Stochastic Processes and Orthogonal Polynomials

The main focus of this book is the relationship between orthogonal polynomials and stochastic processes. In this chapter we review the relevant background of orthogonal polynomials. We start with some preliminaries and introduce the concept of an orthogonal polynomial. After classifying the so-called classical orthogonal polynomials, we describe the Askey scheme. This scheme contains the most important orthogonal polynomials and stresses the limit relations between them.

One important type of stochastic process is a Markov process, a stochastic process that has a limited form of “historical” dependency. To precisely define this dependency, let \(\{ {X_t},t \in \mathcal{T}\}\) be a stochastic process defined on the parameter set \(\mathcal{T}\). We think of \(\mathcal{T} \subset {\text{[0,}}\infty {\text{)}}\) in terms of time, and the values that X _t can assume are called the states which are elements of a state space S ⊂ ℝ. A stochastic process is called a Markov process if it satisfies for any value of t0, t₁> 0. To interpret (2.1), we think of t_o as being the present time. Equation (2.1) states that the evolution of a Markov process at a future time, conditioned on its present and past values, depends only on its present value. Expressed differently, the present value of X _t0 contains all the information about the past evolution of the process that is needed to determine the future distribution of the process. The condition (2.1) that defines a Markov process is sometimes termed the Markov property.

The study of the time-dependent behavior of birth and death processes (BDP) involves many intricate and interesting orthogonal polynomials, such as Charlier, Meixner, Laguerre, Krawtchouk, and other polynomials from the Askey scheme; other famous orthogonal polynomials not in the Askey scheme could appear also, e.g., orthogonal polynomials related to the Roger-Ramanujan continued fraction [88]. In fact, the three-term recurrence relation lies at the heart of continued fractions, orthogonal polynomials, and birth and death processes. For birth and death processes with complicated birth and death rates, for example, when rates are state dependent or nonlinear, it is almost impossible to find closed form solutions of the transition functions. Due to the difficulties involved in analytical methods, it is pertinent to develop other techniques to gain insight into the behavior of the various system characteristics such as system size probabilities, expected system size, etc. The Karlin and McGregor representation [59], [74] of the transition probabilities, which uses a system of orthogonal polynomials satisfying a three-term recurrence relation involving the birth and death rates, is very useful in understanding the asymptotic behavior of the birth and death process. In fact these polynomials appear in some important distributions, namely, the (doubly) limiting conditional distributions. Also these polynomials play a fundamental role in the study of exponential ergodicity (see, e.g., [33] through [36]).

Lévy processes appear in many areas, such as in models for queues, insurance risks, and more recently in mathematical finance Historically, the first research goes back to the late 1920s with the study of infinitely divisible distributions. In mathematical finance an important role is played by martingales. They are, for example, used with the interpretation of a risk-neutral market. The famous option-pricing model of Black and Scholes uses special martingales, which are related to a generating function of Hermite polynomials. In this chapter we give a large class of new martingales based on Sheffer polynomials for some important Lévy processes.

In this chapter we study orthogonal polynomials in the theory of stochastic integration. Some orthogonal polynomials in stochastic theory will play the role of ordinary monomials in deterministic theory. A consequence is that related polynomial transformations of stochastic processes involved will have very simple chaotic representations. In a different context, the orthogonalization of martingales, the coefficients of some other orthogonal polynomials will play an important role. We start with a reference to deterministic integration and then search for stochastic counterparts. We look at integration with respect to Brownian motion, the compensated Poisson process, and the binomial process. Next we develop a chaotic and predictable representation theory for general Lévy processes satisfying some weak condition on its Lévy measure. It is in this representation theory that we need the concept of strongly orthogonal martingales and orthogonal polynomials come into play. Examples include the Gamma process, where again the Laguerre polynomials turn up, the Pascal process, with again the Meixner polynomials, and the Meixner process, with as expected the Meixner-Pollaczek polynomials. When we look at combinations of pure-jump Lévy processes and Brownian motion, an inner product with some additional weight in zero plays a major role. We give an example with the Laguerre-type polynomials introduced by L. Littlejohn.

Stein’s method provides a way of finding approximations to the distribution of a random variable, which at the same time gives estimates of the approximation error involved. The strengths of the method are that it can be applied in many circumstances in which dependence plays a part. A key tool in Stein’s theory is the generator method developed by Barbour [10]. In this chapter we show how orthogonal polynomials appear in this context in a natural way. They are used in spectral representations of the transition probabilities of Barbour’s Markov processes. A good introduction to Stein’s approximation method can be found in [95].