the solution to the stochastic equation in (5.1) is simply .
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