On a Multiple Stochastic Integral with Respect to a Strictly Semistable Random Measure

The concept of multiple stochastic integration with respect to Brownian motion was introduced by Wiener (1938). Ito (1951) gave a more general construction of multiple stochastic integrals with regard to Brownian motion. Later the study of multiple stochastic integrals with respect to non-Gaussian processes were considered by some authors (e.g., Lin (1981), Surgailis (1981), Engel (1982)). Multiple stochastic integrals have found their applications in areas such as statistics and quantum mechanics. Recently, several authors (e.g., Szulga and Woyczynski (1983), Krakowiak and Szulga (1985), Rosinski and Woyczynski (1986), and Surgailis (1985)), using different approaches, have constructed multiple stochastic integrals with respect to symmetric stable random measures. This dissertation is concerned with the development of the multiple stochastic integrals with respect to semistable random measures.

One of the above mentioned approaches used to construct the multiple stochastic integrals with respect to stable random measures is the Lebesgue-Dunford type construction. This approach reduces the problem of stochastic integration to the problem of integration with respect to a vector measure. Using this approach Krakowiak and Szulga (1985) developed multiple stochastic integrals of Banach valued functions with respect to symmetric and also nonsymmetric stable random measures. In this dissertation, using an approach similar to that of Krakowiak and Szulga (1985), we develop multiple stochastic integrals with respect to symmetric as well as with respect to (nonsymmetric) strictly semistable random measures with index of stability α ∈ (1, 2). Our methods, in the nonsymmetric case, yield results on multiple stochastic integrals relative to strictly stable random measure with index α ∈ (1, 2) considered in [10, 13].

The most crucial role in the development of the integrals here is played by the inequalities (2.29). In these inequalities we establish a comparison theorem between the moments of the integrals of certain simple functions relative to the strictly semistable random measure and the corresponding moments of integrals of these functions relative to symmetric stable random measure. Once these inequalities are established, the methods of construction of the integrals here are similar to those used by Krakowiak and Szulga in [10, 13] to develop the integrals relative to symmetric stable random measure.

In Chapter I, we collect the notation, definitions, and known results that are basic to this dissertation. In Chapter II, we develop necessary tools and prove the crucial inequalities mentioned above. In the first part of Chapter II, we prove a comparison theorem for tail probabilities of nonsymmetric semistable random measures. This uses a distributional property of a strictly semistable random variable. In Chapter III, we define the multiple stochastic integrals of certain Banach valued Borel measurable functions with respect to a strictly semistable random measure of index a. Then, we show that the class of Banach valued integrable functions relative to a semistable random measure of index α coincides with the class of Banach valued integrable functions relative to a symmetric stable random measure of index α.