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Let Φ′ denotes the strong dual of a nuclear space Φ. In this article, we introduce sufficient conditions for the convergence uniform on compacts in probability for a sequence of Φ′-valued processes with continuous or càdlàg paths. We illustrate the usefulness of our results by considering two applications to stochastic analysis. First, we introduce a topology on the space of Φ′-valued semimartingales which are good integrators and show that this topology is complete and that the stochastic integral mapping is continuous on the integrators. Second, we introduce sufficient conditions for the convergence uniform on compacts in probability of the solutions to a sequence of stochastic evolution equations; we do this both in the case of linear equations with semimartingale noise and nonlinear equations with multiplicative cylindrical martingale-valued measure noise. [ABSTRACT FROM AUTHOR]

The objective of this article is to introduce and study Itô type stochastic integrals with respect to tempered fractional Brownian motion (TFBM) of Hurst index H ∈ (1 2 , 1) and tempering parameter λ > 0 , by using the Wick product. The main tools are fractional calculus and Malliavin calculus. The Itô formula for this stochastic integral is established for the Itô type processes driven by TFBM. Based on this new Itô formula, we analyze the stability of stochastic differential equations driven by TFBM in the sense of p -th moment. A numerical example is given to illustrate our stability results. [ABSTRACT FROM AUTHOR]

Let (V t) t ∈ [ 0 , L ] be a stochastic process with quadratic variation on a probability space (Ω , F , P) and Q (∋ 0) a dense subset of [ 0 , L ] , where [ 0 , L ] is regarded as the infinite interval [ 0 , ∞) when L = ∞. First, we introduce the L 0 (Ω) -module D Q (V) of V-differentiable noncausal processes on Q and V-derivative operator D V , Q = d Q d Q V defined on D Q (V) , which enjoys the modularity: D V , Q (α X + β Y) = α D V , Q X + β D V , Q Y for any X , Y ∈ D Q (V) and α , β ∈ L 0 (Ω). Second, we show that the class Q V , Q = { X ∈ D Q (V) | d [ X ] Q , t d [ V ] Q , t = | d Q X t d Q V t | 2 } forms an L 0 (Ω) -module, where [ ] Q , t stands for the quadratic variation on Q. As a result, we have the isometry: 〈 X , Y 〉 Q , t = 〈 D V , Q X , D V , Q Y 〉 L 2 ([ 0 , t ] , [ V ]) for any X , Y ∈ Q V , Q , where 〈 , 〉 Q , t stands for the quadratic covariation on Q. Finally, we present universal properties and examples of the stochastic integral I with D V , Q ∘ I = i d D (I) . This result is essentially used for solving the identification problem from the stochastic Fourier coefficients. [ABSTRACT FROM AUTHOR]

We consider extreme value analysis for independent but nonidentically distributed observations. In particular, the observations do not share the same extreme value index. Assuming continuously changing extreme value indices, we provide a nonparametric estimate for the functional extreme value index. Besides estimating the extreme value index locally, we also provide a global estimator for the trend and its joint asymptotic theory. The asymptotic theory for the global estimator can be used for testing a prespecified parametric trend in the extreme value indices. In particular, it can be applied to test whether the extreme value index remains at a constant level across all observations. [ABSTRACT FROM AUTHOR]

In this article, we provide predictable and chaotic representations for Itô-Markov additive processes X. Such a process is governed by a finite-state continuous time Markov chain J which allows one to modify the parameters of the Itô-jump process (in so-called regime switching manner). In addition, the transition of J triggers the jump of X distributed depending on the states of J just prior to the transition. This family of processes includes Markov modulated Itô-Lévy processes and Markov additive processes. The derived chaotic representation of a square-integrable random variable is given as a sum of stochastic integrals with respect to some explicitly constructed orthogonal martingales. We identify the predictable representation of a square-integrable martingale as a sum of stochastic integrals of predictable processes with respect to Brownian motion and power-jumps martingales related to all the jumps appearing in the model. This result generalizes the seminal result of Jacod-Yor and is of importance in financial mathematics. The derived representation then allows one to enlarge the incomplete market by a series of power-jump assets and to price all market-derivatives. [ABSTRACT FROM AUTHOR]

In statistical inference on the drift parameterain the fractional Brownian motionWHtwith the Hurst parameterH∈ (0, 1) with a constant driftYHt=at+WHt, there is a large number of options how to do it. We may, for example, base this inference on the properties of the standard normal distribution applied to the differences between the observed values of the process at discrete times. Although such methods are very simple, it turns out that more appropriate is to use inverse methods. Such methods can be generalized to non constant drift. For the hypotheses testing about the drift parametera, it is more proper to standardize the observed process, and to use inverse methods based on the first exit time of the observed process of a pre-specified interval until some given time. These procedures are illustrated, and their times of decision are compared against the direct approach. Other generalizations are possible when the random part is a symmetric stochastic integral of a known, deterministic function with respect to fractional Brownian motion. [ABSTRACT FROM PUBLISHER]

In this article, we introduce a Lévy analogue of the spatially homogeneous Gaussian noise of [5], and we construct a stochastic integral with respect to this noise. The spatial covariance of the noise is given by a tempered measure μ on, whose density is given byfor a symmetric complex-valued functionh. Without assuming that the Fourier transform of μ is a non-negative function, we identify a large class of integrands with respect to this noise. As an application, we examine the linear stochastic heat and wave equations driven by this type of noise. [ABSTRACT FROM PUBLISHER]

In this paper, we present an elementary and self-contained proof of the stochastic Fubini theorem, which states that one can interchange a Lebesgue integral and a stochastic integral. The integrability conditions we use are weaker and more natural than the usual conditions in the literature. In particular, we do not need integrability in , and we use -integrability instead of -integrability in the additional parameter. [ABSTRACT FROM AUTHOR]

