The Heston stochastic volatility model in Hilbert space

We extend the Heston stochastic volatility model to a Hilbert space framework. The tensor Heston stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein–Uhlenbeck process with itself. The volatility process is then defined by a Cholesky decomposition of the variance process. We define a Hilbert-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional and covariance operator of this process. This process is then applied to the modeling of forward curves in energy and commodity markets. Finally, we compute the dynamics of the tensor Heston volatility model when the generator is bounded, and study its projection down to the real line for comparison with the classical Heston dynamics.     

Local time and Tanaka formula for a multitype Dawson–Watanabe superprocess

In [3] Dynkin defined the local time of a continuous superprocess as a stochastic integral and gave a criterion for existence of local time. Here we prove that the conditions in Dynkin's existence criterion are satisfied by the multitype Dawson–Watanabe superprocess, and give a Tanaka formula‐like representation of the local time which is used to show that the occupation measure of the multitype superprocess is absolutely continuous with respect to an appropriate reference measure, and that the corresponding density coincides a.s. with the local time. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An extension of the Wong-Zakai theorem for stochastic evolution equations in Hilbert spaces

In this paper we examine an approximation theorem of the Wong–Zakai type for stochastic evolution equations in a Hilbert space with the noise being the generalized derivative of the Wiener process with values in another Hilbert space. As a consequence of the approximation of the Wiener process we get in the limit equation the Ito correction term for the infinite dimensional case. The obtained result includes the case of stochastic delay equations. The uniqueness and existence of solutions are guaranteed by known theorems for the mild solutions     

Self-intersection Local Time of Planar Brownian Motion Based on a Strong Approximation by Random Walks

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka?CRosen?CYor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-intersection local time is more transparent and elementary than other existing ones.     

Self-intersection local times and collision local times of bifractional Brownian motions

In this paper, we consider the local time and the self-intersection local time for a bifractional Brownian motion, and the collision local time for two independent bifractional Brownian motions. We mainly prove the existence and smoothness of the self-intersection local time and the collision local time, through the strong local nondeterminism of bifractional Brownian motion, L2 convergence and Chaos expansion.     

Approximation of Hopkins equation in Hilbert space and its application in polarized illumination modelling

In this article, we derive an analytical approximation to compute the aerial images on general geometric domains. Under the framework in Hilbert space, we develop a general approximation model for aerial image intensity, which is particularly valid for the computation of the aerial image with polarized illumination source. Our mathematical result is a generalization of classical Mercer's theorem provided that the integral kernel is Hilbert–Schmidt. By applying Synopsys's simulation tool Progen, we demonstrate the effectiveness of our proposed approximation with polarized illumination sources (65 nm and below). Since our approach setting is quite general, the proposed approximation in this article is suitable for modification in various applications.     

A Series Approach to Stochastic Differential Equations with Infinite Dimensional Noise

This paper considers semilinear stochastic differential equations in Hilbert spaces with Lipschitz nonlinearities and with the noise terms driven by sequences of independent scalar Wiener processes (Brownian motions). The interpretation of such equations requires a stochastic integral. By means of a series of Itô integrals, an elementary and direct construction of a Hilbert space valued stochastic integral with respect to a sequence of independent scalar Wiener processes is given. As an application, existence and strong and weak uniqueness for the stochastic differential equation are shown by exploiting the series construction of the integral.     

Bivariate occupation measure dimension of multidimensional processes

Bivariate occupation measure dimension is a new dimension for multidimensional random processes. This dimension is given by the asymptotic behavior of its bivariate occupation measure. Firstly, we compare this dimension with the Hausdorff dimension. Secondly, we study relations between these dimensions and the existence of local time or self-intersection local time of the process. Finally, we compute the local correlation dimension of multidimensional Gaussian and stable processes with local Hölder properties and show it has the same value that the Hausdorff dimension of its image have. By the way, we give a new a.s. convergence of the bivariate occupation measure of a multidimensional fractional Brownian or particular stable motion (and thus of a spatial Brownian or Lévy stable motion).     

Upper and lower bounds for the Hilbert–Schmidt norm of a potential operator

Archiv der Mathematik - We give upper and lower bounds for the Hilbert–Schmidt norm of logarithmic potential on a planar domain in terms of its area and inradius.     

