asymptotic properties of monte carlo estimators of diffusion processes abstract this paper studies the limit distributions of monte carlo estimators of diffusion processes we examine two types of estimators based on the euler scheme one applied to the original processes the other to a doss transformation of the processes we show that the transformation increases the speed of convergence of the euler scheme we also study estimators of conditional expectations of diffusions after characterizing expected approximation errors we construct second order bias corrected estimators we also derive new convergence results for the mihlstein scheme illustrations of the results are provided in the context of simulation based estimation of diffusion processes

on the long time integration of stochastic gradient systems this article addresses the weak convergence of numerical methods for brownian dynamics typical analyses of numerical methods for stochastic differential equations focus on properties such as the weak order which estimates the asymptotic stepsize h 0 convergence behaviour of the error of finite time averages recently it has been demonstrated by study of fokker planck operators that a non markovian numerical method generates approximations in the long time limit with higher accuracy order second order than would be expected from its weak convergence analysis finite time averages are first order accurate in this article we describe the transition from the transient to the steady state regime of this numerical method by estimating the time dependency of the coefficients in an asymptotic expansion for the weak error demonstrating that the convergence to second order is exponentially rapid in time moreover we provide numerical tests of the theory including comparisons of the efficiencies of the euler maruyama method the popular second order heun method and the non markovian method

variance reduction methods for simulation of densities on wiener space we develop a general error analysis framework for the monte carlo simulation of densities for functionals in wiener space we also study variance reduction methods with the help of malliavin derivatives for this we give some general heuristic principles which are applied to diffusion processes a comparison with kernel density estimates is made

exact simulation of stochastic volatility and other affine jump diffusion processes the stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated discretization methods can be used for simulating security prices under these models however discretization introduces bias into the simulation results and a large number of time steps may be needed to reduce the discretization bias to an acceptable level this paper suggests a method for the exact simulation of the stock price and variance under hestons stochastic volatility model and other affine jump diffusion processes the sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security we compare our method with the more conventional euler discretization method and demonstrate the faster convergence rate of the error in our method specifically our method achieves an o s 1 2 convergence rate where s is the total computational budget the convergence rate for the euler discretization method is o s 1 3 or slower depending on the model coefficients and option payoff function

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

split step forward methods for stochastic differential equations in this paper we discuss split step forward methods for solving ito stochastic differential equations sdes eight fully explicit methods the drifting split step euler drsse method the diffused split step euler disse method and the three stage milstein tsm 1a tsm 1f methods are constructed based on euler maruyama method and milstein method respectively in this paper their order of strong convergence is proved the analysis of stability shows that the mean square stability properties of the methods derived in this paper are improved on the original methods the numerical results show the effectiveness of these methods in the pathwise approximation of ito sdes

functional inequalities for the incomplete gamma function abstract we present several inequalities for f a x a x a 0 a 0 x 0 where a x is the incomplete gamma function one of our theorems states that the inequalities f a s p x 1 x n f a x 1 f a x n f a s q x 1 x n p q 0 hold for all nonnegative real numbers x 1 x n n 2 if and only if p m i n a 1 and q m a x a 1 here s t x 1 x n denotes the power sum of order t this extends and complements a result published by ismail and laforgia in 2006

discretization and simulation of the zakai equation this paper is concerned with numerical approximations for a class of nonlinear stochastic partial differential equations zakai equation of nonlinear filtering problem and mckean vlasov type equations the approximation scheme is based on the re pre sentation of the solutions as weighted conditional distributions we first accurately analyse the error caused by an euler type scheme of time discretization sharp error bounds are calculated we show that the rate of convergence is in general of order sqrt delta delta is the time step but in the case when there is no correlation between the signal and the observation for the zakai equation the order of convergence becomes delta this result is obtained by carefully employing techniques of malliavin calculus in a second step we propose a simulation of the time discretization euler scheme by a quantization approach this formally consists in an approximation of the weighted conditional distribution by a conditional discrete distribution on finite supports we provide error bounds and rate of convergence in terms of the number n of the grids of this support these errors are minimal at some optimal grids which are computed by a recursive method based on monte carlo simulations finally we illustrate our results with some numerical experiments arising from correlated kalman bucy filter and burgers equation

