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This paper studies the limiting behavior of the count of subgraphs isomorphic to a graph $G$ with $m\geq 0$ fixed endpoints in the random-connection model, as the intensity $\lambda$ of the underlying Poisson point process tends to infinity. When connection probabilities are of order $\lambda^{-\alpha}$ we identify a phase transition phenomenon via a critical decay rate $\alpha^\ast_m (G)>0$ such that normal approximation for subgraph counts holds when $\alpha \in (0,\alpha^\ast_m (G) )$, and a Poisson limit result holds if $\alpha = \alpha^\ast_m (G)$. Our method relies on cumulant growth rates derived by the convex analysis of planar diagrams that list the partitions involved in cumulant identities. As a result, by the cumulant method we obtain normal approximation results with convergence rates in the Kolmogorov distance, and a Poisson limit theorem, for subgraph counts.

We provide a probabilistic representation of the solutions of ordinary differential equations (ODEs) by random generation of Butcher trees. This approach complements and simplifies a recent probabilistic representation of ODE solutions, by removing the need to generate random branching times. The random sampling of trees allows one to improve numerical accuracy by Monte Carlo iterations whereas the finite order truncation of Butcher series can have higher time complexity, as shown in examples.

We consider the count of subgraphs with an arbitrary configuration of endpoints in the random-connection model based on a Poisson point process on ${\Bbb R}^d$. We present combinatorial expressions for the computation of the cumulants and moments of all orders of such subgraph counts, which allow us to estimate the growth of cumulants as the intensity of the underlying Poisson point process goes to infinity. As a consequence, we obtain a central limit theorem with explicit convergence rates under the Kolmogorov distance, and connectivity bounds. Numerical examples are presented using a computer code in SageMath for the closed-form computation of cumulants of any order, for any type of connected subgraph and for any configuration of endpoints in any dimension $d\geq 1$. In particular, graph connectivity estimates, Gram-Charlier expansions for density estimation, and correlation estimates for joint subgraph counting are obtained.

We provide sufficient conditions for the existence of viscosity solutions of fractional semilinear elliptic PDEs of index $\alpha \in (1,2)$ with polynomial gradient nonlinearities on $d$-dimensional balls, $d\geq 2$. Our approach uses a tree-based probabilistic representation of solutions and their partial derivatives using $\alpha$-stable branching processes, and allows us to take into account gradient nonlinearities not covered by deterministic finite difference methods so far. In comparison with the existing literature on the regularity of solutions, no polynomial order condition is imposed on gradient nonlinearities. Numerical illustrations demonstrate the accuracy of the method in dimension $d=10$, solving a challenge encountered with the use of deterministic finite difference methods in high-dimensional settings., Comment: arXiv admin note: text overlap with arXiv:2110.09941, arXiv:2106.12127

This paper derives normal approximation results for subgraph counts written as multiparameter stochastic integrals in a random-connection model based on a Poisson point process. By combinatorial arguments we express the cumulants of general subgraph counts using sums over connected partition diagrams, after cancellation of terms obtained by M\"obius inversion. Using the Statulevi\v{c}ius condition, we deduce convergence rates in the Kolmogorov distance by studying the growth of subgraph count cumulants as the intensity of the underlying Poisson point process tends to infinity. Our analysis covers general subgraphs in the dilute and full random graph regimes, and tree-like subgraphs in the sparse random graph regime.

We present an algorithm for the numerical solution of systems of fully nonlinear PDEs using stochastic coded branching trees. This approach covers functional nonlinearities involving gradient terms of arbitrary orders, and it requires only a boundary condition over space at a given terminal time $T$ instead of Dirichlet or Neumann boundary conditions at all times as in standard solvers. Its implementation relies on Monte Carlo estimation, and uses neural networks that perform a meshfree functional estimation on a space-time domain. The algorithm is applied to the numerical solution of the Navier-Stokes equation and is benchmarked to other implementations in the cases of the Taylor-Green vortex and Arnold-Beltrami-Childress flow.

We derive Wasserstein distance bounds between the probability distributions of a stochastic integral (It\^o) process with jumps $(X_t)_{t\in [0,T]}$ and a jump-diffusion process $(X^\ast_t)_{t\in [0,T]}$. Our bounds are expressed using the stochastic characteristics of $(X_t)_{t\in [0,T]}$ and the jump-diffusion coefficients of $(X^\ast_t)_{t\in [0,T]}$ 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.

