Martingales and Stochastic Analysis

Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.

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.

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.

The purpose of this chapter is to overview elements of the theory of stochastic differential equations, based on Wiener processes, for use in the subsequent chapters. This theory was founded by K. Itô in 1942 (Itô, Zenkoku Shijo Sugaku Danwakai 244:1352–1400, 1942; Itô, On stochastic differential equations. Memoirs of the American Mathematical Society, vol 4. AMS, New York City, 1951). His aim was to construct Markov processes, governed by Kolmogorov’s differential equations via Wiener processes, and to analyze their sample paths. After that, stochastic differential equations have been used to describe dynamical processes in random environments of various fields. Here we consider stochastic differential equations with random coefficients, because we aim at studying stochastic control problems.The chapter is organized as follows. Section 1.1 is preliminaries. The basic definitions and results on stochastic processes are collected for later use. Stochastic differential equations and stochastic analysis will be introduced in Sect. 1.2. Section 1.3 deals with asset pricing problems as an application of previous results.KeywordsBackward Stochastic Differential EquationsAsset Pricing ProblemsConstructing Markov ProcessesRandom CoefficientsBurkholder Davis Gundy InequalityThese 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.

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.

Fractional Brownian Motions, Fractional Noises and Applications

The purpose of this paper is to examine the relationship between stock price and futures volume. This paper contributes to previous studies of price-volume relationship and the determinants of futures volume by postulating three hypotheses and testing them with data for four stock index futures in Taiwan. The model developed in this article formalizes the price-volume relationship by stochastic calculus and Ito process. First, we find a long-run relationship between stock price and futures volume by cointegration test. If the cointegarted relationship exists, stock price and futures volume are non-stationary in level but stationary in the first differences. That is, stock price and futures volume follow a random-walk process. On the other hand, we extract the short-run and long-run impacts by vector error correction model. Furthermore, we consider three measures for stock price volatility to test the determinants of change and volatility of futures volume. Although the determinant of change and volatility of futures volume are sensitive to the volatility estimate used, we find that absolute stock price change is a more suitable measure for stock index price volatility.

This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.

In this chapter, we survey some of the concepts in the theory of stochastic processes which will be used in the sequel. The material is intended to be only a discussion of some of the main ideas which will be needed and to serve as a convenient reference. It is not intended to be complete in any sense. Additional details, motivation and background can be found in the references. The reader familiar with the basic facts concerning stochastic differential equations can skip the chapter and refer to the results as needed. Section 1 gives some definitions and facts concerning martingales. In Section 2, we define stochastic integrals with respect to a Wiener process and state Itô's Formula, the fundamental tool in the stochastic calculus. Section 3 uses the stochastic integrals to define stochastic differential equations (SDE) of the diffusion type, and obtains bounds on their solutions which will be used later in the study of properties of the solution processes, as well as in the proofs of tightness of sequences of solutions to such equations. In Section 4, we discuss three standard methods for obtaining existence and uniqueness of solutions to SDE's. These sections give some flavor of a few of the basic ideas. But they just touch the surface of the subject. Sections 5 and 6 concern an alternative and very useful method for verifying whether a process satisfies an SDE of the diffusion type. The so-called "martingale problem" method which is discussed in these sections is perhaps the most useful approach to showing that the limit of a weakly convergent sequence of processes is, indeed, a solution to a SDE.

Random dynamical systems can be generated by stochastic differential equations (sde) on the one hand, and by random differential equations (rde), i.e. randomly parametrized ordinary differential equations on the other hand. Due to conflicting concepts in stochastic calculus and ergodic theory, asymptotic problems for systems associated with sde are harder to treat. We show that both objects are basically identical, modulo a stationary coordinate change (cohomology) on the state space. This observation opens completely new opportunities for the treatment of asymptotic problems for systems related to sde: just study them for the conjugate rde, which is often possible by simple path-by-path classical arguments. This is exemplified for the problem of local linearization of random dynamical systems, the classical analogue of which leads to the Hartman–Grobman theorem.

Roughly speaking, stochastic differential equations are differential equations driven by Gaussian white noise. Here, we are using the term “stochastic differential equations” in a restricted sense and not merely to denote differential equations with some probabilistic aspects. The importance of. stochastic differential equations is largely due to the fact that the solution of such an equation is a sample-continuous Markov process, and conversely, a large and important class of sample-continuous Markov processes can be modeled by the solutions of stochastic differential equations. From the point of view of applications, this is a direct benefit of using white noise as a noise model, and this fact accounts for its popularity. After all, white noise is, at best, a tolerable abstraction and is never a completely faithful representation of a physical noise source. Its raison d’être is the simplicity of analysis that it brings about. We have seen this in connection with Kalman filtering, and we shall see it again in the Markovian nature of solutions to stochastic differential equations.

A Study on Uncertainty of Discharge in River Channel Using Stochastic Partial Differential Equation

Using results from work on shuffle algebras, we show that Lyndon words provide an algebraic basis for the sets of iterated Stratonovich or Ito integrals that appear in the stochastic Taylor series expansion of the solution to a stochastic differential equation and give a method for rewriting these stochastic integrals in terms of the basis. This basis is similar to, but simpler than, that obtained by Sussmann ([19]) using Hall words. We also show how the shuffle product can be used to obtain moments of stochastic integrals and hence reprove a result of Kloeden and Platen ([9]) using a purely combinatorial approach.

The tenth special issue of the Japanese Journal of Statistics and Data Science (JJSD) focuses on statistics for stochastic processes. With the advancement of information and communication technology and measurement technology, a vast amount of diverse time series data have been readily obtained. Consequently, the significance of stochastic processes as mathematical models for random phenomena evolving in time or space has increased dramatically, and statistics for various types of stochastic processes have been rapidly developing. In fact, a multitude of studies on statistics for stochastic processes such as ARMA, CARMA, COGARCH, diffusion processes, fractional Brownian motions, GARCH, jump-diffusion processes, Lèvy processes, Markov processes, point processes, random fields, stochastic differential equations and stochastic partial differential equations have been conducted. Nonetheless, because of the mathematical complexity of statistics for stochastic processes, numerous theoretical and computational challenges remain to be addressed. This special feature contains ten original papers contributed by leading mathematical statisticians and data scientists on current topics in statistics for stochastic processes. The first paper examines the efficient particle Markov chain Monte Carlo method. The second and third papers respectively address the estimation problems of Hawkes processes and Cox proportional excess hazard models. The fourth and fifth papers investigate parametric estimation for probability partial differential equation models. The latter five papers focus on statistical inference for stochastic differential equation models driven by various stochastic processes, including the Wiener process, fractional Brownian motion, β-stable process and Lèvy process. The ten papers in the special feature are summarized as follows: Awaya and Omori propose an efficient particle Markov chain Monte Carlo methodology for the rolling-window estimation of state space models. They provide theoretical justifications for their proposed method and demonstrate the efficiency of B Masayuki Uchida

We develop a stochastic integration theory with respect to creation, annihilation and gauge operators on the full Fock space. This is done by using a kernel representation for a large class of bounded operators on the full Fock space. It is shown that the kernels form a Banach algebra. Having established the definition of processes and stochastic integrals we go on to prove an Ito formula and use this for examining stochastic evolutions and constructing dilations of special completely positive semigroups. Explicit solutions of the corresponding stochastic differential equations are given.