Top PDF Stochastic integration for Lévy processes with values in Banach spaces

Stochastic integration for Lévy processes with values in Banach spaces

Stochastic integration for Lévy processes with values in Banach spaces

A real valued stochastic integral with respect to a real valued Wiener process can be defined in the classical sense of K. Itˆo. By augmenting only a small amount of operator theory this approach can be easily generalized to integrands with values in Hilbert spaces and Hilbert space valued Wiener processes, which is accomplished in Da Prato and Zabczyk [ 9 ]. Their approach has been extended to L´evy processes by Peszat and Zabczyk in [ 14 ]. For Banach spaces, even in the case of Wiener processes, there seemed to be no general method for introducing a rigorously defined stochastic integral without making special assumptions on the geometry of the Banach space. But more recently, van Neerven and Weis introduced in [ 18 ] for deterministic Banach space valued integrands a stochastic integral with respect to Wiener processes on Banach spaces without any conditions on the underlying Banach space; see also [ 7 , 17 ]. The main point in their construction is the case of a Banach space valued integrand and a scalar Wiener process, which is then extended to Banach space valued Wiener processes. Together with Veraar they continued this work in [ 19 ] for random integrands on UMD Banach spaces. But already the integral for deterministic integrands turned out to be very helpful for dealing with evolution equations on infinite dimensional spaces.
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A study of stochastic processes in Banach spaces

A study of stochastic processes in Banach spaces

This chapter contains essential preliminary material for chapter 4. We study Gaussian random vectors, Wiener processes and It^o stochastic integrals (for deterministic inte- grands) with values in a separable complex Banach space E . We observe that, for all E valued cylindrical Q -Wiener processes on a probability space ( ; F ; P ), Q factors through

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Stochastic antiderivational equations on non Archimedean Banach spaces

Stochastic antiderivational equations on non Archimedean Banach spaces

In Section 2, suitable analogs of Gaussian measures are considered. Cer- tainly they do not have any complete analogy with the classical one, some of their properties are similar and some are different. They are used for the definition of the standard (Wiener) stochastic process. Integration by parts for- mula for the non-Archimedean stochastic processes is studied. Some particular cases of the general Itô formula from [8] are discussed here more concretely. In Section 3, with the help of them, stochastic antiderivational equations are de- fined and investigated. Analogs of theorems about existence and uniqueness of solutions of stochastic antiderivational equations are proved. Generating operators of solutions of stochastic equations are investigated. All results of this paper are obtained for the first time.
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Stochastic processes on non Archimedean Banach spaces

Stochastic processes on non Archimedean Banach spaces

This work treats the case that was not considered by other authors and that is suitable and helpful for the investigation of stochastic processes and quasi- invariant measures on non-Archimedean topological groups. Here are consid- ered spaces of functions with values in Banach spaces over non-Archimedean local fields, in particular, with values in the field Q p of p-adic numbers. For this,

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Levy processes and stochastic integrals in Banach spaces

Levy processes and stochastic integrals in Banach spaces

In section 2.4, we describe the L´evy-Itˆo decomposition alluded to above which gives the sample path structure of a generic L´evy process in terms of Gaussian and jump components. Following Dettweiler [13], we give an account of “strong” stochastic integration in section 2.4. Geometric consid- erations again play a role in limiting the types of Banach spaces in which such integrals can be defined and despite the beautiful mathematics which so arises, this might be seen as a major drawback for stochastic evolution equations. In section 2.5, we indicate how recent work on weaker types of stochastic integration can overcome this obstacle, as they are not tied to the Banach space geometry.
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option pricing under Lévy processes

option pricing under Lévy processes

A numerical method is developed that can price options, including exotic options that can be priced recursively such as Bermudan options, when the underlying process is an exponential Lévy process with closed form conditional characteristic function. The numerical method is an extension of a recent quadrature option pricing method so that it can be applied with the use of fast Fourier transforms. Thus the method possesses desirable features of both transform and quadrature option pricing techniques since it can be applied for a very general set of underlying Lévy processes and can handle certain exotic features. To illustrate the method it is applied to European and Bermudan options for a log normal process, a jump diffusion process, a variance gamma process and a normal inverse Gaussian process.
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A Semigroup Approach to Nonlinear Lévy Processes

A Semigroup Approach to Nonlinear Lévy Processes

1 ∧ |y| 2 dµ(y)  < ∞, (1.4) which does not exclude any L´ evy triplet at all. In particular, L´ evy processes with non- integrable jumps can be considered, see e.g. Example 3.6, and for finite Λ the condition (1.4) is always satisfied. In order to obtain uniqueness for the viscosity solution of (1.2) one additionally needs the second condition in (1.3) and tightness of the family of L´ evy measures {µ : (b, Σ, µ) ∈ Λ}, which is due to [17]. In Hollender [12] the results of [19] are generalized to upper expectations over state-dependent L´ evy triplets, see also K¨ uhn [15] for existence results on the respective integro-differential equations under fairly general conditions. A related concept to nonlinear L´ evy processes are second order backward stochastic differential equation with jumps, see Kazi-Tani et al. [13], [14] and also Soner et al. [27].
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Weak Subordination of Multivariate Lévy Processes

