### Friday, July 28, 2017 (11:00 a.m. in Yost 306)

**Title: **Stochastic Integrals with respect to Tempered alpha-Stable Levy Process

**Student:** Yukun Song

**Advisor:** Wojbor Woyczynski (Professor, Case Western Reserve University)

**Abstract: **As we know, there are many equalities and inequalities for stochastic integrals. Some equalities and inequalities hold when the stochastic integrator has very nice characteristics, like Brownian motion. What will happen to the equalities or inequalities valid for the Brownian motion in the case of other stochastic process, as like proper tempered alpha-stable Levy process? A proper tempered alpha-stable Levy process combines both the alpha-stable and Gaussian trends. In a short time frame it is close to an alpha-stable process while in a long time frame it approximates a Brownian motion. So, we can find the keys that make analogs of these equalities and inequalities hold.

First I found two equalities and two inequalities that hold for the Brownian motion integral. There are two aspects to prove them. First of all, we consider when the integrand is predictable step process. Base on that it is a finite sum, we can get it from normal random variable inequalities. On the other hand, we need to extend the situation to where integrand is a general predictable process. It involves the problem whether what we deal with is integrable. Secondly, I will research if these equalities and inequalities hold for proper tempered alpha-stable Levy process. In this step, I will find the space of functions which are integrable for proper tempered alpha-stable Levy process at first. So that we can find the predictable process which is integrable for proper tempered alpha-stable Levy process then. And I will research if these equalities and inequalities still hold for integrals of predictable step integrands with respect to proper tempered alpha-stable Levy processes. At last, I will extend the predictable step condition to general predictable condition. Based on the research about proper tempered alpha-stable Levy process, there are some tools that make these equalities and inequalities hold. I will prove that a process X(t) which is a Levy process and martingale with EX(t)^2=t satisfies these equalities and inequalities.

The results are based on the the book by Kwapien and Woyczynski and paper by Rosinski.