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We obtain sufficient conditions for the existence of conditional densities of functionals of the Poisson process, and expressions of these densities via the Malliavin calculus on Poisson space. These results are applied to enlargement of filtrations on Poisson space with explicit examples of computations via the Clark formula which is presented as a consequence of the Itô formula and the martingale property. The gradient operators of stochastic analysis on Poisson space are classified into three families, and each type of gradient is considered separately. [ABSTRACT FROM AUTHOR]

Using a Clark formula for the predictable representation of random variables in discrete time and adapting the method presented in [Electron. Commun. Probab. 2 (1997) 71–81] in the Brownian case, we obtain a proof of modified and L1 logarithmic Sobolev inequalities for Bernoulli measures. We also prove a bound that improves these inequalities as well as the optimal constant inequality of [J. Funct. Anal. 156 (2) (1998) 347–365]. To cite this article: F. Gao, N. Privault, C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]

We show that for the binomial process (or Bernoulli random walk) the orthogonal functionals constructed in Kroeker, J.P. (1980) "Wiener analysis of functionals of a Markov chain: application to neural transformations of random signals", Biol. Cybernetics 36 , 243-248, [14] for Markov chains can be expressed using the Krawtchouk polynomials, and by iterated stochastic integrals. This allows to construct a chaotic calculus based on gradient and divergence operators and structure equations, and to establish a Clark representation formula. As an application we obtain simple infinite dimensional proofs of covariance identities on the discrete cube. [ABSTRACT FROM AUTHOR]

The compensated Poisson noise is expressed as a composite sum (splitting) of creation and annihilation operators, whose probabilistic interpretation relies on time changes. We construct an Itô table for this decomposition and obtain continuous and discrete time realizations of Lévy processes on the finite difference algebra fd and on sl[sub 2], e.g. the space-time dual of the Poisson process (compensated gamma process), and the continuous binomial process. [ABSTRACT FROM AUTHOR]

Proposes a stochastic calculus with respect to a class ... of anticipating processes which is wider than the class of Skorokhod integral processes. Notation and preliminaries on the Skorokhod integral operator; Extension of the Skorokhod integral as a continuous operator on Sobolev spaces; Conditions for the existence of pathwise integrals.

Discusses a study which examined the gradient of Riemannian manifold in configuration spaces. Details of probability and Poisson measures; Application of differential calculus on configuration space; Sample equation of integrals and stochastic integration.

We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any orders, by combining the use of neural networks with a Monte Carlo branching algorithm. In comparison with other deep learning PDE solvers, it also allows us to check the consistency of the learned neural network function. Numerical experiments presented show that this algorithm can outperform deep learning approaches based on backward stochastic differential equations or the Galerkin method, and provide solution estimates that are not obtained by those methods in fully nonlinear examples. [ABSTRACT FROM AUTHOR]

We propose an extension of the Cox-Ross-Rubinstein (CRR) model based on q-binomial (or Kemp) random walks, with application to default with logistic failure rates. This model allows us to consider time-dependent switching probabilities varying according to a trend parameter on a non-self-similar binomial tree. In particular, it includes tilt and stretch parameters that control increment sizes. Option pricing formulas are written using q-binomial coefficients, and we study the convergence of this model to a Black-Scholes type formula in continuous time. A convergence rate of order O(N-1/2) is obtained. [ABSTRACT FROM AUTHOR]

Provides correction to the article titled 'Skorohod Stochastic Integration With Respect to Non-Adapted Processes on Weiner Space,' published in 'Stochastics and Stochastic Reports.'

We derive convex comparison inequalities for stochastic integrals of the formand, whereare adapted processes with respect to the filtration generated by a standard Brownian motion, andis an independent Brownian motion. Our method uses forward–backward stochastic integration and the Malliavin calculus, and is also applied to jump–diffusion processes. [ABSTRACT FROM AUTHOR]