Weak Subordination of Multivariate Lévy Processes

For weak subordination, we derive characteristics (Section 2.3.1), marginal com- ponent consistency (Proposition 2.3.7), sample path properties (Proposition 2.3.21) and moment formulas (Proposition 2.3.22). We also give results for weak subordina- tion in the case of a superposition of independent univariate subordinators travelling along rays (Section 2.3.4). This is a model for common and idiosyncratic time change, and our results allow for the law of weakly subordinated processes to be easily determined and understood in this situation. In addition, we show that when the subordinator has independent components, the weakly subordinated process does too (Proposition 2.3.18). There are also differences between strong and weak subordination. For instance, the time marginals of the weakly subordinated process X T(t), t ≥ 0, coincide with that of the strongly subordinated process X ◦ T(t) when T is assumed to have monotonic components (Proposition 2.3.26), but not in general. In fact, there may be no L´ evy process with time marginals that match X ◦ T(t) in distribution for all t ≥ 0 (Proposition 2.3.29).
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Modelling sovereign debt with Lévy Processes

Modelling sovereign debt with Lévy Processes

The classical structural approach to corporate credit risk modeling describes the asset value process as a geometric Brownian motion and defines default either as the equity value drop- ping to zero at maturity (Merton 1974) or as the first passage time of an exogenous default barrier (Black and Cox 1976). It establishes an intuitive relationship between default and the value of a firm’s assets, and its dynamics allows the straightforward computation of survival probabilities and the credit spread term structure. However, it is known that the Black- Scholes framework used is unable to capture several well-grounded empirical evidences, such as the skewed and leptokurtic distribution of returns. These shortcomings are rooted in the assumption of Gaussian increments, that imply continuous sample paths. We can overcome them by extending the modeling dynamics to the wider class of L´ evy processes. In particu- lar, we can then capture sudden shocks through the introduction of jumps in the asset value process, thereby removing the local predictability of default.
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A unifying approach to fractional Lévy processes

A unifying approach to fractional Lévy processes

In this paper we now provide a unifying approach to fractional L´ evy processes. We derive sufficient conditions on the exponent of the kernel function leading to a larger class of processes, especially for the short range dependent case. We will see that the upper bound of the exponent depends on the existing moment of the underlying L´ evy process and the lower bound on the Blumenthal-Getoor index, i.e. the jump activity. In some circumstances only an appropriate choice of the drift component in the L´ evy process ensures the existence of fractional L´ evy processes. In addition we provide both distributional and path properties of the constructed processes, e.g. regularity of the sample paths and semimartingale property, and compare them to fractional Brownian motion. Especially we see that for fractional Brownian motion and fractional L´ evy processes the characteristic quantities, i.e. exponent of the kernel function, exponent in the correlation function, maximal H¨ older exponent and self-similarity index do not stay in the same functional relationship. While for fractional Brownian motion one parameter H is sufficient to describe them all, for fractional L´ evy processes in general we need three parameters, the exponent of the kernel function, the Blumenthal-Getoor index and the maximal existing moment, if it is less than two.
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Isometric embeddings of compact spaces into Banach spaces

Isometric embeddings of compact spaces into Banach spaces

We show the existence of a compact metric space K such that whenever K embeds isometrically into a Banach space Y , then any separable Banach space is linearly isometric to a subspace of Y . We also address the following related question: if a Banach space Y contains an isometric copy of the unit ball or of some special compact subset of a separable Banach space X, does it necessarily contain a subspace isometric to X? We answer positively this question when X is a polyhedral finite-dimensional space, c 0 or  1 . © 2008 Elsevier Inc. All rights reserved.

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Yosida approximations for multivalued stochastic differential equations on Banach spaces via a Gelfand triple

Yosida approximations for multivalued stochastic differential equations on Banach spaces via a Gelfand triple

In this chapter, we recall the necessary fundamentals in the theory of stochas- tic processes. We gather some well-known facts on stochastic processes and define the martingale on general Banach spaces. These statements are valid for c´ adl´ ag processes. Of course, this covers the special case of continuous processes. Furthermore, we introduce the Q-Wiener process as well as the Poisson random measure. In Chapters 4 and 5, these processes will serve as integrators for the stochastic integral. For more details on the theory of stochastic processes, we refer to [Kno05], [IW81], [Pro05], [PR07], [App09].
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Maximal inequalities for Stochastic convolutions driven by compensated Poisson random measures in Banach spaces

Maximal inequalities for Stochastic convolutions driven by compensated Poisson random measures in Banach spaces

