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The Ito-Stratonovich theory of stochastic integration and stochastic differential equations has several shortcomings, especially when it comes to existence and consistency with the theory of Lebesque-Stieltjes integration and ordinary differential equations. An attempt is made firstly, to isolate the path property, possessed by almost all Brownian paths, that makes the stochastic theory of integration work. Secondly, to construct a new concept of solutions for differential equations, which would have the required consistency and continuity properties, within a class of deterministic noise functions, large enough to include almost all Brownian paths. The algebraic structure of iterated path integrals for smooth paths leads to a formal definition of a solution for a differential equation in terms of generalized path integrals for more general noises. This suggests a way of constructing solutions to differential equations in a large class of paths as limits of operators. The concept of the driving noise is extended to include the generalized path integrals of the noise. Less stringent conditions on the Holder continuity of the path can be compensated by giving more of its iterated integrals. Sufficient conditions for the solution to exist are proved in some special cases, and it is proved that almost all paths of Brownian motion as well as some other stochastic processes can be included in the theory.

This thesis is concerned with stochastic heat equation with memory and nonlinear energy supply. The main motivation to study such systems comes from Thermodynamics, see [85]. The main objective of this work is to study the existence and uniqueness of solutions to such equations and to investigate some fundamental properties of solutions like continuous dependence on initial conditions. In our approach we follow the seminal papers by Da Prato and Clement [10], where the stochastic heat equation with memory is tranformed into an integral equation in a function space and the so-called mild solutions are studied. In the aforementioned papers only linear equations with additive noise were investigated. The main contribution of this work is the extension of this approach to nonlinear equations. Our main tools are the theory of stochastic convolutions as developed in [33] and the theory of resolvent kernels for deterministic linear heat equations with memory, see[10]. Since the solution at time t depends on the whole history of the process up to time t, the resolvent kernel does not define a semigroup of operators in the state space of the process and therefore a ???standard??? theory of stochastic evolution equations as presented in the monograph [33] does not apply. A more delicate analysis of the resolvent kernles and the associated stochastic convolutions is needed. We will describe now content of this thesis in more detail. Introductory Chapters 1 and 2 collect some basic and essentially well known facts about the Wiener process, stochastic integrals, stochastic convolutions and integral kernels. However, some results in Chapter 2 dealing with stochastic convolution with respect to non-homogenous Wiener process are extensions of the existing theory. The main results of this thesis are presented in Chapters 3 and 4. In Chapter 3 we prove the existence and uniqueness of solutions to heat equations with additive noise and either Lipschitz or dissipative nonlinearities. In both cases we prove the continuous dependence of solutions on initial conditions. In Chapter 4 we prove the existence and uniqueness of solutions and continuous dependence on initial conditions for equations with multiplicative noise. The diffusion coefficients defined by unbounded operators are allowed.

This thesis is concerned with stochastic heat equation with memory and nonlinear energy supply. The main motivation to study such systems comes from Thermodynamics, see [85]. The main objective of this work is to study the existence and uniqueness of solutions to such equations and to investigate some fundamental properties of solutions like continuous dependence on initial conditions. In our approach we follow the seminal papers by Da Prato and Clement [10], where the stochastic heat equation with memory is tranformed into an integral equation in a function space and the so-called mild solutions are studied. In the aforementioned papers only linear equations with additive noise were investigated. The main contribution of this work is the extension of this approach to nonlinear equations. Our main tools are the theory of stochastic convolutions as developed in [33] and the theory of resolvent kernels for deterministic linear heat equations with memory, see[10]. Since the solution at time t depends on the whole history of the process up to time t, the resolvent kernel does not define a semigroup of operators in the state space of the process and therefore a ???standard??? theory of stochastic evolution equations as presented in the monograph [33] does not apply. A more delicate analysis of the resolvent kernles and the associated stochastic convolutions is needed. We will describe now content of this thesis in more detail. Introductory Chapters 1 and 2 collect some basic and essentially well known facts about the Wiener process, stochastic integrals, stochastic convolutions and integral kernels. However, some results in Chapter 2 dealing with stochastic convolution with respect to non-homogenous Wiener process are extensions of the existing theory. The main results of this thesis are presented in Chapters 3 and 4. In Chapter 3 we prove the existence and uniqueness of solutions to heat equations with additive noise and either Lipschitz or dissipative nonlinearities. In both cases we prove the continuous dependence of solutions on initial conditions. In Chapter 4 we prove the existence and uniqueness of solutions and continuous dependence on initial conditions for equations with multiplicative noise. The diffusion coefficients defined by unbounded operators are allowed.

