Weak second-order stochastic Runge–Kutta methods for non-commutative stochastic differential equations

A new explicit stochastic Runge–Kutta scheme of weak order 2 is proposed for non-commutative stochastic differential equations (SDEs), which is derivative-free and which attains order 4 for ordinary differential equations. The scheme is directly applicable to Stratonovich SDEs and uses 2m-1$2m-1$ random variables for one step in the m-dimensional Wiener process case. It is compared with other derivative-free and weak second-order schemes in numerical experiments.     

Regularity properties of viscosity solutions of integro-partial differential equations of Hamilton–Jacobi–Bellman type

We study the regularity properties of integro-partial differential equations of Hamilton–Jacobi–Bellman type with the terminal condition, which can be interpreted through a stochastic control system, composed of a forward and a backward stochastic differential equation, both driven by a Brownian motion and a compensated Poisson random measure. More precisely, we prove that, under appropriate assumptions, the viscosity solution of such equations is jointly Lipschitz and jointly semiconcave in (t,x)∈Δ×R^d$(t,x)\in \Delta \times {\mathbb{R}}^d$, for all compact time intervals Δ$\Delta$ excluding the terminal time. Our approach is based on the time change for the Brownian motion and on Kulik’s transformation for the Poisson random measure.     

Approximations of Euler–Maruyama type for stochastic differential equations with Markovian switching,under non-Lipschitz conditions

We develop the Euler–Maruyama scheme for a class of stochastic differential equations with Markovian switching (SDEwMSs) under non-Lipschitz conditions  . Both L¹$L^1$ and L²$L^2$-convergence are discussed under different non-Lipschitz conditions. To overcome the mathematical difficulties arisen from the Markovian switching as well as the non-Lipschitz coefficients, several new analytical techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems.     

The interrelation between stochastic differential inclusions and set-valued stochastic differential equations

In this paper we connect the well established theory of stochastic differential inclusions with a new theory of set-valued stochastic differential equations. Solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty, bounded, convex and closed subsets of the space L²$L^2$ consisting of square integrable random vectors. We show that for the solution X$X$ to a set-valued stochastic differential equation corresponding to a stochastic differential inclusion, there exists a solution x$x$ for this inclusion that is a _‖⋅‖L2${\Vert \cdot \Vert }_{L^2}$-continuous selection of X$X$. This result enables us to draw inferences about the reachable sets of solutions for stochastic differential inclusions, as well as to consider the viability problem for stochastic differential inclusions.     

Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations

In this paper we establish lower and upper Gaussian bounds for the solutions to the heat and wave equations driven by an additive Gaussian noise, using the techniques of Malliavin calculus and recent density estimates obtained by Nourdin and Viens in [17]. In particular, we deal with the one-dimensional stochastic heat equation in [0, 1] driven by the space-time white noise, and the stochastic heat and wave equations in R^d${\mathbb{R}}^d$ (d≥1$d\ge 1$ and d≤3$d\le 3$, respectively) driven by a Gaussian noise which is white in time and has a general spatially homogeneous correlation.     

Comparison theorems for anticipated BSDEs with non-Lipschitz coefficients

We study comparison theorems for one dimensional anticipated backward stochastic differential equations under one kind of non-Lipschitz assumption. In the results, the generator functions are allowed to contain the anticipated term of z, neither generator function needs to be necessarily monotone in the anticipated term of y  , and the anticipated times of the anticipated terms of (y,z)$(y,z)$ in one generator function can differ from those in the other.     

Hybrid function method for solving Fredholm and Volterra integral equations of the second kind

Numerical solutions of Fredholm and Volterra integral equations of the second kind via hybrid functions, are proposed in this paper. Based upon some useful properties of hybrid functions, integration of the cross product, a special product matrix and a related coefficient matrix   with optimal order, are applied to solve these integral equations. The main characteristic of this technique is to convert an integral equation into an algebraic; hence, the solution procedures are either reduced or simplified accordingly. The advantages of hybrid functions are that the values of n$n$ and m$m$ are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function, for the solutions of integral equations. We propose that the available optimal values of n$n$ and m$m$ can minimize the relative errors of the numerical solutions. The high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples. The hybrid function method is superior to other piecewise constant orthogonal functions [W.F. Blyth, R.L. May, P. Widyaningsih, Volterra integral equations solved in Fredholm form using Walsh functions, Anziam J. 45 (E) (2004) C269–C282; M.H. Reihani, Z. Abadi, Rationalized Haar functions method for solving Fredholm and Volterra integral equations, J. Comp. Appl. Math. 200 (2007) 12–20] for these problems.     

Stability of IMEX Runge–Kutta methods for delay differential equations

Stability of IMEX (implicit–explicit) Runge–Kutta methods applied to delay differential equations (DDEs) is studied on the basis of the scalar test equation du/dt=λu(t)+μu(t-τ)$\mathrm{d}u/\mathrm{d}t=\lambda u(t)+\mu u(t-\tau )$, where τ$\tau$ is a constant delay and λ,μ$\lambda ,\mu$ are complex parameters. More specifically, P-stability regions of the methods are defined and analyzed in the same way as in the case of the standard Runge–Kutta methods. A new IMEX method which possesses a superior stability property for DDEs is proposed. Some numerical examples which confirm the results of our analysis are presented.     

Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

We develop a notion of nonlinear expectation–G$G$-expectation–generated by a nonlinear heat equation with infinitesimal generator G$G$. We first study multi-dimensional G$G$-normal distributions. With this nonlinear distribution we can introduce our G$G$-expectation under which the canonical process is a multi-dimensional G$G$-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô’s type with respect to our G$G$-Brownian motion, and derive the related Itô’s formula. We have also obtained the existence and uniqueness of stochastic differential equations under our G$G$-expectation.     

On the width of hybrid zones

Hybrid zones occur when two species are found in close proximity and interbreeding occurs, but the species’ characteristics remain distinct. These systems have been treated in the biology literature using partial differential equations models. Here we investigate a stochastic spatial model and prove the existence of a stationary distribution that represents the hybrid zone in equilibrium. We calculate the width of the hybrid zone, which agrees with the PDE formula only in dimensions d≥3$d\ge 3$. Our results also give insight into properties of hybrid zones in patchy environments.     

Correlations and bounds for stochastic volatility models

We investigate here, systematically and rigorously, various stochastic volatility models used in Mathematical Finance. Mathematically, such models involve coupled stochastic differential equations with coefficients that do not obey the natural and classical conditions required to make these models “well-posed”. And we obtain necessary and sufficient conditions on the parameters, such as correlation, of these models in order to have integrable or Lp$L^p$ solutions (for 1<p<∞$1<p<\infty$).     

Regularity for solutions of nonlocal,nonsymmetric equations

We study the regularity for solutions of fully nonlinear integro differential equations with respect to nonsymmetric kernels. More precisely, we assume that our operator is elliptic with respect to a family of integro differential linear operators where the symmetric parts of the kernels have a fixed homogeneity σ and the skew symmetric parts have strictly smaller homogeneity τ  . We prove a weak ABP estimate and C^1,α$C^{1,\alpha }$ regularity. Our estimates remain uniform as we take σ→2$\sigma \to 2$ and τ→1$\tau \to 1$ so that this extends the regularity theory for elliptic differential equations with dependence on the gradient.     

Resolution of the Transport equation subject to constraint

We present a method for solving the Transport equation when its solution has to belong to a constrained set which is not required to be convex. An autonomous formulation of the characteristics method allows us to use the tangency condition which has been introduced for ordinary differential equations. Thus we obtain a sufficient condition for existence of solutions, which shows the interplay between the geometry of the constraints set K   and the velocity field β$\beta$. A numerical method is proposed for solving the problem when the sufficient condition is not satisfied. A numerical experiment is presented showing the efficiency of the algorithm proposed.     

One-dimensional stochastic differential equations with generalized and singular drift

Introducing certain singularities, we generalize the class of one-dimensional stochastic differential equations with so-called generalized drift. Equations with generalized drift, well-known in the literature, possess a drift that is described by the semimartingale local time of the unknown process integrated with respect to a locally finite signed measure ν$\nu$. The generalization which we deal with can be interpreted as allowing more general set functions ν$\nu$, for example signed measures which are only σ$\sigma$-finite. However, we use a different approach to describe the singular drift. For the considered class of one-dimensional stochastic differential equations, we derive necessary and sufficient conditions for existence and uniqueness in law of solutions.     

Mean-square convergence of stochastic multi-step methods with variable step-size

We study mean-square consistency, stability in the mean-square sense and mean-square convergence of drift-implicit linear multi-step methods with variable step-size for the approximation of the solution of Itô stochastic differential equations. We obtain conditions that depend on the step-size ratios and that ensure mean-square convergence for the special case of adaptive two-step-Maruyama schemes. Further, in the case of small noise we develop a local error analysis with respect to the h$h$–ε$\epsilon$ approach and we construct some stochastic linear multi-step methods with variable step-size that have order 2 behaviour if the noise is small enough.     

On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain

We study the global existence of weak solutions to a multi-dimensional simplified Ericksen–Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, where N=2 or 3$N=2\text{or}3$. By exploiting a maximum principle, Nirenberg?s interpolation inequality and a smallness condition imposed on the N  -th component of initial direction field _d0${\mathbf{d}}_0$ to overcome the difficulties induced by the supercritical nonlinearity |∇d|²d${|\mathrm{\nabla }\mathbf{d}|}^2\mathbf{d}$ in the equations of angular momentum, and then adapting a modified three-dimensional approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity.     

Linearly repetitive Delone sets are rectifiable

We show that every linearly repetitive Delone set in the Euclidean d  -space R^d${\mathbb{R}}^d$, with d?2$d?2$, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice Z^d${\mathbb{Z}}^d$. In the particular case when the Delone set X   in R^d${\mathbb{R}}^d$ comes from a primitive substitution tiling of R^d${\mathbb{R}}^d$, we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from X   to the lattice βZ^d$\beta {\mathbb{Z}}^d$ for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.