We derive exact analytical expressions for the cumulants of any orders of neuronal membrane potentials driven by spike trains in a multivariate Hawkes process model with excitation and inhibition. Such expressions can be used for the prediction and sensitivity analysis of the statistical behavior of the model over time, and to estimate the probability densities of neuronal membrane potentials using Gram-Charlier expansions. Our results are shown to provide a better alternative to Monte Carlo estimates via stochastic simulations, and computer codes based on combinatorial recursions are included.

Recent work on Path-Dependent Partial Differential Equations (PPDEs) has shown that PPDE solutions can be approximated by a probabilistic representation, implemented in the literature by the estimation of conditional expectations using regression. However, a limitation of this approach is to require the selection of a basis in a function space. In this paper, we overcome this limitation by the use of deep learning methods, and we show that this setting allows for the derivation of error bounds on the approximation of conditional expectations. Numerical examples based on a two-person zero-sum game, as well as on Asian and barrier option pricing, are presented. In comparison with other deep learning approaches, our algorithm appears to be more accurate, especially in large dimensions.

We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and non-negative integrands, our estimates involve only the third moment of integrand in addition to a variance term using a square norm of the integrand. As a consequence, we are able to observe a "third moment phenomenon" in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time, or have been obtained only for specific kernels such as the exponential or Erlang kernels.

We propose an algorithm for the closed-form recursive computation of joint moments and cumulants of all orders for k-hop counts in the 1D unit disk random graph model with Poisson distributed vertices. Our approach uses decompositions of k-hop counts into multiple Poisson stochastic integrals. As a consequence, using the Stein method we derive Berry-Esseen bounds for the asymptotic convergence of renormalized k-hop path counts to the normal distribution as the density of Poisson vertices tends to infinity. Computer codes for the recursive symbolic computation of moments and cumulants are provided in appendix.

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.

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.

We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations. This algorithm extends the classical Feynman-Kac formula to fully nonlinear partial differential equations, by using random trees that carry information on nonlinearities on their branches. It applies to functional, non-polynomial nonlinearities that are not treated by standard branching arguments, and deals with derivative terms of arbitrary orders. A Monte Carlo numerical implementation is provided.

The purpose of this paper is to present a recursive algorithm and its implementation in Maple and Mathematica for the computation of joint moments and cumulants of Hawkes processes with exponential kernels. Numerical results and computation times are also discussed. Obtaining closed form expressions can be computationally intensive, as joint fifth cumulant and moment formulas can be respectively expanded into up to 3,288 and 27,116 summands.

We prove the existence of viscosity solutions for fractional semilinear elliptic PDEs on open balls with bounded exterior condition in dimension $d\geq 1$. Our approach relies on a tree-based probabilistic representation based on a (2s)-stable branching processes for all $s\in (0,1)$, and our existence results hold for sufficiently small exterior conditions and nonlinearity coefficients. In comparison with existing approaches, we consider a wide class of polynomial nonlinearities without imposing upper bounds on their maximal degree or number of terms. Numerical illustrations are provided in large dimensions.

We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on $\mathbb{R}^d$ with polynomial nonlinearities in $(u, \nabla u)$, using a tree-based probabilistic representation. This probabilistic representation applies to the solution of the equation itself, as well as to its partial derivatives by associating one of d marks to the initial tree branch. Partial derivatives are dealt with by integration by parts and subordination of Brownian motion. Numerical illustrations are provided in examples for the fractional Laplacian in dimension up to 10, and for the fractional Burgers equation in dimension two.

A 1d random geometric graph (1d RGG) is built by joining a random sample of $n$ points from an interval of the real line with probability $p$. We count the number of $k$-hop paths between two vertices of the graph in the case where the space is the 1d interval $[0,1]$. We show how the $k$-hop path count between two vertices at Euclidean distance $|x-y|$ is in bijection with the volume enclosed by a uniformly random $d$-dimensional lattice path joining the corners of a $(k-1)$-dimensional hyperrectangular lattice. We are able to provide the probability generating function and distribution of this $k$-hop path count as a sum over lattice paths, incorporating the idea of restricted integer partitions with limited number of parts. We therefore demonstrate and describe an important link between spatial random graphs, and lattice path combinatorics, where the $d$-dimensional lattice paths correspond to spatial permutations of the geometric points on the line., Comment: 27 pages, 11 figures

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 (BSDEs). 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.

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.

Using finite difference operators, we define a notion of boundary and surface measure for configuration sets under Poisson measures. A Margulis-Russo type identity and a co-area formula are stated with applications to deviation inequalities and functional inequalities, and bounds are obtained on the associated isoperimetric constants., Comment: A shortened version of this paper has been published in Statistics & Probability Letters 78 (2008) 2154-2164, see also the Preprint 2003-04, Universite de La Rochelle. As is, our intent is not for it to be published but to make it available since it has been referenced on various instances in the literature

We apply general moment identities for Poisson stochastic integrals with random integrands to the computation of the moments of Markovian growth-collapse processes. This extends existing formulas for mean and variance available in the literature to closed form moments expressions of all orders. In comparison with other methods based on differential equations, our approach yields polynomial expressions in the time parameter. We also treat the case of the associated embedded chain.

