Convolution semigroups of states

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C ₀-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.     

Regularity theory for fully nonlinear integro‐differential equations

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C^{1, α} regularity for general fully nonlinear integro‐differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. © 2008 Wiley Periodicals, Inc.     

On quantum stochastic differential equations

Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional initial space or, more generally, for coefficients satisfying a finite localisability condition. Necessary and sufficient conditions are obtained for a conjugate pair of quantum stochastic cocycles on a finite-dimensional operator space to strongly satisfy such a quantum stochastic differential equation. This gives an alternative approach to quantum stochastic convolution cocycles on a coalgebra.     

Remark on pathwise uniqueness of stochastic differential equations driven by Lévy processes

Consider real-valued processes determined by stochastic differential equations driven by Lévy processes. The jump parts of the driving Lévy process are not always α-stable ones, nor symmetric ones. In the present article, we shall study the pathwise uniqueness of the solutions to the stochastic differential equations under the conditions on the coefficients that the diffusion and the jump terms are Hölder continuous, while the drift one is monotonic. Our approach is based on Gronwall’s inequality.     

Random attractors for stochastic differential equations driven by two-sided Lévy processes

Abstract

In this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor.     

Non-Standard Skorokhod Convergence of Lévy-Driven Convolution Integrals in Hilbert Spaces

We study the convergence in probability in the non-standard M₁ Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type to a process driven by a Lévy process L. In Banach spaces, we introduce strong, weak. and product modes of -convergence, prove a criterion for the -convergence in probability of stochastically continuous càdlàg processes in terms of the convergence in probability of the finite dimensional marginals and a good behavior of the corresponding oscillation functions, and establish criteria for the convergence in probability of Lévy driven stochastic convolutions. The theory is applied to the infinitely dimensional integrated Ornstein–Uhlenbeck processes with diagonalizable generators.     

Noncommutative dynamics and generalized master equations

We review the basic concepts of quantum probability and stochastics using the universal Itô B*-algebra approach. The main notions and results of classical and quantum stochastics are reformulated in this unifying approach. The general Lévy process is defined in terms of the modular B*-Itô algebra, and the corresponding quantum stochastic master equation on the predual space of theW*-algebra is derived as a noncommutative version of the Zakai equation driven by the process. This is done by a noncommutative analog of the Girsanov transformation, which we introduce here in full generality.     

M-Estimation for Discretely Observed Ergodic Diffusion Processes with Infinitely Many Jumps

We study parametric inference for multidimensional stochastic differential equations with jumps from some discrete observations. We consider a case where the structure of jumps is mainly controlled by a random measure which is generated by a Lévy process with a Lévy measure f_θ(z)dz, and we admit the case ∫ f_θ(z)dz = ∞ in which infinitely many small jumps occur even in any finite time intervals. We propose an estimating function under this complicated situation and show the consistency and the asymptotic normality. Although the estimators in this paper are not completely efficient, the method can be applied to comparatively wide class of stochastic differential equations, and it is easy to compute the estimating equations. Therefore, it may be useful in applications. We also present some simulation results for some simple models. Final version 25 December 2004     

Carr's randomization for American options in regime-switching models

In the paper, we solve the pricing problem for American put-like options in Markov-modulated Lévy models. The early exercise boundaries and prices are calculated using a generalization of Carr's randomization for regime-switching models. An efficient iteration pricing procedure is developed. The computational time is of order m², where m is the number of states, and of order m, if the parallel computations are allowed. The payoffs, riskless rates and class of Lévy processes may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modelled as finite-state Markov chains. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sub-Markovian C 0-Semigroups Generated by Fractional Laplacian with Gradient Perturbation

We present a sufficient condition for fractional Laplacian with gradient perturbation to generate a sub-Markovian C ₀-semigroup on ${L^1(\mathbb{R}^d, dx)}$ . The condition also yields the ultracontractivity of the semigroup and an upper on-diagonal estimate of the associated transition kernel. Based on the subordination technique, the extension to general pure jump Lévy process with gradient perturbation is studied. As a direct application, we obtain sufficient conditions for the strong Feller property of stochastic differential equations driven by additive Lévy process.     

Simulation of stochastic integrals with respect to Lévy processes of type G

We study the simulation of stochastic processes defined as stochastic integrals with respect to type G Lévy processes for the case where it is not possible to simulate the type G process exactly. The type G Lévy process as well as the stochastic integral can on compact intervals be represented as an infinite series. In a practical simulation we must truncate this representation. We examine the approximation of the remaining terms with a simpler process to get an approximation of the stochastic integral. We also show that a stochastic time change representation can be used to obtain an approximation of stochastic integrals with respect to type G Lévy processes provided that the integrator and the integrand are independent.     

Periodic averaging method for impulsive stochastic differential equations with Lévy noise

The purpose of this paper is to present a periodic averaging method for impulsive stochastic differential equations with Lévy noise under non-Lipschitz condition. It is shown that the solutions of impulsive stochastic differential equations with Lévy noise converge to the solutions of the corresponding averaged stochastic differential equations without impulses     

Inhomogeneous Lévy Processes in Lie Groups and Homogeneous Spaces

We obtain a representation of an inhomogeneous Lévy process in a Lie group or a homogeneous space in terms of a drift, a matrix function and a measure function. Since the stochastic continuity is not assumed, our result generalizes the well-known Lévy–Itô representation for stochastic continuous processes with independent increments in ? ^d and its extension to Lie groups.     

Derivative Formula and Harnack Inequality for SDEs Driven by Lévy Processes

By using lower bound conditions of the Lévy measure, derivative formulae and Harnack inequalities are derived for linear stochastic differential equations driven by Lévy processes. As applications, explicit gradient estimates and heat kernel inequalities are presented. As byproduct, a new Girsanov theorem for Lévy processes is derived.     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

Successful couplings for a class of stochastic differential equations driven by Lévy processes

By constructing proper coupling operators for the integro-differential type Markov generator, we establish the existence of a successful coupling for a class of stochastic differential equations driven by Lévy processes. Our result implies a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying Markov semigroups, and it is sharp for Ornstein-Uhlenbeck processes driven by ??-stable Lévy processes.     

Local Malliavin calculus for Lévy processes and applications

The Malliavin derivative operator for the Poisson process introduced by Carlen and Pardoux [Differential calculus and integration by parts on a Poisson space, in Stochastics, Algebra and Analysis in Classical and Quantum Dynamics, S. Albeverio et al. (eds), Kluwer, Dordrecht, 1990, pp. 63–73] is extended to Lévy processes. It is a true derivative operator (in the sense that it satisfies the chain rule), and we deduce a sufficient condition for the absolute continuity of functionals of the Lévy process. As an application, we analyse the absolute continuity of the law of the solution of some stochastic differential equations with jumps.     

Maximal Inequalities for Fractional Lévy and Related Processes

In this article we study processes that are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding “convoluted martingale” is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.     

Representations of homogeneous quantum Lévy fields

We study homogeneous quantum Lévy processes and fields with independent additive increments over a noncommutative *-monoid. These are described by infinitely divisible generating state functionals, invariant with respect to an endomorphic injective action of a symmetry semigroup. A strongly covariant GNS representation for the conditionally positive logarithmic functionals of these states is constructed in the complex Minkowski space in terms of canonical quadruples and isometric representations on the underlying pre-Hilbert field space. This is of much use in constructing quantum stochastic representations of homogeneous quantum Lévy fields on Itô monoids, which is a natural algebraic way of defining dimension free, covariant quantum stochastic integration over a space-time indexing set.