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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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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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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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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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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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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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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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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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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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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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.

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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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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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,

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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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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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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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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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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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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