Stochastic Integrals and Random Sums of Power Contractions in Systemics

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.

In this chapter we give a brief overview of the basic mathematical concepts used for pricing financial products. Modern pricing theory is mainly based on describing the ups and downs of market prices via stochastic processes. Therefore, we start with the definition and characteristics of stochastic processes in Section 2.1. Usually the price-changes of the most basic financial instruments are modelled in terms of a so-called stochastic differential equation (SDE). The price of a financial derivative is considered to be a function of the basic financial instruments’ prices. One of the central tools in stochastic calculus which can be applied to determine the stochastic differential equation for the prices of these derivatives, Itô’s lemma, is described in Section 2.4. This concept is closely related to the stochastic integral, which is defined in Section 2.3 to be the mean square limit of some random sums, carefully put together so that the resulting limit is a (local) martingale. And in fact, martingales are one of the most important elements needed for the evaluation of financial instruments. They are defined in Section 2.5 and are used to describe the prices of financial derivatives in terms of conditional expectations. The Feynman-Kac formula provides a partial differential equation (PDE) that corresponds to such conditional expectations as described in Section 2.6. The advantage of dealing with partial differential equations is that they can be solved numerically. Also, PDE methods come naturally to applied mathematicians and physicists. This approach led Black and Scholes [BS73] to their famous equation for pricing European options. Their model will be discussed in Section 3.5. By contrast, martingales and Itô (stochastic) calculus come naturally to probabilists, and this approach led Merton [Mer74] to his option pricing formula.KeywordsStochastic ProcessStochastic Differential EquationConditional ExpectationWiener ProcessLocal MartingaleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A comprehensive treatment of stochastic systems beginning with the foundations of probability and ending with stochastic optimal control. The book divides into three interrelated topics. First, the concepts of probability theory, random variables and stochastic processes are presented, which leads easily to expectation, conditional expectation, and discrete time estimation and the Kalman filter. With this background, stochastic calculus and continuous-time estimation are introduced. Finally, dynamic programming for both discrete-time and continuous-time systems leads to the solution of optimal stochastic control problems resulting in controllers with significant practical application. This book will be valuable to first year graduate students studying systems and control, as well as professionals in this field.

It is known the strong practical support of the consideration of fundamental probabilistic concepts as structural factors for the implementations of stochastic modeling operations. It is widely acknowledged that random sums, stochastic integrals, and stochastic models are probabilistic concepts of exceptional significance. The paper focuses on the development of stochastic models by establishing interconnections between the very important probabilistic concepts of random sums and stochastic integrals. In addition, the paper demonstrates how such stochastic models can be implemented to incorporate the crucial concept of proactivity in decision making related to risk treatment operations. Clearly, evidence is presented regarding the effectiveness of making proactive decisions in facilitating the design and implementation of risk treatment.

In this dissertation, we focus mainly on the further study of the new stochastic integral introduced by Ayed and Kuo in 2008. Several properties of this new stochastic integral are obtained. We first introduce the concept of near-martingale for non-adapted stochastic processes. This concept is a generalization of the martingale property for adapted stochastic processes in the It\^o theory. We prove a special case of It\^o isometry for the stochastic integral of certain instantly independent processes. We obtain some formulas for expressing a new stochastic integral in terms of It\^o integrals and Riemann integrals. Several generalized versions of It\^o's formula for the new stochastic integral obtained by Ayed and Kuo are given. We also provide some examples to illustrate the ideas.

Stochastic Integrals and Abelian Processes

We consider a process Xt1 with independent increments without positive jumps in the state space (−∞; +∞) VarXt1=+∞. For a stopped process in the space E0 we construct its continuation in E0 U {0}.

The celebrated Ito theory of stochastic integration deals with stochastic integrals of adapted stochastic processes. The Ito formula and Girsanov theorem in this theory are fundamental results which are used in many applied fields, in particular, the finance and the stock markets, e.g. the Black-Scholes model. In chapter 1 we will briefly review the Ito theory. In recent years, there have been several extension of the Ito integral to stochastic integrals of non-adapted stochastic processes. In this dissertation we will study an extension initiated by Ayed and Kuo in 2008. In Chapter 2 we review this new stochastic integral and some results. In chapter 3, we prove the Ito formula for the Ayed-Kuo integral. In chapter 4, we prove the Girsanov theorem for this new stochastic integral. In chapter 5, we present an application of our results.