weak approximation of stochastic partial differential equations the nonlinear case we study the error of the euler scheme applied to a stochastic partial dierential equation we prove that as it is often the case the weak order of convergence is twice the strong order a key ingredient in our proof is malliavin calculus which enables us to get rid of the irregular terms of the error we apply our method to the case a semilinear stochastic heat equation driven by a space time white noise

on the regularity of subelliptic p harmonic functions in carnot groups abstract in this paper we prove second order horizontal differentiability and c 1 regularity results for subelliptic p harmonic functions in carnot groups for p close to 2

the euler scheme for levy driven stochastic differential equations limit theorems we study the euler scheme for a stochastic differential equation driven by a levy process y more precisely we look at the asymptotic behavior of the normalized error process un xn x where x is the true solution and xn is its euler approximation with stepsize 1 n and un is an appropriate rate going to infinity if the normalized error processes converge or are at least tight we say that the sequence un is a rate which in addition is sharp when the limiting process or processes is not trivial we suppose that y has no gaussian part otherwise a rate is known to be u n sqrt n then rates are given in terms of the concentration of the levy measure of y around 0 and further we prove the convergence of the sequence un xn x to a nontrivial limit under some further assumptions which cover all stable processes and a lot of other levy processes whose levy measure behave like a stable levy measure near the origin for example when y is a symmetric stable process with index 0 2 a sharp rate is un n logn 1 when y is stable but not symmetric the rate is again un n logn 1 when 1 but it becomes un n logn 2 if 1 and un n if 1

learning topology and dynamics of large recurrent neural networks large scale recurrent networks have drawn increasing attention recently because of their capabilities in modeling a large variety of real world phenomena and physical mechanisms this paper studies how to identify all authentic connections and estimate system parameters of a recurrent network given a sequence of node observations this task becomes extremely challenging in modern network applications because the available observations are usually very noisy and limited and the associated dynamical system is strongly nonlinear by formulating the problem as multivariate sparse sigmoidal regression we develop simple to implement network learning algorithms with rigorous convergence guarantee in theory for a variety of sparsity promoting penalty forms a quantile variant of progressive recurrent network screening is proposed for efficient computation and allows for direct cardinality control of network topology in estimation moreover we investigate recurrent network stability conditions in lyapunov s sense and integrate such stability constraints into sparse network learning experiments show excellent performance of the proposed algorithms in network topology identification and forecasting

log concavity for bernstein type operators using stochastic orders abstract this paper aims to study the preservation of log concavity for bernstein type operators in particular attention is focused on positive linear operators defined on the positive semi axis admitting a probabilistic representation in terms of a process with independent increments this class includes the classical gamma szasz and szasz durrmeyer operators with respect to the first and second operators the results of this paper correct two erroneous counterexamples in 10 as a main tool in our results we use stochastic order techniques our results include as a particular case the log concavity of certain functions related to the incomplete gamma function

inequalities for the chi square distribution function abstract letpa x 1 x a0 e ttx 1 dtbe the chi square distribution function and letmt u v be the weighted power mean of ordert we prove the following ifx y 0 x y and 0 1 are fixed real numbers then the inequalities formula are valid for all real numbersa 0 if and only ifr 0 ands in particular we obtain that for alla 0 the functionx pa x is log concave on 0 this proves a conjecture of m merkle

hypoelliptic diffusions filtering and inference from complete and partial observations the statistical problem of parameter estimation in partially observed hypoelliptic diffusion processes is naturally occurring in many applications however because of the noise structure where the noise components of the different co ordinates of the multi dimensional process operate on different timescales standard inference tools are ill conditioned we propose to use a higher order scheme to approximate the likelihood such that the different timescales are appropriately accounted for we show consistency and asymptotic normality with non typical convergence rates when only partial observations are available we embed the approximation in a filtering algorithm for the unobserved co ordinates and use this as a building block in a stochastic approximation expectation maximization algorithm we illustrate on simulated data from three models the harmonic oscillator the fitzhugh nagumo model used to model membrane potential evolution in neuroscience and the synaptic inhibition and excitation model used for determination of neuronal synaptic input

high order numerical approximation of the invariant measure of ergodic sdes we introduce new sufficient conditions for a numerical method to approximate with high order of accuracy the invariant measure of an ergodic system of stochastic differential equations independently of the weak order of accuracy of the method we then present a systematic procedure based on the framework of modified differential equations for the construction of stochastic integrators that capture the invariant measure of a wide class of ergodic sdes brownian and langevin dynamics with an accuracy independent of the weak order of the underlying method numerical experiments confirm our theoretical findings