Intersection Local Times as Generalized White Noise Functionals

For any dimension we present the expansions of Brownian motion self-intersection local times in terms of multiple Wiener integrals. Suitably subtracted, they exist in the sense of generalized white noise functionals; their kernel functions are given in closed (and remarkably simple) form.     

Absolute continuity of the law of an infinite dimensional Wiener functional with respect to the Wiener probability

Summary In this paper we study conditions ensuring that the law of aC([0, 1])-valued functional defined on an abstract Wiener space is absolutely continuous with respect to the Wiener measure onC([0,1]). These conditions extend those established byP. Malliavin [12, 13] for finite-dimensional Wiener functionals, and those of [15] for Hilbert-valued functionals.     

An approximation theorem of wong-zakai type for nonlinear stochastic partial differential equations

We present an extension of the Wong-Zakai approximation theorem for nonlinear 984 given by the Wiener process and a martingale. By approximating these disturbances we obtain in the limit equation the Ito correction term for the infinite dimensional case. Such form of the correction term connected with the Wiener process was proved in the author's papers [21] and [22], where the approximation theorem for semilinear stochastic evolution equations in Hilbert spaces was studied. Our model here is similar as the one considered by Pardoux [17]     

Optimal Approximation of the Second Iterated Integral of Brownian Motion

Abstract

In this article, a theorem is proved that describes the optimal approximation (in the L ²(?)-sense) of the second iterated integral of a standard two-dimensional Wiener process, W, by a function of finitely many elements of the Gaussian Hilbert space generated by W. This theorem has some interesting corollaries: First of all, it implies that Euler's method has the optimal rate of strong convergence among all algorithms that depend solely on linear functionals of the Wiener process, W; second, it shows that the approximation of the second iterated integral based on Karhunen–Loève expansion of the Brownian bridge is asymptotically optimal.     

Third order trace formula

In J. Funct. Anal. 257 (2009) 1092–1132, Dykema and Skripka showed the existence of higher order spectral shift functions when the unperturbed self-adjoint operator is bounded and the perturbation is Hilbert–Schmidt. In this article, we give a different proof for the existence of spectral shift function for the third order when the unperturbed operator is self-adjoint (bounded or unbounded, but bounded below).     

On the regularity of the local times of a class of symmetric lévy processes

We provevia Dynkin's isomorphism theorem, that spatial trajectories of local times of a class of symmetric Lévy processes, with regularly varying Lévy exponent ψ at infinity, belong to a class of Besov spaces. Our result generalizes the case of symmetric stable Lévy processes treated in [5]     

Regularity of local times of random fields

In this paper, we study the fractional smoothness of local times of general processes starting from the occupation time formula, and obtain the quasi-sure existence of local times in the sense of the Malliavin calculus. This general result is then applied to the local times of N-parameter d-dimensional Brownian motions, fractional Brownian motions and the self-intersection local time of the 2-dimensional Brownian motion, as well as smooth semimartingales.     

Some of my favorite results with Endre Csáki and Pál Révész - A survey

Summary Some topics of our twenty some years of joint work is discussed. Just to name a few; joint behavior of the maximum of the Wiener process and its location, global and local almost sure limit theorems, strong approximation of the planar local time difference, a general Strassen type theorem, maximal local time on subsets.     

Integral representation of renormalized self-intersection local times

In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1). As a consequence, we derive the existence of some exponential moments for this random variable.     

Stationary distributions of second order stochastic evolution equations with memory in Hilbert spaces

In this paper, we consider stationarity of a class of second-order stochastic evolution equations with memory, driven by Wiener processes or Lévy jump processes, in Hilbert spaces. The strategy is to formulate by reduction some first-order systems in connection with the stochastic equations under investigation. We develop asymptotic behavior of dissipative second-order equations and then apply them to time delay systems through Gearhart–Prüss–Greiner’s theorem. The stationary distribution of the system under consideration is the projection on the first coordinate of the corresponding stationary results of a lift-up stochastic system without delay on some product Hilbert space. Last, two examples of stochastic damped wave equations with memory are presented to illustrate our theory.