We introduce exact methods for the simulation of sample paths of one-dimensional diffusions with a discontinuity in the drift function. Our procedures require the simulation of finite-dimensional candidate draws from probability laws related to those of Brownian motion and its local time, and are based on the principle of retrospective rejection sampling. A simple illustration is provided.

In this paper we consider linear functions constructed on two different weighted branching processes and provide explicit bounds for their Kantorovich–Rubinstein distance in terms of couplings of their corresponding generic branching vectors. Motivated by applications to the analysis of random graphs, we also consider a variation of the weighted branching process where the generic branching vector has a different dependence structure from the usual one. By applying the bounds to sequences of weighted branching processes, we derive sufficient conditions for the convergence in the Kantorovich–Rubinstein distance of linear functions. We focus on the case where the limits are endogenous fixed points of suitable smoothing transformations.

Various numerical methods have been developed in order to solve complex systems with uncertainties, and the stochastic collocation method using l1-minimisation on low discrepancy point sets is investigated here. Halton and Sobol' sequences are considered, and low discrepancy point sets and random points are compared. The tests discussed involve a given target function in polynomial form, high-dimensional functions and a random ODE model. Our numerical results show that the low discrepancy point sets perform as well or better than random sampling for stochastic collocation via l1-minimisation.

Upon a set of backward orthogonal polynomials, we propose a novel multi-step numerical scheme for solving the decoupled forward-backward stochastic differential equations (FBSDEs). Under Lipschtiz conditions on the coefficients of the FBSDEs, we first get a general error estimate result which implies zero-stability of the proposed scheme, and then we further prove that the convergence rate of the scheme can be of high order for Markovian FBSDEs. Some numerical experiments are presented to demonstrate the accuracy of the proposed multi-step scheme and to numerically verify the theoretical results.

We present a new approach to Davie’s theorem on the uniqueness of solutions to the equation

for almost all Brownian paths. A generalization of this result and a discussion of some closely related problems are given.

We discuss simulation of sensitivities or Greeks of multi-asset European style options under a special Lévy process model: that is, the subordinated Brownian motion model. The Malliavin calculus method combined with Monte Carlo and quasi-Monte Carlo methods is used in the simulations. Greeks are expressed in terms of the expectations of the option payoff functions multiplied by the weights involving Malliavin derivatives for multi-asset options. Numerical results show that the Malliavin calculus method is usually more efficient than the finite difference method for options with nonsmooth payoffs. The superiority of the former method over the latter is even more significant when both are combined with quasi-Monte Carlo methods.

Left-invariant PDE-evolutions on the roto-translation group SE(2)(and their resolvent equations) have been widely studied in the fields of cortical modeling and image analysis. They include hypo-elliptic diffusion (for contour enhancement) proposed by Citti & Sarti, and Petitot, and they include the direction process (for contour completion) proposed by Mumford. This paper presents a thorough study and comparison of the many numerical approaches, which, remarkably, are missing in the literature. Existing numerical approaches can be classified into 3 categories: Finite difference methods, Fourier based methods (equivalent to SE(2)-Fourier methods), and stochastic methods (Monte Carlo simulations). There are also 3 types of exact solutions to the PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in previous works by Duits and van Almsick in 2005. Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches. We compute relative errors of all numerical approaches to the exact solutions, and the Fourier based methods show us the best performance with smallest relative errors. We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions, crucial in implementations of the exact solutions. Furthermore, we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of underlying stochastic processes that overcomes the singular behavior in the origin of time-integrated kernels. Finally, we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.

An explicit difference scheme is described, analyzed and tested for numerically approximating stochastic elastic equation driven by infinite dimensional noise. The noise processes are approximated by piecewise constant random processes and the integral formula of the stochastic elastic equation is approximated by a truncated series. Error analysis of the numerical method yields estimate of convergence rate. The rate of convergence is demonstrated with numerical experiments.

