Abstract not found

density estimation.

Torsion free connections and a notion of curvature are introduced on the infinite dimensional nonlinear configuration space \Gamma of a Riemannian manifold M under a Poisson measure. This allows to state identities of Weitzenbock type and energy identities for anticipating stochastic integral operators. The onedimensional Poisson case itself gives rise to a non-trivial geometry, a de RhamHodge -Kodaira operator, and a notion of Ricci tensor under the Poisson measure. The methods used in this paper have been so far applied to d-dimensional Brownian path groups, and rely on the introduction of a particular tangent bundle and associated damped gradient. Key words: Configuration spaces, Poisson spaces, covariant derivatives, curvature, connections. Mathematics Subject Classification (1991). Primary: 60H07, 58G32, 53B21. Secondary: 53B05, 58A10, 58C35, 60H25. 1

The gradient on a Riemannian manifold X is lifted to the configuration space \Upsilon X on X via a pointwise identity. This entails a norm equivalence that either holds under any probability measure or characterizes the Poisson measures, depending on the tangent space chosen on \Upsilon X . More generally, this approach links carr'e du champ operators on X to their counterparts on \Upsilon X , and also includes structures that do not admit a gradient. Key words: Configuration spaces, Poisson measures, Stochastic analysis. Mathematics Subject Classification (1991): 58G32, 60H07, 60J45, 60J75. 1 Introduction Stochastic analysis under Poisson measures, cf. [5], [6], has been developed in several different directions. This is mainly due to the fact that, unlike on the Wiener space, the gradient on Fock space and the infinitesimal Poisson gradient do not coincide under the identification of the Fock space to the L 2 space of the Poisson process. - The gradient on Fock space is in...

. We use a white noise approach to Malliavin calculus to prove the following white noise generalization of the Clark-Haussmann-Ocone formula F(#) = E[F] + Z T 0 E[D t F|F t ] #W (t)dt Here E[F] denotes the generalized expectation, D t F(#) = dF d# is the (generalized) Malliavin derivative,# is the Wick product and W(t) is 1-dimensional Gaussian white noise. The formula holds for all f # G # # L 2 (), where G # is a space of stochastic distributions and is the white noise probability measure. We also establish similar results for multidimensional Gaussian white noise, for multidimensional Poissonian white noise and for combined Gaussian and Poissonian noise. Finally we give an application to mathematical finance: We compute the replicating portfolio for a European call option in a Poissonian Black & Scholes type market. Journal of Economic Literature index numbers: G12 Mathematics Subject Classification (MSC): 60H40, 60G20 1. Introduction Let B t (#) = B(t, #)...

The gradient and divergence operators of stochastic analysis on Riemannian manifolds are expressed using the gradient and divergence of the flat Brownian motion. By this method we obtain the almost-sure version of several useful identities that are usually stated under expectations. The manifoldvalued Brownian motion and random point measures on manifolds are treated successively in the same framework, and stochastic analysis of the Brownian motion on a Riemannian manifold turns out to be closely related to classical stochastic calculus for jump processes. In the setting of point measures we introduce a damped gradient that was lacking in the multidimensional case. Key words: Stochastic calculus of variations, Brownian motion, random measures, Riemannian manifolds. Mathematics Subject Classification (1991). 60H07, 60H25, 58-99, 58C20, 58G32, 58G99. 1 Introduction The IR d -valued Brownian motion on the Wiener space (W; F W ; ) gathers many properties that are important in stoch...

We study the convergence of Monte Carlo estimators of derivatives when the transition density of the underlying state variables is unknown. Three types of estimators are compared. These are respectively based on Malliavin derivatives, on the covariation with the driving Wiener process, and on finite difference approximations of the derivative. We analyze two different estimators based on Malliavin derivatives. The first one, the Malliavin path estimator, extends the path derivative estimator of Broadie and Glasserman (1996) to general diffusion models. The second one, the Malliavin weight estimator, proposed by Fournié et. al. (1999), is based on an integration by parts argument and generalizes the likelihood ratio derivative estimator. It is shown that for discontinuous payoff functions only the estimators based on Malliavin derivatives attain the optimal convergence rate for Monte Carlo schemes. Estimators based on the covariation or on finite difference approximations are found to converge at slower rates. Their asymptotic distributions are shown to depend on additional second order biases even for smooth payoff functions.

This article is an introduction to Malliavin Calculus for practitioners.

Abstract. We consider an SDE in R m of the type dX(t) = a(X(t))dt + dUt with a Lévy process U and study the problem for the distribution of a solution to be regular in various senses. We do not impose any specific conditions on the Lévy measure of the noise, and this is the main difference between our method and the known methods by J.Bismut or J.Picard. The main tool in our approach is the stochastic calculus of variations for a Lévy process, based on the time-stretching transformations of the trajectories. Three problems are solved in this framework. First, we prove that if the drift coefficient a is non-degenerated in an appropriate sense, then the law of the solution to the Cauchy problem for the initial equation is absolutely continuous, as soon as the Lévy measure of the noise satisfies one of the rather weak intensity conditions, for instance the so-called wide cone condition. Secondly, we provide the sufficient conditions for the density of the distribution of the solution to the Cauchy problem to be smooth in the terms of the family of the so-called order indices of the Lévy measure of the noise (the drift again is supposed to be non-degenerated). At last, we show that an invariant distribution to the initial equation, if exists, possesses a C ∞-density provided the drift is non-degenerated and the Lévy measure of the noise satisfies the wide cone condition.

Abstract. We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations with a polynomially growing quasi-monotone nonlinearity and multiplicative Poisson noise. We also study existence and uniqueness of invariant measures for the associated semigroup in the Markovian case. A key role is played by a new maximal inequality for stochastic convolutions in Lp spaces. 1.