Let X, Y be local martingales such that Y is differentially subordinate to X. The article contains the proof of the sharp estimates As an application, we establish sharp inequalities for stochastic integrals with respect to Brownian motion in ℝ d , derive some maximal bounds for harmonic functions and Riesz system and present tight estimates for a stopped three-dimensional Bessel process. [ABSTRACT FROM AUTHOR]

The goal of this article is to introduce strike Asian options on stochastic average and to price them using a variable reduction technique. The problem of solving the associated Black–Scholes equation is reduced to finding the heat kernel of the operator . This is done by two approaches: using a geometrical method and then applying the van Vleck formula; using a moments method to get a double infinite series formula involving Hermite polynomials. [ABSTRACT FROM PUBLISHER]

In this article, we consider a Levy process under condition Γ1 = x where {Γk} is a sequence of arrivals of a Poisson process with unit arrival rate. We show that under condition Γ1 = x Levy process can be decomposed into a simple process and a Levy process. These two processes are independent. As an application of this decomposition we consider α-stable Levy processes. We give a closed form of the process [image omitted] and evaluate its expected value where Z is an α-stable Levy process with 0 < α < 2. We show that X(t) = 0 for 1 < α < 2 and this expectation is equal infinity for α ≤ 1. [ABSTRACT FROM AUTHOR]

In this paper we provide a discrete approximation for the stochastic integral with respect to the fractional Brownian motion of Hurst index H>1/2 defined in terms of the divergence operator. To determine the suitable class of integrands for which the approximation holds, we also investigate the relations among the spaces of Malliavin differentiable processes in the fractional and standard case. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Ito-type formula for the process X. [ABSTRACT FROM AUTHOR]

In this paper, we consider complex-valued Brownian motion with p-adic time index and the associated abstract Wiener space. We define symmetric stochastic integrals with respect to p-adic Brownian motion. We also provide a sufficient condition for the existence of symmetric stochastic integrals and present a relation to the adjoint of the Malliavin derivatives. [ABSTRACT FROM AUTHOR]

We study several properties of the sub-fractional Brownian motion (fBm) introduced by Bojdecki et al. related to those of the fBm. This process is a self-similar Gaussian process depending on a parameter H ∈ (0, 2) with non stationary increments and is a generalization of the Brownian motion (Bm). The strong variation of the indefinite stochastic integral with respect to sub-fBm is also discussed. [ABSTRACT FROM AUTHOR]

In many practical situations, such as cross-sectional sampling, the observed random sample suffers from selection bias in the sense that it does not represent the target population. This paper considers the problem of estimating the target survivor function when the data for two independent competing risks having proportional hazards is observed under such a selection bias. The large sample behavior of the proposed estimator is studied through martingale representation and stochastic integrals. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and research the vague convergence of semimartingale random measures in distribution. The conditions are provided for the vague convergence of semimartingale random measures and the convergence of stochastic integrals with respect to semimartingale random measures in distribution. [ABSTRACT FROM AUTHOR]

In connection with a symmetric α stable random measure Φ on a measurable space (F, F) with values in Rd a complete metric space of symmetric finite measures on Sd-1 is constructed, and is employed to characterize the law of Φ by a unique positive measure on F and a unique function on F x Rd. The stochastic integral ∫FfdΦ is also defined for certain d x d matrix valued functions f, which for α = 2 reduces to the Wiener-Masani integral. [ABSTRACT FROM AUTHOR]

We consider a filtered probability space with a standard Brownian motion W, a simple Poisson process N with constant intensity γ > 0, and we consider the process Y such that Yo ε R and dYt=atdt+σtdWt+γ,dNt, t>0, (1) where a, σ are predictable bounded stochastic processes, and γ is a predictable process which is bounded away from zero. A discrete record of n + 1 observations Y0, Yt1, Ytn-1, Ytn) is available, with ti = ih. Using such observations, we construct estimators of Nti, i = 1, ..., n, γ and γθj, where θj are the instants of jump within [0, nh]. They are consistent and asymptotically controlled when the number of observations increases and the step h tends to zero. [ABSTRACT FROM AUTHOR]

Motivated by the theory of bond markets, we consider an infinite assets model driven by marked point process and Wiener process. The self-financed wealth processes are defined by using measure-valued strategies. Going further on the works of Bjork et al. ["Bond market structure in the presence of marked point processes", Mathematical Finance , 7 (1997a) pp. 211-239; "Towards a general theory of bond markets", Finance and Stochastics , 1 (1997b) pp. 141-174] who focus on the existence of martingale measures and market completeness questions, we study here the incompleteness case. Our main result is a predictable decomposition theorem for supermartingales in this infinite assets model context. The concept of approximate wealth processes is introduced, and we show in an example that the space of measure-valued strategies is not complete with respect to the semimartingale topology. As in the case of stock markets, one can then derive a dual representation of the super-replication cost and study the problem of utility maximization by duality methods. [ABSTRACT FROM AUTHOR]

We develop a stochastic calculus for the fractional Brownian motion with Hurst parameter H >1/2 using the techniques of the Malliavin calculus. We establish estimates in L p , maximal inequalities and a continuity criterion for the stochastic integral. Finally, we derive an Itô's formula for integral processes. [ABSTRACT FROM AUTHOR]

We approximate certain stochastic integrals, typically appearing in Stochastic Finance, by stochastic integrals over integrands, which are path-wise constant within deterministic, but not necessarily equidistant, time intervals. We ask for rates of convergence if the approximation error is considered in L 2 . In particular, we show that by using non-equidistant time nets, in contrast to equidistant time nets, approximation rates can be improved considerably. [ABSTRACT FROM AUTHOR]