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

generalized spectral testing for multivariate continuous time models we develop an omnibus specification test for multivariate continuous time models using the conditional characteristic function which often has a convenient closed form or can be accurately approximated for many multivariate continuous time models in finance and economics the proposed test fully exploits the information in the joint conditional distribution of underlying economic processes and hence is expected to have good power in a multivariate context a class of easy to interpret diagnostic procedures is supplemented to gauge possible sources of model misspecification our tests are also applicable to discrete time distribution models simulation studies show that the tests provide reliable inference in finite samples

adaptive weak approximation of stochastic differential equations adaptive time stepping methods based on the monte carlo euler method for weak approximation of ito stochastic differential equations are developed the main result is new expansions of the computational error with computable leading order term in a posteriori form based on stochastic flows and discrete dual backward problems the expansions lead to efficient and accurate computation of error estimates adaptive algorithms for either stochastic time steps or deterministic time steps are described numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps stochastic time steps use brownian bridges and require more work for a given number of time steps deterministic time steps may yield more time steps but require less work for example in the limit of vanishing error tolerance the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps c 2001 john wiley sons inc

approximation for non smooth functionals of stochastic differential equations with irregular drift abstract this paper aims at developing a systematic study for the weak rate of convergence of the euler maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients we apply our method to obtain the rates of approximation for the expectation of various non smooth functionals of both stochastic differential equations and killed diffusion we also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is holder continuous

variance reduction for simulated diffusions this article develops some variance reduction techniques for the monte carlo integration of functionals of the solutions of ito stochastic differential equations sdes the monte carlo method for sdes offers a means of calculating solutions to certain types of parabolic partial differential equation and so has applications in various fields including stochastic control particle physics and econometrics it involves the representation of the required integrals as means of random variables defined on infinite dimensional wiener spaces which cannot be simulated directly sthey must at some stage be approximated by variables defined on high but finite dimensional spaces the approach taken here is to construct variance reduced random variables on the infinite dimensional spaces which can subsequently be approximated by any of a number of known finite difference methods the methods of control variates and importance sampling are developed in both cases a perfect variate i e one which is unbiased and h

convergence of numerical time averaging and stationary measures via poisson equations numerical approximation of the long time behavior of a stochastic differential equation sde is considered error estimates for time averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the sde the error analysis is based on using an associated poisson equation for the underlying sde the main advantages of this approach are its simplicity and universality it works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables and for hypoelliptic sdes to simplify the exposition we consider only the case where the state space of the sde is a torus and we study only smooth test functions however we anticipate that the approach can be applied more widely an analogy between our approach and stein s method is indicated some practical implications of the results are discussed

high order discretization schemes for the cir process application to affine term structure and heston models this paper presents weak second and third order schemes for the cox ingersoll ross cir process without any restriction on its parameters at the same time it gives a general recursive construction method to get weak second order schemes that extends the one introduced by ninomiya and victoir cite nv combining these both results this allows to propose a second order scheme for more general affine diffusions simulation examples are given to illustrate the convergence of these schemes on cir and heston models algorithms are stated in a pseudocode language

