Stochastic Calculus with Respect to Gaussian Processes

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

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.

STOCHASTIC INTEGRATION

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.

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.

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.

It is known that any functional of Brownian motion with finite second moment can be expressed as the sum of a constant and an Ito stochastic integral. It is also known that homogeneous additive functionals of Brownian motion with finite expectations have a similar representation. This paper extends these results in several ways. It is shown that any finite functional of Brownian motion can be represented as a stochastic integral. This representation is not unique, but if the functional has a finite expectation it does have a unique representation as a constant plus a stochastic integral in which the process of indefinite integrals is a martingale. A corollary of this result is that any martingale (on a closed interval) that is measurable with respect to the increasing family of $\sigma$-fields generated by a Brownian motion is equal to a constant plus an indefinite stochastic integral. Sufficiently well-behaved Frechet-differentiable functionals have an explicit representation as a stochastic integral in which the integrand has the form of conditional expectations of the differential.

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.

The purpose of this paper is to establish the existence of analytic Wiener and Feynman integrals for a class of certain cylinder functions which is of the form: urn:x-wiley:01611712:media:ijmm132848:ijmm132848-math-0001 on the abstract Wiener space, and to establish the relationship between the Wiener integral and the analytic Feynman integral for such cylinder functions on the abstract Wiener space. We then establish a change of scale formula for Wiener integrals of such cylinder functions on the abstract Wiener space.

Various types of stochastic integrals with respect to fractional Brownian sheet and their applications

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.

Degradation tests are used when there is a quality characteristic related to the life of a product. In this paper, we investigate the model mis-specification effect on the estimation precision of product's mean time to failure (MTTF) and consider a degradation test design problem. The Wiener and inverse Gaussian (IG) processes are two possible models considered. We derive expressions for the mean and variance of the estimated product's MTTF when the true model is an IG process, but is wrongly fitted by a Wiener process. We further discuss the experimental design problem and derive the explicit functional form of the estimation variances. Using the derived functions, optimal degradation test plans assuming a Wiener process model is correctly or wrongly specified are both proposed. The derived plans are applied to a laser data example. We evaluate the test efficiency of the plans derived from a Wiener process assumption when the model is mis-specified. For many optimization criteria, we observe that the obtained plans are robust even when the fitted model is mis-specified. For some criteria that may result in very different test plans under different models, we use a weighted ratio criterion to find practically useful degradation plans under model uncertainty.

Let be a standard d-dimensional Brownian motion and L a smooth functional on the space C = Cd[0, 1] of continuous functions from [0,1] to Rd, such that Denote Thenis a square integrable martingale which can be represented as a stochastic integral; consequently the random variable has the representationFor Fréchet differentiable functions L, Clark(1) gave the following formula for the integrand e:

We define a fractional Itô stochastic integral with respect to a randomly scaled fractional Brownian motion via an S -transform approach. We investigate the properties of this stochastic integral, prove an Itô formula for functions of such stochastic integrals and apply this Itô formula to the investigation of related generalized time-fractional evolution equations. We show that the constructed stochastic integrals serve as stochastic solutions for a large class of integro-differential evolution equations containing non-local operators acting on the time variable. Our results cover such special cases of randomly scaled fractional Brownian motion as gamma-grey Brownian motion, generalized grey Brownian motion, superstatistical fractional Brownian motion with random diffusion coefficient having generalized gamma distribution, generalized Fox- H processes and some other processes discussed in the literature.

In the present paper, we obtain an explicit product formula for products of multiple integrals w.r.t. a random measure associated with a Lévy process. As a building block, we use a representation formula for products of martingales from a compensated-covariation stable family. This enables us to consider Lévy processes with both jump and Gaussian part. It is well known that for multiple integrals w.r.t. the Brownian motion such product formulas exist without further integrability conditions on the kernels. However, if a jump part is present, this is, in general, false. Therefore, we provide here sufficient conditions on the kernels which allow us to establish product formulas. As an application, we obtain explicit expressions for the expectation of products of iterated integrals, as well as for the moments and the cumulants for stochastic integrals w.r.t. the random measure. Based on these expressions, we show a central limit theorem for the long time behaviour of a class of stochastic integrals. Finally, we provide methods to calculate the number of summands in the product formula.