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We present an algorithm for the numerical solution of ordinary differential equations by random enumeration of the Butcher trees used in the implementation of the Runge–Kutta method. Our Monte Carlo scheme allows for the direct numerical evaluation of an ODE solution at any given time within a certain interval, without iteration through multiple time steps. In particular, this approach does not involve a discretization step size, and it does not require the truncation of Taylor series. [ABSTRACT FROM AUTHOR]

We derive Wasserstein distance bounds between the probability distributions of a stochastic integral (Itô) process with jumps (X t) t ∈ [ 0 , T ] and a jump-diffusion process (X t ∗) t ∈ [ 0 , T ] . Our bounds are expressed using the stochastic characteristics of (X t) t ∈ [ 0 , T ] and the jump-diffusion coefficients of (X t ∗) t ∈ [ 0 , T ] evaluated in X t , and apply in particular to the case of different jump characteristics. Our approach uses stochastic calculus arguments and L p integrability results for the flow of stochastic differential equations with jumps, without relying on the Stein equation. [ABSTRACT FROM AUTHOR]

We derive normal approximation bounds in the Wasserstein distance for sums of generalized U-statistics, based on a general distance bound for functionals of independent random variables of arbitrary distributions. Those bounds are applied to normal approximation for the combined weights of subgraphs in the Erdős–Rényi random graph, extending the graph counting results of Barbour et al. (A central limit theorem for decomposable random variables with applications to random graphs, J. Combin. Theory Ser. B 47(2) (1989), pp. 125–145) to the setting of weighted graphs. Our approach relies on a general stochastic analytic framework for functionals of independent random sequences. [ABSTRACT FROM AUTHOR]

We extend the construction of equilibria for linear-quadratic and mean-variance portfolio problems available in the literature to a large class of mean-field time-inconsistent stochastic control problems in continuous time. Our approach relies on a time discretization of the control problem via n-person games, which are characterized via the maximum principle using backward stochastic differential equations. The existence of equilibria is proved by applying weak convergence arguments to the solutions of n-person games. A numerical implementation is provided by approximating n-person games using finite Markov chains. [ABSTRACT FROM AUTHOR]

We derive a characterization of equilibrium controls in continuous-time, time-inconsistent control (TIC) problems using the Malliavin calculus. For this, the classical duality analysis of adjoint BSDEs is replaced with the Malliavin integration by parts. This results into a necessary and sufficient maximum principle which is applied to a linear-quadratic TIC problem, recovering previous results obtained by duality analysis in the mean-variance case, and extending them to the linear-quadratic setting. We also show that our results apply beyond the linear-quadratic case by treating the generalized Merton problem. [ABSTRACT FROM AUTHOR]

We construct unbiased estimators for the distribution of the number of points inside random stopping sets based on a Poisson point process. Our approach is based on moment identities for stopping sets, showing that the random count of points inside the complement S̅ of a stopping set S has a Poisson distribution conditionally to S. The proofs do not require the use of set-indexed martingales, and our estimators have a lower variance when compared to standard sampling. Numerical simulations are presented for examples such as the convex hull and the Voronoi flower of a Poisson point process, and their complements. [ABSTRACT FROM AUTHOR]

This paper considers a general stochastic SIR epidemic model driven by a multidimensional Lévy jump process with heavy-tailed increments and possible correlation between noise components. In this framework, we derive new sufficient conditions for disease extinction and persistence in the mean. Our method differs from previous approaches by the use of Kunita’s inequality instead of the Burkholder–Davis–Gundy inequality for continuous processes, and allows for the treatment of infinite Lévy measures by the definition of new threshold R ¯ 0 . An SIR model driven by a tempered stable process is presented as an example of application with the ability to model sudden disease outbreak, illustrated by numerical simulations. The results show that persistence and extinction are dependent not only on the variance of the processes increments, but also on the shapes of their distributions. [ABSTRACT FROM AUTHOR]

We construct unbiased estimators for the distribution of the number of points inside random stopping sets based on a Poisson point process. Our approach is based on moment identities for stopping sets, showing that the random count of points inside the complement S of a stopping set S has a Poisson distribution conditionally to S. The proofs do not require the use of set-indexed martingales, and our estimators have a lower variance when compared to standard sampling. Numerical simulations are presented for examples such as the convex hull and the Voronoi flower of a Poisson point process, and their complements. [ABSTRACT FROM AUTHOR]

The probability hypothesis density (PHD) filter, which is used for multi-target tracking based on sensor measurements, relies on the propagation of the first-order moment, or intensity function, of a point process. This algorithm assumes that targets behave independently, an hypothesis which may not hold in practice due to potential target interactions. In this paper, we construct a second-order PHD filter based on determinantal point processes which are able to model repulsion between targets. Such processes are characterized by their first- and second-order moments, which allows the algorithm to propagate variance and covariance information in addition to first-order target count estimates. Our approach relies on posterior moment formulas for the estimation of a general hidden point process after a thinning operation and a superposition with a Poisson point process, and on suitable approximation formulas in the determinantal point process setting. The repulsive properties of determinantal point processes apply to the modeling of negative correlation between distinct measurement domains. Monte Carlo simulations with correlation estimates are provided. [ABSTRACT FROM AUTHOR]