Let us finish this Introduction by commenting that the results presented are applicable to non- linear SPDEs, e.g. stochastic Euler Equations. In the case of similar problems with the Gaussian noise, the paper [5] on which to a large extent our current research is based on, was in some sense a byproduct of a previous study by the same authors for stochastic Euler Equations in [6]. It turns out that applications to stochastic Navier-Stokes Equations of our paper even before it’s publication have been found in a recent paper by Fernando et al. [11]. For related results for stochastic reaction diffusion equations obtained by different approach one can consult a paper [23] by Marinelli and R¨ ockner.
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20. Some strong convergence results for Mann and Ishikawa iterative processes in Banach spaces

20. Some strong convergence results for Mann and Ishikawa iterative processes in Banach spaces

Abstract. In this paper, we establish some strong convergence results for Mann and Ishikawa iterative processes in a Banach space setting by employing some general contractive conditions as well as weakening further the conditions on the parameter sequence {𝛼 𝑛 } ⊂ [0, 1]. In addition, in some of our results,

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A note on nonfragmentability of Banach spaces

A note on nonfragmentability of Banach spaces

The notion of fragmentability was originally introduced in [11] as an abstraction of phenomena often encountered, for example, in Banach spaces with the Radon- Nikodym property, in weakly compact subsets of Banach spaces and in the dual of Banach spaces. The notion of σ -fragmentability appeared in [10] in order to extend the study of compact fragmented space to noncompact spaces. It turns out that the question of whether a given Banach space with weak topology is sigma-fragmented by the norm is closely connected with the question of the existence of an equivalent Kadec and locally uniformly convex norm. The reader may refer to [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] for some application of fragmentability and its variants in other topics of Banach spaces.
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Nonlinear classification of Banach spaces

Nonlinear classification of Banach spaces

is that ϕ 1 (t) → ∞ as t → ∞. These maps were introduced by Gromov [Gr1] and are called coarse embeddings. They were introduced in order to study groups as geometric objects. Finitely generated groups are considered as metric spaces under the word distance. In relation to algebraic topology, Yu [Y] proved that a metric space with bounded geometry that coarsely embeds into a Hilbert space satisfies the coarse geometric Novikov conjecture. Later, Kasparov and Yu [KaY] strengthened this result by showing that it is enough for the metric space in question to admit a coarse embedding into a uniformly convex Banach space. Whether this was really a strengthening was not very clear though, because it is not apparent at first sight that there are uniformly convex Banach spaces that do not coarsely embed into a Hilbert space.
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Upper Estimates for Banach Spaces

Upper Estimates for Banach Spaces

characterizations are all based on the relatively recent tools of weakly null trees. One important result in particular for us is a characterization of subspaces of reflexive spaces with an FDD satisfying subsequential V upper block estimates and subse- quential U lower block estimates where V is an unconditional, block stable, and right dominant basic sequence and U is an uncondition, block stable, and left dominant basic sequence [OSZ2]. This characterization when applied to Tzirelson’s spaces was shown to have strong applications to the Szlenk index of reflexive spaces [OSZ3]. Our main result adds to this theory with the following theorem which extends the results in [OSZ2] and [OSZ3] to spaces with separable dual. The notions and concepts used, will be introduced in the next section.
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Nonlinear Quotients of Banach Spaces

Nonlinear Quotients of Banach Spaces

The operator T is called an ε-Fr´ echet derivative of f at x. Such an operator may not be unique, but it is not hard to check that if a Lipschitz quotient map has a point of ε-Fr´ echet differentiability for small enough ε, then any such ε-Fr´ echet derivative is a linear quotient map from X onto Y . Now the question is plain: when does a Lipschitz map f : X → Y have points of ε-Fr´ echet differentiability? It is known that there are points at which f is Gˆ ateaux differentiable provided X is separable and Y has the Radon-Nikod´ ym property (RNP) (see, e.g., Theorem 6.42 in [8]). Additional asymptotic structures are needed for the spaces to prove a similar existence theorem for ε-Fr´ echet derivatives.
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Stochastic nonautonomous Gompertz model with Lévy jumps

Stochastic nonautonomous Gompertz model with Lévy jumps

This paper deals with stochastic nonautonomous Gompertz model with Lévy jumps. To begin with, the existence of a global positive solution and an explicit solution have been derived. In addition, asymptotic moment properties are discussed. Besides, sufficient conditions for extinction, persistence in mean, and weak persistence are obtained. It is proved that the variability of Lévy jumps can affect the asymptotic property of the system.

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A pythagorean approach in Banach spaces

A pythagorean approach in Banach spaces

H } denotes the diameter of H. A Banach space X is said to have normal structure if every bounded, convex subset of X has normal structure. A Banach space X is said to have weak normal structure if each weakly compact convex set K in X that contains more than one point has normal structure. X is said to have uniform normal structure if there exists 0 < c < 1 such that for any subset K as above, there exists x 0 ∈ K such that sup { x 0 −

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