K. Ito invented his famous stochastic calculus on Brownian motion in the 1940s. In the 1960s and 1970s, the “Strasbourg school,” headed by P.A. Meyer, developed a modern theory of martingales, the general theory of stochastic processes, and stochastic calculus on semimartingales. It turned out soon that semimartingales constitute the largest class of right continuous adapted integrators with respect to which stochastic integrals of simple predictable integrands satisfy the theorem of dominated convergence in probability. Stochastic calculus on semimartingales not only became an important tool for modern probability theory and stochastic processes but also has broad applications to many branches of mathematics, physics, engineering, and mathematical finance.

Fractional Brownian Motions, Fractional Noises and Applications

In this thesis, stochastic processes closely related to a prominent process called Brownian motion is introduced. The processes are called local Brownian motions. Two rich families are constructed, which can be used to model stochastic dynamical systems in such fields as finance, biology, and physics. Stochastic calculus with respect to local Brownian motion is developed, and strong solutions are presented for some stochastic differential equations driven by the noise generated by local Brownian motions.

Part 1 Stochastic processes: generated theta-algebras stochastic processes stopping times convergence in Lp and uniform integrability. Part 2 Martingales: martingale, submartingale and supermartingale fundamental submartingale inequalities convergence of submartingales uniformly integrable submartingales regularity of sample functions of submartingales increasing processes. Part 3 Stochastic integrals: L2-martingales and quadratic variation processes stochastic integrals with respect to martingales Ft-Brownian motions local martingales and extensions of the stochastic integral Ito's formula Ito's stochastic calculus. Part 4 Stochastic differential equations: the space of continuous functions on R++ definition and function space representation of solutions existence and uniqueness of solutions strong solutions.

Recently, a new concept for some stochastic process called fractional G-Brownian motion (fGBm) was developed, which generalizes the concepts of standard Brownian motion (Bm), fractional Brownian motion (fBm) and G-Brownian motion (GBm) under the framework of sublinear expectation. The fGBm can exhibit long-range dependence and feature the volatility uncertainty of financial markets simultaneously. Thus it can be a better alternative stochastic process in the financial applications. In this paper, some financial markets driven by the fGBm were considered and the corresponding arbitrage opportunities were discussed. Specifically, the financial market M that consists a risk free asset and a risky asset will admit some arbitrage opportunity if the stochastic integrals with respect to the fGBm were established in the path-wise sense, and the arbitrage opportunity can be excluded if the stochastic integrals were established in the Wick calculus sense. What is more, the arbitrage opportunity for the financial market ( M , C ) , consisting of the original market M and some contingent claim C were also investigated in the general case, and some interval of free arbitrage prices for the claim were derived by applying the generalized fractional G-Girsanov theorem and fractional G-Clark-Ocone theorem. This study generalized the well-known existing results of arbitrage theory in financial markets driven by the standard Bm, fBm and GBm.

Limit theorems in the sense of a.s. convergence, convergence in L 1 {L_1} -norm and convergence in distribution are proved for variational series. In the first two cases, if g is a bounded, nonnegative continuous function satisfying an additional assumption at zero, and if { X ( t ) , 0 ≤ t ≤ T } \{ X(t),0 \leq t \leq T\} is a stochastically continuous stochastic process with independent increments, with no Gaussian component and whose trend term is of bounded variation, then the sequence of variational sums of the form Σ k = 1 n g ( X ( t n k ) − X ( t n , k − 1 ) ) \Sigma _{k = 1}^ng(X({t_{nk}}) - X({t_{n,k - 1}})) is shown to converge with probability one and in L 1 {L_1} -norm. Also, under the basic assumption that the distribution of the centered sum of independent random variables from an infinitesimal system converges to a (necessarily) infinitely divisible limit distribution, necessary and sufficient conditions are obtained for the joint distribution of the appropriately centered sums of the positive parts and of the negative parts of these random variables to converge to a bivariate infinitely divisible distribution. A characterization of all such limit distributions is obtained. An application is made of this result, using the first theorem, to stochastic processes with (not necessarily stationary) independent increments and with a Gaussian component.

The purpose of this paper is twofold: first, to extend the definition of the stochastic integral for processes with values in Banach spaces; and second, to define the stochastic integral as a genuine integral, with respect to a measure, that is, to provide a general integration theory for vector measures, which, when applied to stochastic processes, yields the stochastic integral along with all its properties. For the reader interested only in scalar stochastic integration, our approach should still be of interest, since it sheds new light on the stochastic integral, enlarges the class of integrable processes and presents new convergence theorems involving the stochastic integral.