STOCHASTIC INTEGRATION

Brownian motion is an important stochastic process used in modelling the random evolution of stock prices. In their 1973 seminal paper—which led to the awarding of the 1997 Nobel prize in Economic Sciences—Fischer Black and Myron Scholes assumed that the random stock price process is described (i.e., generated) by Brownian motion. Despite its relative simplicity, the description of Brownian motion in advanced textbooks sometimes lacks an intuitive basis. The present exposition attempts to provide a simplified construction of standard Brownian motion based on a gambling analogy. This is followed by a description and explicit solution of two stochastic differential equations (known as arithmetic and geometric Brownian motion processes) that are driven by the standard Brownian motion process. The paper also illustrates the use of the Maple computer algebra system to simulate the standard and geometric Brownian motion processes.

This thesis consists of two parts, each part concentrating on a different problem from the theory of Stochastic Integration. Chapter 1 contains the introduction explaining the results in this dissertation in general terms. We use the infinite dimensional space S'R endowed with the gaussian measure mu. The Hilbert space ( L2) is defined as (L2) (L2) ≡ L2( S'R , mu) and our results are based on the Gel'fand triple ( S )beta ⊂ (L2) ⊂ S* b . The necessary preliminary background in white noise analysis are well elaborated in Chapter 2. In Chapter 3 we present a generalization of the Ito Formula to anticipating processes in the white noise framework. We first introduce an extension of the Ito integral to anticipating processes called the Hitsuda Skorokhod integral. We then use the anticipating Ito formula for processes of the type theta(X(t), B(c)), where X(t) is a Wiener integral and B(t) = 〈·, 1&sqbl0;0,t&parr0; 〉 is Brownian motion for a ≤ c ≤ b, which was obtained in Professor H. H. Kuo's book, White Noise Distribution Theory, as a motivation to obtain our first main result. We generalize the formula in Prof. Kuo's book to processes of the form theta(X(t),F) where X(t) is a Wiener integral and F ∈ W 1/2, a special subspace of (L 2). Chapter 4 contains the second part of our work. We first state the Clark-Ocone (C-O) formula in the white noise framework. We then verify the formula for Brownian functionals using the very important white noise tool, the S-transform. As our second main result, we extend the formula to generalized Brownian functionals of the form F=Sinfinityn=0 :˙⊗n:,fn where fn ∈ L 2 R ⊗&d4;n . As examples we verify the formula for the Hermite Brownian functional: B(t)n:t using the Ito formula. We also compute the C-O formula for the Donsker's delta function using the result for the Hermite Brownian functional. Finally, the formula is generalized to compositions of tempered distributions with Brownian motion, i.e, to functionals of the form f( B(t)) where f ∈ S'R and B(t) = B( t) = 〈·, 1&sqbl0;0,t&parr0; 〉 is Brownian motion.

Stochastic integration for inhomogeneous Wiener process in the dual of a nuclear space

The Wiener process (or the Brownian motion) is the starting point and the basis for all the following chapters. This is why we will consider this process more explicitly. It is a continuous-time process having as prominent a position in stochastic calculus as the Gaussian distribution in statistics. After introducing its defining properties intuitively, we will discuss important characteristics in the third section. Examples derived from the Wiener process will conclude the exposition.

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.

We consider Riemann sum approximations of stochastic integrals with respect to the fractional Browian motion of index H≥1 2. We show the convergence of these schemes at first and second order. The processes obtained in the limit in the second case are stochastic integrals with respect to the Rosenblatt process if H>3 4 and the standard Brownian motion otherwise. These results are obtained under the assumption that the integrand is a “controlled” process. We provide many examples of such processes, in particular fractional semimartingales and multiple Wiener-Itô integrals.

The theory of random walks holds significance across various domains. To this end, this paper discusses the application of random walk theory in the modeling and analysis of complex systems, including the Brownian motion, Wiener process, and Pu migration. The first part mainly explains the concepts of Brownian motion and random walks, as well as their relationship. The core characteristic is the independence and normal distribution of increments, which makes it a special model of random walks. The next part is the Wiener process, which is a continuous time stochastic process. It can describe random phenomena and solve complex problems in finance, encompassing areas such as stock price prediction, option pricing, and risk management. The last part is about Pu migration, with the clear introduction of Pu migration in biology. The authors discuss why the random walk theory can be regarded as the better approach to analysis this problem. Thus, this paper highlights the importance of the random walk in solving many scientific issues.

