On Riccati Matrix Differential Equations

In this paper square Riccati matrix differential equations are considered. The coefficients can be arbitrary time—dependent matrices and need not satisfy any symmetry conditions. Contributions to the basic problems — existence and asymptotic behaviour of solutions — are presented based on two new methods. The first one is the usage of maximum principles for second order linear differential equations, the second one is a variety of possibilities for the parametric representation of solutions of Riccati differential equations.     

Level Hitting Probabilities and Extremal Indexes for Some Particular Dynamical Systems

We establish an exact formula for the distribution of the partial maximum sequence generated by the stationary process obtained by iterations of the Rényi map x → βx mod 1, β = 2, 3, .... We thus obtain a simple proof of some asymptotic behaviour of the extremes and the values of the extremal index. A numerical application is presented.     

Stroock–Varadhan Theorem for Flows Generated by Stochastic Differential Equations with Interaction

We prove a theorem that characterizes the support of a flow generated by a system of stochastic differential equations with interaction.     

Riccati方法与微分方程的渐近积分

考虑二阶常微分方程x″ f(t)x=0,t≥a,(1)假设应用Riccati方法得到方程(1)的主解(principal solution)的一个渐近积分并研究其副解(nonprincipal solutions)的三种不同的渐近性质.主要结果如下:定理1 若(Ⅰ)成立,则方程(1)有解x_1满足及另一解x_2满足x_2(t)=t[1 o(1)]. 反之,若方程(1)有解x(t)→1,t→∞,则(Ⅰ)成立. 定理2 设(Ⅰ)成立.(i)若(Ⅱ)成立,则方程(1)有解x_2使x_2’(t)=1 [tF(t) G(t)][1 o(1)] o(1). (ii) 反之,若方程(1)有解x使x’→1,t→∞,则(Ⅱ)成立. 定理3 若(Ⅲ)和(Ⅳ)成立,则方程(1)有解x_1满足(2)及解x_2满足     

The Link between Stochastic Differential Equations with Non-Markovian Coefficients and Backward Stochastic Partial Differential Equations

In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extension of such kind of link in Markovian framework to non-Markovian framework.Different from Markovian framework, where the corresponding partial differential equation is deterministic, the backward stochastic partial differential equation here has a pair of adapted solutions, and thus the link has a much different form. Moreover, two examples are given to demonstrate the applications of the derived link.     

Backward Doubly Stochastic Differential Equations with Jumps and Stochastic Partial Differential-Integral Equations

Backward doubly stochastic differential equations driven by Brownian motions and Poisson process(BDSDEP) with non-Lipschitz coeffcients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations(SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.     

Backward Stochastic Differential Equations in the Plane

This paper is devoted to study backward stochastic differential equations in the plane driven by a Brownian sheet, where the value of the solution at the corner (s ₀,t ₀) is fixed. The existence and uniqueness of a solution is obtained by means of Picard's approximation scheme and a suitable two-parameter Gronwall's type lemma.     

Stochastic Viability and Comparison Theorems for Mixed Stochastic Differential Equations

For a mixed stochastic differential equation containing both Wiener process and a Hölder continuous process with exponent γ?>?1/2, we prove a stochastic viability theorem. As a consequence, we get a result about positivity of solution and a pathwise comparison theorem. An application to option price estimation is given.     

A New Approach to Linearly Perturbed Riccati Equations Arising in Stochastic Control

In this paper a linearly perturbed version of the well-known matrix Riccati equations which arise in certain stochastic optimal control problems is studied. Via the concepts of mean square stabilizability and mean square detectability we improve previous results on both the convergence properties of the linearly perturbed Riccati differential equation and the solutions of the linearly perturbed algebraic Riccati equation. Furthermore, our approach unifies, in some way, the study for this class of Riccati equations with the one for classical theory, by eliminating a certain inconvenient assumption used in previous works (e.g., [10] and [26]). The results are derived under relatively weaker assumptions and include, inter alia, the following: (a) An extension of Theorem 4.1 of [26] to handle systems not necessarily observable. (b) The existence of a strong solution, subject only to the mean square stabilizability assumption. (c) Conditions for the existence and uniqueness of stabilizing solutions for systems not necessarily detectable. (d) Conditions for the existence and uniqueness of mean square stabilizing solutions instead of just stabilizing. (e) Relaxing the assumptions for convergence of the solution of the linearly perturbed Riccati differential equation and deriving new convergence results for systems not necessarily observable. Accepted 30 July 1996     

The Milstein Scheme for Stochastic Delay Differential Equations Without Using Anticipative Calculus

The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one. The scheme has been extended to the stochastic delay differential equations but the analysis of the convergence is technically complicated due to anticipative integrals in the remainder terms. This article employs an elementary method to derive the Milstein scheme and its first order strong rate of convergence for stochastic delay differential equations.     