We propose a recursive method for the computation of the cumulants of self-exciting point processes of Hawkes type, based on standard combinatorial tools such as Bell polynomials. This closed-form approach is easier to implement on higher-order cumulants in comparison with existing methods based on differential equations, tree enumeration or martingale arguments. The results are corroborated by Monte Carlo simulations, and also apply to the computation of joint cumulants generated by multidimensional self-exciting processes.

We derive Berry-Esseen approximation bounds for general functionals of independent random variables, based on chaos expansions methods. Our results apply to $U$-statistics satisfying the weak assumption of decomposability in the Hoeffding sense, and yield Kolmogorov distance bounds instead of the Wasserstein bounds previously derived in the special case of degenerate $U$-statistics. Linear and quadratic functionals of arbitrary sequences of independent random variables are included as particular cases, with new fourth moment bounds, and applications are given to Hoeffding decompositions, weighted $U$-statistics, quadratic forms, and random subgraph weighing. In the case of quadratic forms, our results recover and improve the bounds available in the literature, and apply to matrices with non-empty diagonals.

We derive normal approximation bounds in the Wasserstein distance for sums of weighted 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 Erd\H{o}s-R\'enyi random graph, extending the graph counting results of [1] to the setting of graph weighting. Our approach relies on a general stochastic analytic framework for functionals of independent random sequences.

We derive sufficient conditions for the convex and monotonic g-stochastic ordering of diffusion processes under nonlinear g-expectations and g-evaluations. Our approach relies on comparison results for forward-backward stochastic differential equations and on several extensions of convexity, monotonicity and continuous dependence properties for the solutions of associated semilinear parabolic partial differential equations. Applications to contingent claim price comparison under different hedging portfolio constraints are provided.

This paper considers a general stochastic SIR epidemic model driven by a multidimensional Levy 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 Levy measures by the definition of new threshold values. 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.

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 (DPPs) 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 (PPP), 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.

In recent years, a number of functional inequalities have been derived for Poisson random measures, with a wide range of applications. In this paper, we prove that such inequalities can be extended to the setting of marked temporal point processes, under mild assumptions on their Papangelou conditional intensity. First, we derive a Poincar\'e inequality. Second, we prove two transportation cost inequalities. The first one refers to functionals of marked point processes with a Papangelou conditional intensity and is new even in the setting of Poisson random measures. The second one refers to the law of marked temporal point processes with a Papangelou conditional intensity, and extends a related inequality which is known to hold on a general Poisson space. Finally, we provide a variational representation of the Laplace transform of functionals of marked point processes with a Papangelou conditional intensity. The proofs make use of an extension of the Clark-Ocone formula to marked temporal point processes. Our results are shown to apply to classes of renewal, nonlinear Hawkes and Cox point processes.

We derive moment identities for the stochastic integrals of multiparameter processes in a random-connection model based on a point process admitting a Papangelou intensity. Those identities are written using sums over partitions, and they reduce to sums over non-flat partition diagrams in case the multiparameter processes vanish on diagonals. As an application, we obtain general identities for the moments of $k$-hop counts in the random-connection model, which simplify the derivations available in the literature.

We derive sufficient conditions for the differentiability of all orders for the flow of stochastic differential equations with jumps, and prove related $L^p$-integrability results for all orders. Our results extend similar results obtained in [Kun04] for first order differentiability and rely on the Burkholder-Davis-Gundy inequality for time inhomogeneous Poisson random measures on ${\Bbb R}_+\times {\Bbb R}$, for which we provide a new proof.

We derive normal approximation bounds in the Kolmogorov distance for sums of discrete multiple integrals and $U$-statistics made of independent Bernoulli random variables. Such bounds are applied to normal approximation for the renormalized subgraphs counts in the Erd{\H o}s-R\'enyi random graph. This approach completely solves a long-standing conjecture in the general setting of arbitrary graph counting, while recovering and improving recent results derived for triangles as well as results using the Wasserstein distance.

We obtain Stein approximation bounds for stochastic integrals with respect to a Poisson random measure over ${\Bbb R}^d$, $d\geq 2$. 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.

