## Doctoral Theses

### Time-Space Harmonic Polynomials for Stochastic Processes.

2-28-1998

2-28-1999

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Goswami, Alok (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

The sequence of polynomials of a single variable known as the Hermite polynomialshala) = ), k21, (-1)* ha(z) =has many close links with the Normal distribution. Their association goes very doep, and extends to several connections bet ween the two-variable Hermite polynomialsHll, 2) = the(z/t), . k21.and the prime example of Gaussian processes, that is Brownian motion, as well. Much of this connection stems from what we term the time-space harmonic property of these polynomials for the Brownian motion process. An exact definition of this property follows later. A natural question that arises is, for stochastic processes in general, when do there exist two-variable polynomials that are time space harmonie for the process in question, and in case such polynomials exist, whether they carry algebraic and analytical properties analogous to those of the Hermite polynomials.We have been able to arrive at reasonably satisfactory answers which will be described in detail. Interestingly enough, the investigation has raised several new questions some of which have been answered here while .some others may well be the subject of further research.Throughout, we shall assume as given, on some probability space (, F,P), a process M = (M) indexed by a partially ordered time-set' (7,S) and taking values in some metric space E. We also fix notation for its natural filtration F = a< M, : St>. A real-valued function f on T E will be called time-space harmonic for M if {f (t,Mt):tT} is an {Fi}-martingale. Such functions clearly form a vector space V. For functions in V, their two arguments are referred to as the 'time' and 'space' variables, in that order. The object we shall be interested in studying in this work is the vector subspace of V consisting of only those functions in V which are polynomials in the two variables. We denote this subspace by P and call its elements time-space harmonic polynomials for M. It bears mention that in a majority of situations, T will be either (0,1,.) or (0, 00) and E, the real line R. In such cases, P consists merely of those functions in V which are polynomials in both arguments in the usual sense. Only a few situations not conforming to this setup will occur in our investigation. Foremost among these is the multivariate situation, when the time-set remains cither (0, 1, or (0, 00) and E represents some rth Euclidean space. One can then extend the results in a natural way to the vector-valued case, that is, as for example when E is a llilbert space.Let us now state the main problem more explicitly in the setting to occur most frequently, that is, when T is cither the discrete or the continuous time-set and E = R. As the previous discussion suggests, the subject of this work is the study of conditions for a certain degree of richness of the class P. The following definition makes this clearer. We shall call M polynomially harmonisable, or p-harmonisable in short, if for every positive integer k, P contains a polynomial of degree k in the 'space' variable. We wish to make it clear right away that this definition has no connection with the existing notions of harmonisability of a process, which refer to the representability of its covariance function, or equivalently, sample functions, as Fourier transformis in a certain sense (see (13), p. 474-476). The central question we investigate is, when is a stochastic process M p-harmonisablo, and if it is, how are the various properties of M reflected in a sequence of time-space harmonic polynomials?1.2 Chief examples knownThere are many known examples of processes possessing this property, the most familiar perhaps being Brownian motion, where, as indicated carlier, the two-variable Hermite poly- nomials (Hk) form such a sequence. The standard Poisson process N, is also p-harmonisable, with a sequence of time-space harmonic polynomials being given by the two-variable Charlier polynomialsC(1, 2) =where {}'s (see (11) are the Stirling numbers of the second kind. The Gamma process turns out to be yet another example of a p-harmonisable process.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843000

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