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We derive normal approximation bounds by the Stein method for stochastic integrals with respect to a Poisson random measure over ℝd, d ≥ 2. This approach relies on third cumulant Edgeworth-type expansions based on derivation operators defined by the Malliavin calculus for Poisson random measures. The use of third cumulants can exhibit faster convergence rates than the standard Berry- Esseen rate for some sequences of Poisson stochastic integrals. [ABSTRACT FROM AUTHOR]

Bernstein processes over a finite time interval are simultaneously forward and backward Markov processes with arbitrarily fixed initial and terminal probability distributions. In this paper, a large deviation principle is proved for a family of Bernstein processes (depending on a small parameter ħ which is called the Planck constant) arising naturally in Euclidean quantum physics. The method consists in nontrivial Girsanov transformations of integral forms, suitable equivalence forms for large deviations and the (local and global) estimates on the parabolic kernel of the Schrödinger operator. [ABSTRACT FROM AUTHOR]

We derive closed-form analytical approximations in terms of series expansions for option prices and implied volatilities in a 2-hypergeometric stochastic volatility model with correlated Brownian motions. As in Han et al. (2013), these expansions allow us to recover the well-known skew and smile phenomena on implied volatility surfaces, depending on the values of the correlation parameter. [ABSTRACT FROM AUTHOR]

We derive conditional Gaussian type identities of the form E [ exp ⁡ ( i ∫ 0 T u t d B t ) | ∫ 0 T | u t | 2 d t ] = exp ⁡ ( − 1 2 ∫ 0 T | u t | 2 d t ) , for Brownian stochastic integrals, under conditions on the process ( u t ) t ∈ [ 0 , T ] specified using the Malliavin calculus. This applies in particular to the quadratic Brownian integral ∫ 0 t A B s d B s under the matrix condition A † A 2 = 0 , using a characterization of Yor [6] . [ABSTRACT FROM AUTHOR]

Based on a new multiplication formula for discrete multiple stochastic integrals with respect to non-symmetric Bernoulli random walks, we extend the results of Nourdin et al. (2010) on the Gaussian approximation of symmetric Rademacher sequences to the setting of possibly non-identically distributed independent Bernoulli sequences. We also provide Poisson approximation results for these sequences, by following the method of Peccati (2011). Our arguments use covariance identities obtained from the Clark-Ocone representation formula in addition to those usually based on the inverse of the Ornstein-Uhlenbeck operator. [ABSTRACT FROM AUTHOR]

We derive joint factorial moment identities for point processes with Papangelou intensities. Our proof simplifies previous combinatorial approaches to the computation of moments for point processes. We also obtain new explicit sufficient conditions for the distributional invariance of point processes with Papangelou intensities under random transformations. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the distribution properties of the binomial , where is a gamma random variable. We show in particular that is infinitely divisible for all and , and that for the second order polynomial is a generalized gamma convolution whose Thorin density and Wiener–gamma integral representation are computed explicitly. As a byproduct we deduce that fourth order multiple Wiener integrals are in general not infinitely divisible. [Copyright &y& Elsevier]

Abstract: We propose a formula for the computation of the moments of all orders of Itô and Skorohod stochastic integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. Some characterizations of Gaussian distributions for stochastic integrals are recovered as a consequence. [Copyright &y& Elsevier]

Abstract: Given a divergence operator δ on a probability space such that the law of is infinitely divisible with characteristic exponent we derive a family of Laplace transform identities for the derivative when u is a non-necessarily adapted process. These expressions are based on intrinsic geometric tools such as the Carleman–Fredholm determinant of a covariant derivative operator and the characteristic exponent (0.1), in a general framework that includes the Wiener space, the path space over a Lie group, and the Poisson space. We use these expressions for measure characterization and to prove the invariance of transformations having a quasi-nilpotent covariant derivative, for Gaussian and other infinitely divisible distributions. [Copyright &y& Elsevier]

Abstract: We investigate explosion in finite time of one-dimensional semilinear equations of the form with initial value , where is positive and , are constants. In the free case we provide conditions on φ under which any positive nontrivial solution is non-global. In the case and , , which includes in the special case the equation we use the Feynman–Kac formula for Bessel processes to give conditions on the equation parameters ensuring finite-time blowup and existence of nontrivial positive global solutions. [Copyright &y& Elsevier]

Abstract: We present a moment identity on the Poisson space that extends the Skorohod isometry to arbitrary powers of the Skorohod integral. Applications of this identity are given to the invariance of Poisson measures under intensity preserving random transformations. To cite this article: N. Privault, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: The goal of this paper is to obtain probabilistic representation formulas that are suitable for the numerical computation of the (possibly non-continuous) density functions of infima of reserve processes commonly used in insurance. In particular we show, using Monte Carlo simulations, that these representation formulas perform better than standard finite difference methods. Our approach differs from Malliavin probabilistic representation formulas which generally require more smoothness on random variables and entail the continuity of their density functions. [Copyright &y& Elsevier]

Abstract: In the framework of a nonparametric functional estimation for the drift of a Brownian motion we construct Stein type estimators of the form which are superefficient when is a superharmonic functional on the Wiener space for the Malliavin derivative D. To cite this article: N. Privault, A. Réveillac, C. R. Acad. Sci. Paris, Ser. I 343 (2006). [Copyright &y& Elsevier]

Abstract: We investigate finite-time blow-up and stability of semilinear partial differential equations of the form , , , where Γ is the generator of the standard gamma process and , , are constants. We show that any initial value satisfying , , for some positive constants , , , yields a non-global solution if . If , where , and , then the solution is global and satisfies , for some constant . This complements the results previously obtained in [M. Birkner et al., Proc. Amer. Math. Soc. 130 (2002) 2431; M. Guedda, M. Kirane, Bull. Belg. Math. Soc. Simon Stevin 6 (1999) 491; S. Sugitani, Osaka J. Math. 12 (1975) 45] for symmetric α-stable generators. Systems of semilinear PDEs with gamma generators are also considered. [Copyright &y& Elsevier]

The non-commutative Malliavin calculus on the Heisenberg–Weyl algebra is extended to the affine algebra. A differential calculus and a non-commutative integration by parts are established. As an application we obtain sufficient conditions for the smoothness of Wigner-type laws of non-commutative random variables with gamma or continuous binomial marginals. [Copyright &y& Elsevier]

Let θ be a Young function. Using properties of the Laplace and Legendre transforms, it is shown that white noise measures in the dual of a test function space of θ-exponential growth satisfy an exponential decay property with rate θ. An application to stochastic differential equations is given. To cite this article: H. Ouerdiane, N. Privault, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]

The non-commutative Malliavin calculus on the Heisenberg–Weyl algebra (see (i) C. R. Acad. Sci. Paris, Se´r. I 328 (11) (1999) 1061–1066, (ii) Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (1) (2001) 11–38) is extended to the affine algebra. A differential calculus is established, which generalizes the corresponding commutative integration by parts formulas. As an application we obtain sufficient conditions for the smoothness of Wigner type laws of non-commutative random variables with gamma and continuous binomial marginals. To cite this article: U. Franz et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003). [Copyright &y& Elsevier]

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]