This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric Π-rough paths in our terminology) sketched by Lyons in 1998. Although geometric Π-rough paths can be treated as p-rough paths for a sufficiently large p, and the theory of integration of Lipγ one-forms (γ > p–1) along geometric p-rough paths applies, we prove the existence of integrals of one-forms under weaker conditions. Moreover, we consider differential equations driven by geometric Π-rough paths and give sufficient conditions for existence and uniqueness of solution.

We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean is approximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.

We give an account of the characteristics that result from reflecting a drifting local martingale (i.e. the sum of a local martingale and a multiple of its quadratic variation process) in 0 and b > 0. We present conditions which guarantee the existence of finite moments of what is required to keep the reflected process within its boundaries. Also, we derive an associated law of large numbers and a central limit theorem which apply when the input is continuous. Similar results for integrals of the paths of the reflected process are also presented. These results are in close agreement to what has previously been shown for Brownian motion.

The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows us to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near 0. From a conceptual point of view the proposed method gives a better quality of approximation (from a pathwise point of view) than standard, even exact, simulation of the stochastic differential equation at some deterministic time grid.

We derive Laplace transform identities for the volume content of random stopping sets based on Poisson point processes. Our results are based on anticipating Girsanov identities for Poisson point processes under a cyclic vanishing condition for a finite difference gradient. This approach does not require classical assumptions based on set-indexed martingales and the (partial) ordering of index sets. The examples treated focus on stopping sets in finite volume, and include the random missed volume of Poisson convex hulls.

We study optimal double stopping problems driven by a Brownian bridge. The objective is to maximize the expected spread between the payoffs achieved at the two stopping times. We study several cases where the solutions can be solved explicitly by strategies of a threshold type.

Convergence analysis is presented for recently proposed multistep schemes, when applied to a special type of forward-backward stochastic differential equations (FB-SDEs) that arises in finance and stochastic control. The corresponding k-step scheme admits a k-order convergence rate in time, when the exact solution of the forward stochastic differential equation (SDE) is given. Our analysis assumes that the terminal conditions and the FBSDE coefficients are sufficiently regular.

In this work, we develop a minimum action method (MAM) with optimal linear time scaling, called tMAM for short. The main idea is to relax the integration time as a functional of the transition path through optimal linear time scaling such that a direct optimization of the integration time is not required. The Feidlin-Wentzell action functional is discretized by finite elements, based on which h-type adaptivity is introduced to tMAM. The adaptive tMAM does not require reparametrization for the transition path. It can be applied to deal with quasi-potential: 1) When the minimal action path is subject to an infinite integration time due to critical points, tMAM with a uniform mesh converges algebraically at a lower rate than the optimal one. However, the adaptive tMAM can recover the optimal convergence rate. 2) When the minimal action path is subject to a finite integration time, tMAM with a uniform mesh converges at the optimal rate since the problem is not singular, and the optimal integration time can be obtained directly from the minimal action path. Numerical experiments have been implemented for both SODE and SPDE examples.

In this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in our numerical schemes, one has the flexibility to choose the associated forward SDE, and a suitable choice can significantly reduce the computational complexity. Various numerical examples including the HJB equations are presented to show the effectiveness and accuracy of the proposed numerical schemes.

In this paper we establish the central limit theorem for a class of stochastic partial differential equations and as an application derive this theorem for two widely studied population models: super-Brownian motion and the Fleming-Viot process.

Nonlinear filtering is investigated in a system where both the signal system and the observation system are under non-Gaussian Lévy fluctuations. Firstly, the Zakai equation is derived, and it is further used to derive the Kushner-Stratonovich equation. Secondly, by a filtered martingale problem, uniqueness for strong solutions of the Kushner-Stratonovich equation and the Zakai equation is proved. Thirdly, under some extra regularity conditions, the Zakai equation for the unnormalized density is also derived in the case of α-stable Lévy noise.