We present exact analytical expressions of moments of all orders for neuronal membrane potentials in the multiplicative nonstationary Poisson shot noise model. As an application, we derive closed-form Gram–Charlier density expansions that show how the probability density functions of potentials in such models differ from their Gaussian diffusion approximations. This approach extends the results of Brigham and Destexhe (Preprint, 2015a; Phys Rev E 91:062102, 2015b) by the use of exact combinatorial expressions for the moments of multiplicative nonstationary filtered shot noise processes. Our results are confirmed by stochastic simulations and apply to single- and multiple-noise-source models. [ABSTRACT FROM AUTHOR]

We construct Cournot games with limited demand, resulting into capped sales volumes according to the respective production shares of the players. We show that such games admit three distinct equilibrium regimes, including an intermediate regime that allows for a range of possible equilibria. When information on demand is modeled by a delayed diffusion process, we also show that this intermediate regime collapses to a single equilibrium while the other regimes approximate the deterministic setting as the delay tends to zero. Moreover, as the delay approaches zero, the unique equilibrium achieved in the stochastic case provides a way to select a natural equilibrium within the range observed in the no lag setting. Numerical illustrations are presented when demand is modeled by an Ornstein–Uhlenbeck process and price is an affine function of output. [ABSTRACT FROM AUTHOR]

We derive Mellin integral representations in terms of Macdonald functions for the squared absolute value s ↦ | Γ ⁢ ( a + i ⁢ s ) | 2 {s\mapsto|\Gamma(a+is)|^{2}} of the gamma function and its Fourier transform when a < 0 {a<0} is non-integer, generalizing known results in the case a > 0 {a>0}.This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of a Fokker–Planck equation. [ABSTRACT FROM AUTHOR]

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]

We establish an integration by parts formula for the random functionals of a continuous-time Markov chain, based on partial differentiation with respect to jump times. In comparison with existing methods, our approach does not rely on the Girsanov theorem and it imposes less restrictions on the choice of directions of differentiation, while assuming additional continuity conditions on the considered functionals. As an application we compute sensitivities (Greeks) using stochastic weights in an asset price model with Markovian regime switching. [ABSTRACT FROM PUBLISHER]

We construct differential forms of all orders and a covariant derivative together with its adjoint on the probability space of a standard Poisson process, using derivation operators. In this framewok we derive a de Rham-Hodge-Kodaira decomposition as well as Weitzenböck and Clark-Ocone formulas for random differential forms. As in the Wiener space setting, this construction provides two distinct approaches to the vanishing of harmonic differential forms. [ABSTRACT FROM AUTHOR]

We derive sufficient conditions for the mixing of all orders of interacting transformations of a spatial Poisson point process, under a zero-type condition in probability and a generalized adaptedness condition. This extends a classical result in the case of deterministic transformations of Poisson measures. The approach relies on moment and covariance identities for Poisson stochastic integrals with random integrands. [ABSTRACT FROM PUBLISHER]

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 provide closed-form expressions for bond prices in interest rate models based on compact Lie groups. Our approach uses a Doob transform technique and PDE solutions by the Mathieu periodic functions. As a by-product, we derive formulas for bond option prices as well as new identities for the Laplace transform of periodic functionals of Brownian motion and Brownian diffusion processes. [ABSTRACT FROM AUTHOR]

We consider a random sphere covering model made of random balls with interacting random radii of the product form R(r, ω) = rG(ω), based on a Poisson random measure ω(dy, dr) on ℝd ×ℝ+. We provide sufficient conditions under which the corresponding random ball counting processes are well-defined, and we study the fractional behavior of the associated random fields. The main results rely on moment formulas for Poisson stochastic integrals with random integrands. [ABSTRACT FROM AUTHOR]

Accurate estimation of species divergence times from the analysis of genetic sequences relies on probabilistic models of evolution of the rate of molecular evolution. Importantly, while these models describe the sample paths of the substitution rates along a phylogenetic tree, only the (random) average rate can be estimated on each edge. For mathematical convenience, the stochastic nature of these averages is generally ignored. In this article we derive the probabilistic distribution of the average substitution rate assuming a geometric Brownian motion for the sample paths, and we investigate the corresponding error bounds via numerical simulations. In particular we confirm the validity of the gamma approximation proposed in Guindon (Syst Biol 62(1):22-34, ) for 'small' values of the autocorrelation 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]