In this paper we develop a stochastic calculus with respect to a Gaussian process of the form $B_t = \int^t_0 K(t, s)\, dW_s$, where $W$ is a Wiener process and $K(t, s)$ is a square integrable kernel, using the techniques of the stochastic calculus of variations. We deduce change-of-variable formulas for the indefinite integrals and we study the approximation by Riemann sums.The particular case of the fractional Brownian motion is discussed.

Beginning with an intuitive consideration of sequences of measurements, we define a time-ordered event space representing the collection of all imaginable outcomes for measurement sequences. We then postulate the generalized distributive relation on the event space and examine the class of measurements for which this relation can be experimentally validated. The generalized distributive relation is shown to lead to a σ-additive conditional probability on the event space and to a predictive and retrodictive formalism for stochastic processes. We then show that this formalism has a predictive and a retrodictive representation in a separable Hilbert space H, which has no counterpart in unitary quantum dynamics.

Solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian Motions

A Brownian bridge is a stochastic process derived from standard Brownian motion by requiring an extra constraint. This gives Brownian bridges unique mathematical properties, fascinating, itself, and useful in statistical and mathematical modeling. Here, we define a Brownian bridge and explain some of its unique mathematical properties. These properties make it useful as a mathematical model for real‐world phenomena and as a tool for statistical analysis. We then give some examples of applications. We will see many mathematical ideas that touch Brownian bridges, such as differential equations, linear spaces, topology, metric spaces of functions, measure theory, stochastic integrals, estimation, and hypothesis testing. Copyright © 2009 Wiley Periodicals, Inc.This article is categorized under: Statistical Models > Time Series Models Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

After introducing cylindrical Gaussian random variables and Hilbert space valued Gaussian random variables, we define cylindrical Wiener process and Hilbert space valued Wiener process in a natural way. We develop the Itô stochastic integral with respect to cylindrical and Hilbert space valued Wiener processes simultaneously as they share many common features. Our construction of the Itô integral is consistent with the construction of a stochastic integral with respect to square integrable martingales, however, regarding measurability, we assume only that the integrand is adapted, following the classical ideas as presented by Liptser and Shiryayev (Statistics of Stochastic Processes. Nauka, ) and by Øksendal (Stochastic Differential Equations. Springer, New York, ). We provide detailed proofs of properties of the Itô integral and follow with the Martingale Representation Theorem. Our unified approach to stochastic integration with respect to cylindrical and Hilbert space valued Wiener processes allowed us to present the Stochastic Fubini Theorem and the Itô Formula in both cases in an almost identical way.

Un résultat de A.M. Davie (Int. Math. Res. Not. 24 (2007) rnm124) établit qu’une équation stochastique multi-dimensionnelle $dX_{t}=b(t,X_{t})\,dt+dW_{t}$, $X_{0}=x$, dirigée par un processus de Wiener $W=(W_{t})$ avec un coefficient $b$ qui est seulement borné et mesurable admet une unique solution pour presque tout choix de la trajectoire du processus $W$ la dirigeant. Nous considérons un problème similaire lorsque $W$ est remplacé par un processus de Lévy $L=(L_{t})$ et $b$ est $\beta$-Hölder continu en espace. Nous supposons que $L_{1}$ a un moment fini d’ordre $\theta$ pour un certain $\theta>0$. En utilisant un nouveau résultat de régularité càdlàg, nous prouvons que l’existence et unicité forte pour l’EDS, associées à une $L^{p}$-Lipschitz continuité de la solution forte par rapport à $x$, impliquent une unicité de type Davie pour presque tout choix de la trajectoire de Lévy. Nous appliquons ce résultat à une classe d’EDS dirigées par un processus de Lévy $\alpha$-stable non dégénéré pour $\alpha\in(0,2)$ et $\beta>1-\alpha/2$.

The sub-fractional CEV model

By a well-known theorem of P. Levy, if $(X_t)$ is a standard Brownian motion on $\mathbb{R}$ with $X_0 = 0$ and if $H_t = \min_{u \leq t}X_u$, then $(Y_t) = (X_t - H_t)$ is Brownian motion with 0 as a reflecting lower boundary. More generally, if $X$ is allowed to have nonzero drift or a reflecting lower boundary at $A < 0$, then the process $Y = X - H$ is still a diffusion process. We prove the converse result: If $X$ is a diffusion on an interval $I \subset \mathbb{R}$ which contains 0 as an interior point, and if $(Y_t) = (X_t - H_t)$ is a time homogeneous strong Markov process (when $X_0 = 0$), then $X$ must be a Brownian motion on $I$ (with drift $\mu$, variance parameter $\sigma^2 > 0$, killing rate $c \geq 0$ and reflection at $\inf I$ in case $\inf I > -\infty$).