sensitivity analysis using ito malliavin calculus and martingales and application to stochastic optimal control we consider a multidimensional diffusion process x alpha t 0 leq t leq t whose dynamics depends on a parameter alpha our first purpose is to write as an expectation the sensitivity nabla alpha j alpha for the expected cost j alpha mathbb e f x alpha t in order to evaluate it using monte carlo simulations this issue arises for example from stochastic control problems where the controller is parameterized which reduces the control problem to a parametric optimization one or from model misspecifications in finance previous evaluations of nabla alpha j alpha using simulations were limited to smooth cost functions f or to diffusion coefficients not depending on alpha see yang and kushner siam j control optim 29 1991 pp 1216 1249 in this paper we cover the general case deriving three new approaches to evaluate nabla alpha j alpha which we call the malliavin calculus approach the adjoint approach and the martingale approach to accomplish this we leverage ito calculus malliavin calculus and martingale arguments in the second part of this work we provide discretization procedures to simulate the relevant random variables then we analyze their respective errors this analysis proves that the discretization error is essentially linear with respect to the time step this result which was already known in some specific situations appears to be true in this much wider context finally we provide numerical experiments in random mechanics and finance and compare the different methods in terms of variance complexity computational time and time discretization error

properties of the probability density function of the non central chi squared distribution abstract in this paper we consider the probability density function pdf of a non central 2 distribution with arbitrary number of degrees of freedom for this function we prove that can be represented as a finite sum and we deduce a partial derivative formula moreover we show that the pdf is log concave when the degrees of freedom is greater or equal than 2 at the end of this paper we present some turan type inequalities for this function and an elegant application of the monotone form of l hospital s rule in probability theory is given

stochastic approximation finite element method analytical formulas for multidimensional diffusion process we derive an analytical weak approximation of a multidimensional diffusion process as coefficients are small or time is small our methodology combines the use of gaussian proxies to approximate the law of the diffusion and a finite element interpolation of the terminal function applied to the diffusion we call this the stochastic approximation finite element safe method we provide error bounds of our global approximation depending on the diffusion process coefficients the time horizon and the regularity of the terminal function then we give estimates of the computational cost of our algorithm this shows an improved efficiency compared to monte carlo methods in small and medium dimensions smaller than 10 which is confirmed by numerical experiments

on the connection between the malliavin covariance matrix and hormander s condition abstract a celebrated theorem of hormander gives a sufficient condition for a second order differential operator to be hypoelliptic for operators with analytic coefficients this condition turns out to be also necessary but this is not true for general smooth coefficients on the other hand malliavin conceived a probabilistic approach to the same problem known as malliavin calculus in which a key role is played by the malliavin covariance matrix the aim of our paper is to give several characterizations of the malliavin covariance matrix which are equivalent to hormander s condition and consequently imply the hypoellipticity in this way the distance between hormander s condition and the hypoellipticity property is clearly pointed out in probabilistic terms

the euler scheme for levy driven stochastic differential equations in relation with monte carlo methods to solve some integro differential equations we study the approximation problem of ee g x t by ee g bar x t n where x t 0 leq t leq t is the solution of a stochastic differential equation governed by a levy process z t bar x n t is defined by the euler discretization scheme with step fractn with appropriate assumptions we show that the error ee g x t ee g bar x t n can be expanded in powers of frac1n if the levy measure of z has finite moments of order high enough otherwise the rate of convergence is slower and its speed depends on the behavior of the tails of the levy measure the simulation of the increments of z t is also discussed

estimation of continuous time markov processes sampled at random time intervals we introduce a family of generalized method of moments estimators of the parameters of a continuous time markov process observed at random time intervals the results include strong consistency asymptotic normality and a characterization of standard errors sampling is at an arrival intensity that is allowed to depend on the underlying markov process and on the parameter vector to be estimated we focus on financial applications including tick based sampling allowing for jump diffusions regime switching diffusions and reflected diffusions

adaptive stochastic weak approximation of degenerate parabolic equations of kolmogorov type degenerate parabolic equations of kolmogorov type occur in many areas of analysis and applied mathematics in their simplest form these equations were introduced by kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases more recently equations of kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of asian options the purpose of this paper is to numerically solve the cauchy problem for a general class of second order degenerate parabolic differential operators of kolmogorov type with variable coefficients using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations furthermore we show how to apply these results in the context of mathematical finance and option pricing the approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on for example a finite difference discretization of the partial differential equation in itself these problems are caused by the fact that the natural setting for degenerate parabolic differential operators of kolmogorov type is that of a lie group much more involved than the standard euclidean lie group of translations the latter being relevant in the case of uniformly elliptic parabolic operators