Using forward-backward stochastic calculus, we prove convex concentration inequalities for some additive functionals of the solution of stochastic differential equations with jumps admitting an invariant probability measure. As a consequence, transportation-information inequalities are obtained and bounds on option prices for interest rate derivatives are given as an application. [ABSTRACT FROM AUTHOR]

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]

We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identifies of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Ùstünel and Zakai (Probab. Theory Related Fields 103 (1995) 409-429) on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples of application include transformations conditioned by random sets such as the convex hull of a Poisson random measure [ABSTRACT FROM AUTHOR]

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]

In this paper we provide a Monte Carlo algorithm for the density estimation of functionals of spatial point processes on Lipschitz domains with random marks, using the Malliavin calculus. Our method allows us to explicitly compute the Malliavin weight and is applied to density estimation of the interference in a wireless ad hoc network model. This extends and makes more precise, and under stronger assumptions, some recent results of Privault and Wei [J. Appl. Probab., 44 (2007), pp. 806-823], who dealt with the particular case of the half-line. [ABSTRACT FROM AUTHOR]

In this paper, we derive a formula for the expectation of random Hermite polynomials in Skorohod integrals, extending classical results in the adapted case. As an application we recover, under simple conditions and with short proofs, the anticipative Girsanov identity and quasi-invariance results obtained in Ref. 6 for quasi-nilpotent shifts on the Wiener space. [ABSTRACT FROM AUTHOR]

We present a deformation of a class of elementary classical integrable systems using stochastic diffusion processes. This deformation applies to the solution of the associated classical Newtonian, Hamiltonian, Lagrangian, and variational problems and to the Hamilton-Jacobi method of characteristics. The underlying stochastic action functionals involve dual random times, whose expectations are connected to the new variables of the system after a canonical transformation. [ABSTRACT FROM AUTHOR]

The solution of the Fokker–Planck equation for exponential Brownian functionals usually involves spectral expansions that are difficult to compute explicitly. In this paper, we propose a direct solution based on heat kernels and a new integral representation for the square modulus of the Gamma function. A financial application to bond pricing in the Dothan model is also presented. [ABSTRACT FROM AUTHOR]

Using the Malliavin calculus in time inhomogeneous jump-diffusion models, we obtain an expression for the sensitivity Theta of an option price (with respect to maturity) as the expectation of the option payoff multiplied by a stochastic weight. This expression is used to design efficient numerical algorithms that are compared with traditional finite-difference methods for the computation of Theta. Our proof can be viewed as a generalization of Dupire's integration by parts to arbitrary and possibly non-smooth payoff functions. In the time homogeneous case, Theta admits an expression from the Black-Scholes PDE in terms of Delta and Gamma but the representation formula obtained in this way is different from ours. Numerical simulations are presented in order to compare the efficiency of the finite-difference and Malliavin methods. [ABSTRACT FROM AUTHOR]

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]

A law of iterated logarithm (LIL) in small time and an asymptotic estimate of modulus of continuity are proved for Brownian motion on the loop group ℒ( G ) over a compact connected Lie group G . Upper bounds are obtained via infinite-dimensional deviation inequalities for functionals on the path space ℙ(ℒ( G )) on ℒ( G ), such as the supremum of Brownian motion on ℒ( G ), which are proved from the Clark–Ocone formula on ℙ(ℒ( G )). The lower bounds rely on analog finite-dimensional results that are proved separately on Riemannian path space. [ABSTRACT FROM AUTHOR]

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]

A time reversible probabilistic representation of solutions of the (Euclidean) Schrödinger equation in momentum representation is constructed using Lévy processes and bridges. Each diffusion in the position representation is associated with a jump diffusion in the momentum space. Our method can be looked upon as a rigorous version of Feynman’s path integral approach. Several examples are studied. [ABSTRACT FROM AUTHOR]

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]

Markovian bridges driven by Lévy processes are constructed from the data of an initial and a final distribution, as particular cases of a family of time reversible diffusions with jumps. In this way we construct a large class of not necessarily continuous Markovian Bernstein processes. These processes are also characterized using the theory of stochastic control for jump processes. Our construction is motivated by Euclidean quantum mechanics in momentum representation, but the resulting class of processes is much bigger than the one needed for this purpose. A large collection of examples is included. [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]

A general method for deriving Girsanov or quasi-invariance formulas for classical stochastic processes with independent increments obtained as components of Lévy processes on real Lie algebras is presented. Letting a unitary operator arising from the associated factorizable current representation act on an appropriate commutative subalgebra, a second commutative subalgebra is obtained. Under certain conditions the two commutative subalgebras lead to two classical processes such that the law of the second process is absolutely continuous w.r.t. to the first. Examples include the Girsanov formula for Brownian motion as well as quasi-invariance formulas for the Poisson process, the Gamma process,[sup 15,16] and the Meixner process. [ABSTRACT FROM AUTHOR]

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]