Operator Methods and Constructive Probability Theory

I was led to operator methods while looking at the classification problem for analytically solvable pricing models. An operative definition of analytic solvability can be given in various ways. A stochastic differential equation can be called analytically solvable if the transition probability kernel can be expressed in terms of hypergeometric function. It is also interesting to look at the generating function of a stochastic integral and ask whether that can be expressed by means of hypergeometric functions. I came up with classification results in collaboration with Alexey Kuznetsov and Stephan Lawi around the years 2001-2003. Much more work was done in this area since then. In June 2009 there will be the first conference entirely dedicated to this topic at Leicester University, see the announcement here.

Classifying analytically solvable models taught me two things. Firstly, I learned that functional analysis methods are a very powerful tool in probability theory and it is quite useful to look at stochastic processes in terms of their Markov generator. Secondly, when I attempted to use some of the new models we had discovered, I learned that although one might as well have 14 adjustable paramters, unless these parameters appear in the right locations in the equations the dynamics may not necessarily have any resemblance with econometric evidence. It seemed natural at that point to move on and tackle semi-parametric model specifications which could be made economically realistic by using operator methods.

My first attempts with semi-parametric models were carried out by using numerical spectral analysis. The idea was to use LAPACK routines to numerically diagonalize Markov generators. However, there I stumbled on the problem of pseudospectrum that make it difficult to diagonalize numerically operators which are not normal (and most interesting Markov generators are not). I then overcame this difficulty using fast exponentation, an algorithm which basically boils down to repeated squaring of a kernel by means of a matrix multiplcation routine such as dgemm or sgemm.

Fast exponentiation turned out to be a very robust algorithm. Not only normality restrictions are irrelevant, but the algorithm is empirically robust even under single precision arithmetics. From the engineering standpoint, this is a remarkable property, especially since GPU coprocessors perform best in single precision. I was very surprised when I first witnessed this stability and wanted to understand it mathematically. So I wrote a paper deriving convergence estimates for probability kernels and one on convergence estimates for joint kernels between a stochastic process and an adapted stochastic integral. I considered stochastic differential equations with a drift and a volatility that were not necessarily smooth, but instead just Hoelder continuous. I consider processes on grids and then let the grid spacing tend to zero. In the continuous limit I recover results established in the 1960s by Farbes, Strook and Varadhan and others. However, the proof I provide is entirely constructive and comes with detailed convergence estimates.

One of the side products of these results is that they are examples of an entirely constructive theory of stochastic processes whereby non-trivial existence and smoothness results for stochastic differential equations and stochastic integrals can be obtained out of entirely finatary constructs. In particular, I am not using Kolmogorov's axiom of sigma-additivity. There's to add that the results I established only apply to one dimensional stochastic differential equations and stochastic integrals. Extensions to multi-dimensional cases will require new ideas.