Stochastic Dierential Equations Research Articles

Quantum Entanglement and its Quantification in a Two-Mode Cascade Laser

In this paper, quantum entanglement of correlated two-mode light generated by a three-level laser coupled to a two-mode squeezed vacuum reservoir is thoroughly analyzed using different inseparability criteria, using the master equation, we obtain the stochastic dierential equation and the correlation properties of the noise forces associated with the normal ordering. Next, we study the photon entanglement by considering different inseparability criteria. In particular, the criteria applied are Duan-Giedke-Cirac-Zoller (DGCZ) criterion, logarithmic negativity, Hillery-Zubairy, and Cauchy-Schwartz inequality and we found that the presence of the squeezing parameter leads to an increase in the degree of entanglement. Moreover, the linear gain coecient significantly achieved the degree of entanglement for the cavity radiation

Control óptimo estocásticio LQR con ruido blanco y colorado: La técnica de programación dinámica

This work deals with LQR optimal control problems where the randomness of the dynamic systems evolves according to: (a) multiplicative and additive white noise, (b) mixture white noise and (c) white noise and colored noise. Total cost quadratic and discounted cost quadratic are studied. The white noise is represented by the Wiener process whereas the colored noise is represented by the solution of the Ornstein-Uhlenbeck’s stochastic dierential equation, whit initial condition is constant or normally distributed. In order to find the value functions and optimal policies, as well as algebraic and stochastic Riccati equations the dynamic programming technique was used. The theoretical results are illustrated by two applications. Numerical simulations are carried out using Matlab.

A Method to Deal With the Critical Case in Stochastic Population Dynamics

In numerous papers, the behaviour of stochastic population models is investigated through the sign of a real quantity which is the growth rate of the population near the extinction set. In many cases, it is proven that when this growth rate is positive, the process is persistent in the long run, while if it is negative, the process converges to extinction. However, the critical case when the growth rate is null is rarely treated. The aim of this paper is to provide a method that can be applied in many situations to prove that in the critical case, the process con-gerves in temporal average to extinction. A number of applications are given, for Stochastic Dierential Equations and Piecewise Deterministic Markov Processes modelling prey-predator, epidemilogical or structured population dynamics.

MODELADO DE SISTEMAS USANDO ECUACIONES DIFERENCIALESESTOCÁSTICAS DE ITÔ

This paper deals with the modeling of systems subject to random perturbations. The main objective is to compare the experimentally measured trajectories with the solutions of the ordinary diferential equation (ODE) and the stochastic dierential equation (SDE) which model the systems analyzed with the purpose of verify if the SDEs capture the random perturbations and therefore, are more appropriate to describe the phenomena with random noise. To this end,the Ito’s calculus is used and numerical simulations of the SDEs are done in MATLAB using the Euler-Maruyama method. As an application of the SDEs, an optimal investment problem is solved in analytic form by following the standard dynamic programming technique.

Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by a truncatedα-stable process with indexα∈ (0, 2). We assume that the process depends on a parameterβ= (θ,σ)Tand we study the sensitivity of the density with respect to this parameter. This extends the results of [E. Clément and A. Gloter, Local asymptotic mixed normality property for discretely observed stochastic dierential equations driven by stable Lévy processes.Stochastic Process. Appl.125 (2015) 2316–2352.] which was restricted to the indexα∈ (1, 2) and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density and its derivative as an expectation and a conditional expectation. This permits to analyze the asymptotic behavior in small time of the density, using the time rescaling property of the stable process.

NUMERICAL SCHEME FOR BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS WITH TIME DELAYED COEFFICIENTS

In this paper, we present some assumptions to get the numerical scheme for backward doubly stochastic dierential delay equations (shortly-BDSDDEs), and we propose a scheme of BDSDDEs and discuss the numerical convergence and rate of convergence of our scheme.

Strong solutions to stochastic differential equations with rough coefficients

We study strong existence and pathwise uniqueness for stochastic dierential equations in R d with rough coecients,

Exact Travelling Wave Solutions for Stochastic Fractional Hirota-Satsuma Coupled KdV Equations

A modied fractional sub-equation method is used for constructing exact travelling wave solutions of nonlinear stochastic fractional partial dierential equations. The main idea of this method is to take full advantage of the fractional Riccati equation, which has many exact solutions. Moreover, white noise functional solutions are obtained for the Wick-type stochastic fractional Hirota-Satsuma coupled KdV equations via Hermite transform and white noise analysis. These solutions include stochastic exponential decay, soliton, and periodic wave solutions.