Let H be a separable Hilbert space. Suppose (Ω, F, Ft, P) is a complete stochastic basis with a right continuous filtration and {Wt,t ∈ R} is an H-valued cylindrical Brownian motion with respect to {Ω, F, Ft, P). U(t, s) denotes an almost strong evolution operator generated by a family of unbounded closed linear operators on H. Consider the semilinear stochastic integral equation [formula omitted] where • f is of monotone type, i.e., ft(.) = f(t, w,.) : H → H is semimonotone, demicon-tinuous, uniformly bounded, and for each x ∈ H, ft(x) is a stochastic process which satisfies certain measurability conditions. • gs(.) is a uniformly-Lipschitz predictable functional with values in the space of Hilbert-Schmidt operators on H. • Vt is a cadlag adapted process with values in H. • X₀ is a random variable. We obtain existence, uniqueness, boundedness of the solution of this equation. We show the solution of this equation changes continuously when one or all of X₀, f, g, and V are varied. We apply this result to find stationary solutions of certain equations, and to study the associated large deviation principles. Let {Zt,t ∈ R} be an H-valued semimartingale. We prove an Ito-type inequality and a Burkholder-type inequality for stochastic convolution [formula omitted]. These are the main tools for our study of the above stochastic integral equation. / Science, Faculty of / Mathematics, Department of / Graduate

A novel statistical technique introduced by Besieris is used to study solutions of the nonlinear stochastic complex parabolic equation in the presence of two profiles. Specifically, the randomly modulated linear potential and the randomly perturbed quadratic focusing medium. In the former, a class of solutions is shown to admit an exact statistical description in terms of the moments of the wave function. In the latter, all even-order moments are computed exactly, whereas the odd-order moments are solved asymptotically. Lastly, it is shown that this statistical technique is isomorphic to mappings of nonconstant coefficient partial differential equations to constant coefficient equations. A generalization of this mapping and its inherent restrictions are discussed. / Ph. D. / incomplete_metadata

This work is focused on the problem of filtering of random processes and on the construction of a stochastic integral with a measureable parameter. This integral is used to devise filtration equations for a random process which is based on a model motivated by a financial application. The method used to devise them and the equations themselves are then compared with the so called optional filtering from the book Markov processes and Martingales by Rogers and Williams, while the definition of the optional projection is extended so it is possible to correct a~mistake in a proposition in the aforementioned book. Powered by TCPDF (www.tcpdf.org)

Este texto apresenta alguns dos elementos básicos envolvidos em um estudo introdutório das equações diferencias estocásticas. Tais equações modelam problemas a tempo contínuo em que as grandezas de interesse estão sujeitas a certos tipos de perturbações aleatórias. Em nosso estudo, a aleatoriedade nessas equações será representada por um termo que envolve o processo estocástico conhecido como Movimento Browniano. Para um tratamento matematicamente rigoroso dessas equações, faremos uso da Integral Estocástica de Itô. A construção dessa integral é um dos principais objetivos do texto. Depois de desenvolver os conceitos necessários, apresentaremos alguns exemplos e provaremos existência e unicidade de solução para equações diferenciais estocásticas satisfazendo certas hipóteses. / This text presents some of the basic elements involved in an introductory study of stochastic differential equations. Such equations describe certain kinds of random perturbations on continuous time models. In our study, the randomness in these equations will be represented by a term involving the stochastic process known as Brownian Motion. For a mathematically rigorous treatment of these equations, we use the Itô Stochastic Integral. The construction of this integral is one of the main goals of the text. After developing the necessary concepts, we present some examples and prove existence and uniqueness of solution of stochastic differential equations satisfying some hypothesis.

The aim of this PhD Thesis was to build and develop a stochastic calculus (in particular a stochastic integral) with respect to multifractional Brownian motion (mBm). Since the choice of the theory and the tools to use was not fixed a priori, we chose the White Noise theory which generalizes, in the case of fractional Brownian motion (fBm) , the Malliavin calculus. The first chapter of this thesis presents several notions we will use in the sequel.In the second chapter we present a construction as well as the main properties of stochastic integral with respect to harmonizable mBm.We also give Ito formulas and a Tanaka formula with respect to this mBm. In the third chapter we give a new definition, simplier and generalier of multifractional Brownian motion. We then show that mBm appears naturally as a limit of a sequence of fractional Brownian motions of different Hurst index.We then use this idea to build an integral with respect to mBm as a limit of sum of integrals with respect ot fBm. This being done we particularize this definition to the case of Malliavin calculus and White Noise theory. In this last case we compare the integral hence defined to the one we got in chapter 2. The fourth and last chapter propose a multifractional stochastic volatility model where the process of volatility is driven by a mBm. The interest lies in the fact that we can hence take into account, in the same time, the long range dependence of increments of volatility process and the fact that regularity vary along the time.Using the functional quantization theory in order to, among other things, approximate the solution of stochastic differential equations, we can compute the price of forward start options and then get and plot the implied volatility nappe that we graphically represent.