Linearly Constrained Global Optimization and Stochastic Differential Equations

A stochastic algorithm is proposed for the global optimization of nonconvex functions subject to linear constraints. Our method follows the trajectory of an appropriately defined Stochastic Differential Equation (SDE). The feasible set is assumed to be comprised of linear equality constraints, and possibly box constraints. Feasibility of the trajectory is achieved by projecting its dynamics onto the set defined by the linear equality constraints. A barrier term is used for the purpose of forcing the trajectory to stay within the box constraints. Using Laplace’s method we give a characterization of a probability measure (Π) that is defined on the set of global minima of the problem. We then study the transition density associated with the projected diffusion process and show that its weak limit is given by Π. Numerical experiments using standard test problems from the literature are reported. Our results suggest that the method is robust and applicable to large-scale problems.     

倒向随机微分方程及其应用

本文将介绍一类新的议程：倒向随机微分方程，为了便于理解，我们将首先通过与常微分方程和经典的随机微分方程的对比，并通过数理经济和数学金融学中的一个典型的例子来引入倒向随机微分方程。     

Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings

We develop a systematic matrix-analytic approach, based on intertwinings of Markov semigroups, for proving theorems about hitting-time distributions for finite-state Markov chains—an approach that (sometimes) deepens understanding of the theorems by providing corresponding sample-path-by-sample-path stochastic constructions. We employ our approach to give new proofs and constructions for two theorems due to Mark Brown, theorems giving two quite different representations of hitting-time distributions for finite-state Markov chains started in stationarity. The proof, and corresponding construction, for one of the two theorems elucidates an intriguing connection between hitting-time distributions and the interlacing eigenvalues theorem for bordered symmetric matrices.     

Stochastic Geometric Models with Non-stationary Spatial Correlations in Lagrangian Fluid Flows

Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration’s “Global Drifter Program”, this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015) is reviewed, in which the spatial correlations are time independent. Two new models, called Model 2 and Model 3, introduce two different symmetry breaking mechanisms by which the spatial correlations may be advected by the flow. These models are derived using reduction by symmetry of stochastic variational principles, leading to stochastic Hamiltonian systems, whose momentum maps, conservation laws and Lie–Poisson bracket structures are used in developing the new stochastic Hamiltonian models of GFD.     

Averaging of Backward Stochastic Differential Equations and Homogenization of Partial Differential Equations with Periodic Coefficients

Abstract

We study the limit of the solutions of systems of semi-linear partial differential equations (PDEs) of second order of parabolic type, with rapidly oscillating periodic coefficients, a singular drift, and singular coefficients of the zero and second order terms. Our basic tool is the approach given by Pardoux [14 Pardoux , E. 1999 . Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: a probabilistic approach . J. Funct. Anal. 167 : 498 – 520 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]]. In particular, we use the weak convergence of an associated backward stochastic differential equation (BSDE).     

Shear Flows of an Ideal Fluid and Elliptic Equations in Unbounded Domains

We prove that, in a two‐dimensional strip, a steady flow of an ideal incompressible fluid with no stationary point and tangential boundary conditions is a shear flow. The same conclusion holds for a bounded steady flow in a half‐plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on one‐dimensional symmetry results for solutions of some semilinear elliptic equations. Some related rigidity results of independent interest are also shown in n‐dimensional slabs in any dimension n.© 2016 Wiley Periodicals, Inc.     

Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations

It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct consequence, a specialized form of the Itô formula is derived. When a standard Brownian motion is the original semimartingale, classical Itô stochastic differential equations driven by the Brownian motion with drift extend to a larger class of stochastic differential equations involving a time-change with continuous paths. A form of the general solution of linear equations in this new class is established, followed by consideration of some examples analogous to the classical equations. Through these examples, each coefficient of the stochastic differential equations in the new class is given meaning. The new feature is the coexistence of a usual drift term along with a term related to the time-change.     

Stochastic Integrals and Stochastic Differential Equations over the Field of p-Adic Numbers

We construct, for various classes of p-adic-valued functions, stochastic integrals with respect to the Poisson random measure. This leads to the construction of Markov processes over the field of p-adic numbers by means of stochastic differential equations.