Exact Simulation of Multi-Dimensional Stochastic Differential Equations

We develop a weak exact simulation technique for a process X dened by a multidimensional stochastic dierential equation (SDE). Namely, for a Lipschitz function g, we propose a simulation based approximation of the expectation E[g(Xt1; ;Xtn)], which by-passes the discretization error. The main idea is to start instead from a wellchosen simulatable SDE whose coecients are up-dated at independent exponential times. Such a simulatable process can be viewed as a regime-switching SDE, or as a branching diusion process with one single living particle at all times. In order to

A Parabolic Transform and Some Stochastic Ill-Posed Problems

It is well known that the Cauchy problem for elliptic partial dierential equations is ill-posed. The question, which arises, how a priori knowledge about solutions can bring about stability? A parabolic transform is dened to discuss the stability of the Cauchy problem for some stochastic partial dierential equations under a priori knowledge about solutions. With the help of the parabolic transform, existence results are established for general linear and nonlinear stochastic partial dierential equations, without any restrictions on the characteristic forms. Many physical and engineering problems in areas like seismology, geophysics and biology require the solutions of ill-posed problems. The Cauchy problem for general stochastic dierential equations has many dierent important applications with amazing range.

Conditional distributions, exchangeable particle systems, and stochastic partial differential equations

On considère des équations aux dérivées partielles stochastiques (EDPS) dont les solutions sont des processus à valeurs dans les mesures de probabilité. Des processus à valeurs mesures de ce type apparaissent naturellement comme des mesures de De Finetti de systèmes infinis de particules échangeables et comme solutions de problèmes de filtrage. En particulier nous considérons un modèle de détermination du prix d’un actif par une famille de traders en compétition. L’évaluation de chaque trader sur l’actif est donnée par la solution d’une équation différentielle stochastique et ce système infini d’EDSs, supposé échangeable, est couplé par un bruit commun et par les prix des actifs. Dans le cadre le plus simple à un seul actif, le prix d’équilibre du marché à tout temps $t$ est donné par un quantile de la mesure de De Finetti déterminé par les évaluations du trader individuel. Dans le cadre à plusieurs actifs, les prix sont donnés essentiellement par la solution d’un problème d’attribution introduit par Shapley et Shubik. L’existence de solutions pour le système échangeable infini est obtenue par un argument d’approximation qui nécessite la dépendance continue des distributions des prix par rapport à la mesure de De Finetti associée. Ceci est vrai si la mesure de De Finetti donne une masse positive à tout ouvert non-vide. La solution de l’EDPS satisfaite par la mesure de De Finetti peut être interprétée comme la distribution conditionnelle de la solution d’une seule EDS donnée par le bruit commun et par le processus du prix. Sous des conditions faibles de non-dégénérescence des coefficients de l’EDS, on montre que la distribution conditionnelle donne une masse positive à tout ouvert non-vide, et sous des conditions légèrement plus fortes, on prouve qu’elle est absolument continue par rapport à la mesure de Lebesgue avec une densité strictement positive. Les résultats sur la distribution conditionnelle constituent la contribution technique principale et ils peuvent être aussi utilisés pour étudier les propriétés de la solution de l’équation de filtrage non-linéaire dans un cadre où le bruit du signal et celui de l’observation sont corrélés.

