To all interested parties,

I have written up the steps in this thread (most appropriate place) that I carried out in PDE domain mapping. Please feel free to give comment, pick holes, addenda etc. (BTW this note excludes numerics, that will be the next one).

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PDE Domain Mapping, Part I

The Continuous Problem

Draft 0.8

Summary

This is a description of an approach to mapping a second-order parabolic PDE (convection-diffusion) defined on a semi-infinite space to one that is defined on a unit cube in n dimension. The mapping is nonlinear in general and there are many candidates. The main advantage of using such a transformation is that no boundary conditions are needed (or even allowed) on the boundaries where the quadratic form of the transformed PDE is identically zero. These are precisely the boundaries that are difficult to model when we truncate the domain of integration and employ numerical boundary conditions.

This note excludes a discussion of the numerical solution of the transformed PDE. This is discussed in a forthcoming note.

The Current Situation

One of the major challenges when finding numerical solution of PDEs on semi-infinite regions using the finite difference method (FDM) is to decide how to replace these regions by some bounded regions and consequently determining which kinds of boundary conditions on their boundaries. Even when we have some inkling on how to carry out these activities, there are still a number of major challenges:

. The process for determining numerical boundaries is idiosyncratic and seems to work only in very specific one-dimensional problems

. It is unclear as to how and why numerical boundary conditions are to be defined

. Continuity and smoothness of the resulting solution is not clear; anecdotal evidence suggests that the solution is not smooth at corner points, thus affecting accuracy in the interior of the region of integration

In the light of these difficult challenges we embarked on another approach which can be summarized as ?postponing truncation for as long as possible?.

The Proposed Solution

The new approach seems to be applicable to n-factor PDEs but we reduce the scope to one and two-factor models in this section in order to keep the discussion focused.

The first step (Step 1 in future correspondence) is to find a mapping from a semi-infinite plane to the unit box (a ?box? in one dimension is the unit interval and it is the unit square in two dimensions). There are many possible transformations and it is not yet clear which one is optimal (in some sense) for a given PDE but it does depend on the coefficients of the PDE and the initial condition (this ?optimality? manifests itself when we solve the transformed PDE numerically). Some popular transforms are:

z = tanh(cS) (for the Heston-like model )

z = (1 - cS)/(1 + cS)

z = S/(1 + cS) (used for BS model)

z = exp(-cS) (used for CEV model)

In two dimension we have a variety of options such as z1 = tanh(c1 S1), z2 = tanh(c2 S2), where S1 and S2 are variables in ?user space US?, z1 and z2 are variables in ?transformed space TS? and finally c1 and and c2 are free parameters. Other scenarios are also possible.

The parameter c is fee at this moment. We discuss its usefulness later in this section.

The second step (Step 2) is to create a new PDE, not in the variables (S,t) but in the variables (z,t) where z is the transformed variable from Step 1. The main issues are:

Calculate the first and second derivative of the independent variable (aka u, C, P) with respect to

z in terms of the corresponding derivatives in S (high-school differentiation, but tedious)

Assemble the convection and diffusion of the new PDE

Map the initial condition function to the new coordinates

The third step (Step 3) now addresses the problem of boundary conditions. On non-characteristic boundary we take the boundary conditions corresponding to the original PDE. On the characteristic boundaries we address:

They are indeed characteristic boundaries

Calculate the Fichera function on the characteristic boundaries

Having done this we have now specified the full PDE problem. The Fichera approach leads to a Dirichlet, ODE or lower-order PDE on the boundary (this means, easily solvable).

The fourth step (Step 4) consists of determining ?hotspots? in both user and transformed space (US, TS). Basically, we are interested in a value at certain points in US and from these link these values to given values in TS. Let us take an example in one dimension. We wish to find the solution at S = 100 (for example) in US and we wish to associate this value with the value z = ? in TS. Let use assume that we use the transform z = tanh(c S). Then we can find the parameter c because both z and S are known in this case. This idea can be generalized to arrays of hot spots.

Some other Issues

. Maximum principle /energy inequalities for transformed PDE

. Choose the optimal transformation that is ?consistent? with the PDE?s initial condition

Daniel J. Duffy

November 2008