We derive Stein approximation bounds for functionals of uniform random variables, using chaos expansions and the Clark-Ocone representation formula combined with derivation and finite difference operators. This approach covers sums and functionals of both continuous and discrete independent random variables. For random variables admitting a continuous density, it recovers classical distance bounds based on absolute third moments, with better and explicit constants. We also apply this method to multiple stochastic integrals that can be used to represent U-statistics, and include linear and quadratic functionals as particular cases.

We propose a recursive algorithm for the numerical computation of the optimal value function $\inf_{t\le\tau\le T} E \Big[\sup_{0\le s\le T } Y_s / Y_{\tau} \big| {\cal F}_t\Big]$ over the stopping times $\tau$ with respect to the filtration of a geometric Brownian motion $Y_t$ with Markovian regime switching. This method allows us to determine the boundary functions of the optimal stopping set when no associated Volterra integral equation is available. It applies in particular when regime-switching drifts have mixed signs, in which case the boundary functions may not be monotone.

We construct a tree-based dependence structure for the representation of binomial, Poisson and Gaussian random vectors having a given covariance matrix, using sums of independent random variables. This construction allows us to characterize the supermodular ordering of such random vectors via the componentwise ordering of their covariance matrices. Our method relies on the representation of dependent components using binary trees on the discrete $d$-dimensional hypercube $C_d$, and on M\"obius inversion techniques. In the case of Poisson random vectors this approach involves L\'evy measures on $C_d$, and it is consistent with the approximation of Poisson and multivariate Gaussian random vectors by binomial vectors.

This paper deals with optimal prediction in a regime-switching model driven by a continuous-time Markov chain. We extend existing results for geometric Brownian motion by deriving optimal stopping strategies that depend on the current regime state, and prove a number of continuity properties relating to optimal value and boundary functions. Our approach replaces the use of closed form expressions, which are not available in our setting, with PDE arguments that also simplify the approach of [2] in the classical Brownian case.

We derive joint factorial moment identities for point processes with Papangelou intensities. Our proof simplifies previous approaches to related moment identities and includes the setting of Poisson point processes. Applications are given to random transformations of point processes and to their distribution invariance properties.

We prove a version of the Feynman-Kac formula for Levy processes and integro-differential operators, with application to the momentum representation of suitable quantum (Euclidean) systems whose Hamiltonians involve L\'{e}vy-type potentials. Large deviation techniques are used to obtain the limiting behavior of the systems as the Planck constant approaches zero. It turns out that the limiting behavior coincides with fresh aspects of the semiclassical limit of (Euclidean) quantum mechanics. Non-trivial examples of Levy processes are considered as illustrations and precise asymptotics are given for the terms in both configuration and momentum representations.

Hedging strategies in bond markets are computed by martingale representation and the Clark-Ocone formula under the choice of a suitable of numeraire, in a model driven by the dynamics of bond prices. Applications are given to the hedging of swaptions and other interest rate derivatives, and our approach is compared to delta hedging when the underlying swap rate is modeled by a diffusion process.

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.

We derive an integration by parts formula for functionals of determinantal processes on compact sets, completing the arguments of [4]. This is used to show the existence of a configuration-valued diffusion process which is non-colliding and admits the distribution of the determinantal process as reversible law. In particular, this approach allows us to build a concrete example of the associated diffusion process, providing an illustration of the results of [4] and [30].

We prove a Girsanov identity on the Poisson space for anticipating transformations that satisfy a strong quasi-nilpotence condition. Applications are given to the Girsanov theorem and to the invariance of Poisson measures under random transformations. The proofs use combinatorial identities for the central moments of Poisson stochastic integrals., Comment: Published in at http://dx.doi.org/10.1214/10-AOP640 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We compute the moment of order n of the Poisson stochastic integral of a random process u over a metric space X as a sum that runs over all partitions of {1,...,n} and involves the addition of points to Poisson configurations. This formula recovers known results in case u is a deterministic function on X.

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 identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method applied in [22] 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.

These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time., Comment: Originally published in at http://dx.doi.org/10.1214/08-PS139 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org) This version includes a revision of Section 9

In this paper we consider the nonparametric functional estimation of the drift of Gaussian processes using Paley-Wiener and Karhunen-Lo\`eve expansions. We construct efficient estimators for the drift of such processes, and prove their minimaxity using Bayes estimators. We also construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and stochastic analysis on Gaussian space, in which superharmonic functionals of the process paths play a particular role. Our results are illustrated by numerical simulations and extend the construction of James-Stein type estimators for Gaussian processes by Berger and Wolper.

We prove convex ordering results for random vectors admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. Our method uses forward-backward stochastic calculus and extends previous results in the one-dimensional case. We also study a geometric interpretation of convex ordering for discrete measures in connection with the conditions set on the jump heights and intensities of the considered processes.