Stochastic Integration in Banach Spaces

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 main purpose of this article is to propose a reasonable definition for the stochastic integration (S.I.) of progressive processes w.r.t. semimartingales. This S.I. generalizes that of predictable processes w.r.t. semimartingales as well as the stochastic Stieltjes integration. This S.I. is proposed in §1. We give also in §1 an exponential formula for semimartingales using this S.I.. The rest of this paper consists of several remarks on the theory of stochastic integration which are mostly of pedagogical interest. In §2 we propose a new construction of the S.I. of predictable processes w.r.t. local martingales. A simple proof of the integration by parts formula is given in §3. Finally, we propose in §4 a short proof of Meyer’s theorem on compensated stochastic integrals of local martingales.AMS Subject Classification60H05

We study the metric and w^* -differentiability of pointwise Lipschitz mappings. First, we prove several theorems about metric and w^* -differentiability of pointwise Lipschitz mappings between \mathbb{R}^n and a Banach space X (which extend results due to Ambrosio, Kirchheim and others), then apply these to functions satisfying the spherical Rado–Reichelderfer condition, and to absolutely continuous functions of several variables with values in a Banach space. We also establish the area formula for pointwise Lipschitz functions, and for (n,\lambda) -absolutely continuous functions with values in Banach spaces. In the second part of this paper, we prove two theorems concerning metric and w^* -differentiability of pointwise Lipschitz mappings f:X\mapsto Y where X,Y are Banach spaces with X being separable (resp. X separable and Y=G^* with G separable).

UDC 517.5 Let ( 𝕏 , ‖ . ‖ 𝕏 ) denote a complex Banach space and L ( 𝕏 ) = B C ( ℝ → 𝕏 ) be the set of all 𝕏 -valued bounded continuous functions f : ℝ → 𝕏 . For f ∈ L ( 𝕏 ) we define ‖ f ‖ L ( 𝕏 ) = sup { ‖ f ( x ) ‖ 𝕏 : x ∈ ℝ } . Then ( L ( 𝕏 ) , ‖ . ‖ L ( 𝕏 ) ) itself is a Banach space. The Beurling spectrum S p e c ( f ) of a function f ∈ L ( 𝕏 ) is defined by S p e c ( f ) = { ζ ∈ ℝ : ∀ ϵ > 0 ∃ φ ∈ 𝒮 ( ℝ ) : supp φ ^ ⊂ ( ζ - ϵ , ζ + ϵ ) , φ * f ≢ 0 } . } . We obtain the following Paley-Wiener type theorem for functions with values in Banach spaces: Let f ∈ L ( 𝕏 ) and K be an arbitrary compact set in ℝ . Then Spec ( f ) ⊂ K if and only if for any τ > 0 there exists a constant C τ < ∞ such that < b r > ‖ P ( D ) f ‖ L ( 𝕏 ) ≤ C τ < b r > ‖ f ‖ L ( 𝕏 ) sup x ∈ K ( τ ) | P ( x ) | < b r > for all polynomials with complex coefficients P ( x ) , where the differential operator P ( D ) is obtained from P ( x ) by substituting x → - i d ⅆ x , d ⅆ x is the usual derivative in L ( 𝕏 ) and K ( τ ) is the τ -neighborhood in ℂ of K . Moreover, Paley-Wiener type theorem for integral operators and one for some special compacts K are also given.

In this dissertation, we focus mainly on the further study of the new stochastic integral introduced by Ayed and Kuo in 2008. Several properties of this new stochastic integral are obtained. We first introduce the concept of near-martingale for non-adapted stochastic processes. This concept is a generalization of the martingale property for adapted stochastic processes in the It\^o theory. We prove a special case of It\^o isometry for the stochastic integral of certain instantly independent processes. We obtain some formulas for expressing a new stochastic integral in terms of It\^o integrals and Riemann integrals. Several generalized versions of It\^o's formula for the new stochastic integral obtained by Ayed and Kuo are given. We also provide some examples to illustrate the ideas.

Here we research the univariate multi-composite sigmoid activated quantitative approximation of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued multi-composite sigmoid activated neural network operators. We perform also the related Banach space valued multi-composite sigmoid activated fractional approximation. These multi-composite sigmoid acti-vated approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative or fractional derivatives. Our operators are deÖned by using a multi-composite density function induced by a gen- eral multi-composite sigmoid function. The approximations are pointwise and with respect to the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer. We Önish with a convergence analysis.

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.

In this paper we shall examine the structure of abstract quasi-martingales taking their values in a Banach space E. We are especially interested in the existence of cadlag modifications and in the Doob-Meyer decomposition of such quasimartingales. The motivation of this inquiry was twofold. First, the problems concerning regularity and the Doob-Meyer decomposition of quasimartingales arose naturally in the attempt to develop a stochastic integration theory for Banach space valued processes. The development of this abstract stochastic integral was, in turn, motivated by Kunita’s theory for Hilbert space valued processes [9] and subsequent extensions, using the vector measure approach, by Pellaumail [16], Métivier and Pellaumail [14], Métivier [12] and Kussmaul [10] and [11]. Secondly, we wished to improve on the existing results on the regularity and structure of quasimartingales, as presented by the above mentioned authors.

This paper is concerned with a version of the Lebesgue dominated convergence theorem (DCT) which has been stated for the Kurzweil–Stieltjes integral of real functions. Our objective in this work is to analyze the extension of this result to include vector functions with values in Banach spaces. We establish that the mentioned convergence theorem for the Kurzweil–Stieltjes integral can be formulated in weaker versions for reflexive and separable Banach spaces, and spaces having the Schur property, nonetheless it is not verified in the general case.

Stochastic Integrals and Abelian Processes

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.

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.

Cylindrical fractional Brownian motion in Banach spaces

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.

Laplace Transforms and Integrated, Regularized Semigroups in Locally Convex Spaces