Stochastic Integration in Abstract Spaces

Many questions in functional analysis lead to a consideration of inductive limits of Frechet or even Banach spaces. This chapter utilizes many facts and ideas from the theory of bases, nuclear spaces and the general theory of locally convex spaces. It considers F to be the dual of a nuclear Frechet space E. F will be equipped with the strong topology. The chapter also utilizes a method which consists of first embedding E' in a space which has a basis, and then constructing a certain block basic sequence which can be approximated by a sequence in E', and an old perturbation theorem of L.

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:

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.

In this work we introduce a theory of stochastic integration with respect to general cylindrical semimartingales defined on a locally convex space $\Phi$. Our construction of the stochastic integral is based on the theory of tensor products of topological vector spaces and the property of good integrators of real-valued semimartingales. This theory is further developed in the case where $\Phi$ is a complete, barrelled, nuclear space, where we obtain a complete description of the class of integrands as $\Phi$-valued locally bounded and weakly predictable processes. Several other properties of the stochastic integral are proven, including a Riemann representation, a stochastic integration by parts formula and a stochastic Fubini theorem. Our theory is then applied to provide sufficient and necessary conditions for existence and uniqueness of solutions to linear stochastic evolution equations driven by semimartingale noise taking values in the strong dual $\Phi'$ of $\Phi$. In the last part of this article we apply our theory to define stochastic integrals with respect to a sequence of real-valued semimartingales.

In this chapter, we consider studies involving cadlag processes. We pay particular attention to semi-martingales with respect to a given filtration; these processes will always be taken with cadlag paths. We present the definition of stochastic integrals with respect to a square integrable martingale, and we extend the definition to stochastic integrals with respect to a local martingale. Then, we introduce semi-martingales, quadratic covariation processes for semi-martingales and some general versions of Ito’s formula and Girsanov’s theorem. We give necessary and sufficient conditions for the existence of an equivalent martingale measure. We end the chapter with a brief survey of valuation in an incomplete market.

We give stopped Doob inequalities for p-th moment, 0<p<∞, of stochastic convolution integrals ∫(0,t] U(t,s)φ s − dM s in a Hilbert space, where M is a Hilbert space-valued cadlag square integrable martingale, φ is an operator-valued predictable process and U(t,s) is a contraction-type evolution operator. We also generalize the previous results and try to get smaller constants. 1This research was partially supported by Sharif University of Technology.

It is shown that a block sequence in a nuclear Frechet space with a basis has a block extension if and only if the subspace it generates is complemented. In addition, a short proof is given of the following result of Dubinsky and Robinson: a nuclear Frechet space is isomorphic toΩ = RN, N = {1,2,...} if it has a basis such that any block sequence with blocks of length ≤2 of any permutation of this basis has a block extension. It is shown that a similar result holds without considering permutations of the basis if the length of the blocks is arbitrary.

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.

A general theory of stochastic integral in the abstract topological measurable space is established. The martingale measure is defined as a random set function having some martingale property. All square integrable martingale measures constitute a Hilbert space M 2. For each μ∈M 2, a real valued measure 〈μ〉 on the predictable σ-algebra ℘ is constructed. The stochastic integral of a random function \(\mathfrak{h} \in L^2 \left( {\left\langle \mu \right\rangle } \right)\) with respect to μ is defined and investigated by means of Riesz's theorem and the theory of projections. The stochastic integral operator I μis an isometry from L 2(〈μ〉) to a stable subspace of M 2, its inverse is defined as a random Radon-Nikodym derivative. Some basic formulas in stochastic calculus are obtained. The results are extended to the cases of local martingale and semimartingale measures as well.

Mityagin posed the problem, whether complemented subspaces of nuclear infinite type power series spaces have a basis. A related more general question was asked by Pełczyński. It is well known for a complemented subspace $E$ of a nuclear infinite type power series space, that its diametral dimension can be represented by $\Delta E = \Delta {\Lambda _\infty }(\alpha )$ for a suitable sequence $\alpha$ with ${\alpha _j} \geq \ln (j + 1)$. In this article we prove the existence of a basis for $E$ in case that ${\alpha _j} \geq j$ and $\sup \tfrac {{{\alpha _{2j}}}}{{{\alpha _j}}} < \infty$.

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.

Holomorphic Functions on Nuclear Sequence Spaces

Let $M$ be a locally square integrable martingale with predictable quadratic variance $\\langle M\\rangle$ and let $\\Delta M = M - M_-$ be the jump process of $M$. In this paper, under the various restrictions on $\\Delta M$, the different increasing rates of $M$ in terms of $\\langle M\\rangle$ are obtained. For stochastic integrals $X = B \\cdot M$ of the predictable process $B$ with respect to $M$, the a.s. asymptotic behavior of $X$ is also discussed under restrictions on the rates of increase of $B$ and the restrictions on the conditional distributions of $\\Delta M$ or on the conditional moments of $\\Delta M$. This is applied to some simple examples to determine the convergence rates of estimators in statistics.

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

In this article we discuss the relationship between domains of existence domains of holomorphy, holomorphically convex domains, pseudo convex domains, in the context of locally convex topological vector spaces. By using the method of Hirschowitz for Π n=1 ∞ C and the method used for Banach spaces with a basis we prove generalisations of the Cartan-Thullen-Oka-Norguet-Bremmerman theorem for finitely polynomially convex domains in a variety of locally convex spaces which include the following:1) N-projective limits of Frechet spaces with a basis;2) countable direct sums of Frechet spaces with a basis;3) nuclear spaces.