Relative controllability of semilinear fractional stochastic control systems in Hilbert spaces

This paper is concerned with the relative controllability for a class of dynamical control systems described by semilinear fractional stochastic dierential equations with nonlocal conditions in Hilbert space. Sucient conditions for relative controllability results are obtained using Schaefer's xed point theorem. An example is provided to show the application of our result

Backward stochastic differential equations with respect to general filtrations and applications to insider finance

In this paper, we study backward stochastic dierential equations with respect to general filtrations. The results are used to find the optimal consumption rate for an insider from a cash

Progressive Enlargement of Filtrations and Backward Stochastic Differential Equations with Jumps

This work deals with backward stochastic dierential equation (BSDE) with random marked jumps, and their applications to default risk. We show that these BSDEs are linked with Brownian BSDEs through the decomposition of processes with respect to the progressive enlargement of ltrations. We prove that the equations have solutions if the associated Brownian BSDEs have solutions. We also provide a uniqueness theorem for BSDEs with jumps by giving a comparison theorem based on the comparison for Brownian BSDEs. We give in particular some results for quadratic BSDEs. As applications, we study the pricing and the hedging of a European option in a market with a single jump, and the utility maximization problem in an incomplete market with a nite number of jumps.

Pathwise uniqueness for singular SDEs driven by stable processes

We prove pathwise uniqueness for stochastic dierential equations driven by non-degenerate symmetric -stable L evy processes with values in R d having a bounded and -Holder continuous drift term. We assume > 1 = 2 and 2 (1; 2). The proof requires analytic regularity results for the associated integro-dierenti al operators of Kolmogorov type. We also study dierentia- bility of solutions with respect to initial conditions and the homeomorphism property.

A financial market with interacting investors: does an equilibrium exist?

While trading on a nancial market, the agents we consider take the performance of their peers into account. By maximizing individual utility subject to investment constraints, the agents may ruin each other even unintentionally so that no equilibrium can exist. However, when the agents are willing to waive little expected utility, an approximated equilibrium can be established. The study of the associated backward stochastic dierential equation (BSDE) reveals the mathematical reason for the absence of an equilibrium. Presenting an illustrative counterexample, we explain why such multidimensional quadratic BSDEs may not have solutions despite bounded terminal conditions and in contrast to the one-dimensional case.

Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions

We consider the Cauchy problem in R d for a class of semilinear parabolic partial dier- ential equations that arises in some stochastic control problems. We assume that the coecients are unbounded and locally Lipschitz, not necessarily dierentiable, with continuous data and local uniform ellipticity. We construct a classical solution by approximation with linear parabolic equations. The linear equations involved can not be solved with the traditional results. Therefore, we construct a classical solution to the linear Cauchy problem under the same hypotheses on the coecients for the semilinear equation. Our approach is using stochastic dierential equations and parabolic dierential equations in bounded domains. Finally, we apply the results to a stochastic optimal consumption problem. R

Markov chains and dynamical systems: the open system point of view

Abstract. This article presents several results establishing connections be- tween Markov chains and dynamical systems, from the point of view of open systems in physics. We show how all Markov chains can be understood as the information on one component that we get from a dynamical system on a product system, when losing information on the other component. We show that passing from the deterministic dynamics to the random one is character- ized by the loss of algebra morphism property; it is also characterized by the loss of reversibility. In the continuous time framework, we show that the solu- tions of stochastic dierential equations are actually deterministic dynamical systems on a particular product space. When losing the information on one component, we recover the usual associated Markov semigroup.

Computing conditional Wiener integrals of functionals of a general form

A numerical method of second order of accuracy for computing conditional Wiener integrals of smooth functionals of a general form is proposed. The method is based on simulation of Brownian bridge via the corresponding stochastic dier- ential equations (SDEs) and on ideas of the weak-sense numerical integration of SDEs. A convergence theorem is proved. Special attention is paid to integral-type functionals. A generalization to the case of pinned diusions is considered. Results of some numerical experiments are presented. Keywords. Conditional Wiener measure, Feynman path integrals, Brownian bridge, stochastic dierential equations, weak approximation, Monte Carlo technique, con- ditioned diusions. AMS 2000 subject classi…cation. 65C30, 28C20, 65C05, 60H35.

Set-valued stochastic differential equation in M-type 2 Banach space

Let (›,F,{Ft},P) be a complete probability space with filtra- tion {Ft}, (X ,H,µ) an abstract Wiener space of M-type 2, and {Bt : t ‚ 0} an X -valued Brownian motion such that the distribution of the random func- tion ti1/2Bt : › ! X is µ for any t > 0. We consider the strong solutions to a set-valued stochastic dierential equation with a set-valued drift and a single valued diusion driven by dBt. Under some suitable conditions, the existence and uniqueness of strong solutions are obtained.