Global martingale solutions for a stochastic population cross-diffusion system

The existence of global nonnegative martingale solutions to a stochastic cross-diffusion system for an arbitrary but finite number of interacting population species is shown. The random influence of the environment is modeled by a multiplicative noise term. The diffusion matrix is generally neither symmetric nor positive definite, but it possesses a quadratic entropy structure. This structure allows us to work in a Hilbert space framework and to apply a stochastic Galerkin method. The existence proof is based on energy-type estimates, the tightness criterion of Brzeźniak and co-workers, and Jakubowski’s generalization of the Skorokhod theorem. The nonnegativity is proved by an extension of Stampacchia’s truncation method due to Chekroun, Park, and Temam.     

Backward stochastic differential equations with Azéma's martingale

The aim of this paper is to study backward stochastic differential equations (BSDE) driven by Azéma's martingale and the associated deterministic functional equations. More precisely, we introduce BSDE's vs. Azéma's martingale in a general frame, then we prove that the existence of a solution to a Markovian BSDE implies the existence of a solution to a deterministic functional equation of a new type. Uniqueness for the functional equation is proved in a particular case. Then we discuss BSDE's vs. an asymmetric martingale: half Brownian motion/half Azéma's martingale, which leads to an asymmetric deterministic functional equation.     

EXISTENCE OF PERIODIC SOLUTIONS OF A CLASS OF SECOND-ORDER NON-AUTONOMOUS SYSTEMS

In this paper, we are concerned with the existence of periodic solutions of second-order non-autonomous systems. By applying the Schauder''s fixed point theorem and Miranda''s theorem, a new existence result of periodic solutions is established.     

Martingale and stationary solutions for stochastic Navier-Stokes equations

Summary We prove the existence of martingale solutions and of stationary solutions of stochastic Navier-Stokes equations under very general hypotheses on the diffusion term. The stationary martingale solutions yield the existence of invariant measures, when the transition semigroup is well defined. The results are obtained by a new method of compactness.     

Different types of spdes in the eyes of girsanov's theorem

We prove Girsanov's theorem for continuous orthogonal martingale measures. We then define space-time SDEs, and use Girsanov's theorem to establish a oneto- one correspondence between solutions of two space-time SDEs differing only by a drift coefficient. For such stochastic equations, we give necessary conditions under which the laws of their solutions are absolutely continuous with respect to each other. Using Girsanov's theorem again, we prove additional existence and uniqueness results for space-time SDEs. The same one-to-one correspondence and absolute continuity theorems are also proved for the stochastic heat and wave equations     

On Weak Solutions to Stochastic Differential Inclusions Driven by Semimartingales

Abstract

In this paper we consider weak solutions to stochastic inclusions driven by a general semimartingale. We prove the existence of weak solutions and equivalence with the existence of solutions to the martingale problem formulated to such inclusion. Using this we then analyze compactness property of solutions set. Presenting results extend some of those being known for stochastic differential inclusions of Itô's type.     

DISCRETE-TIME MARTINGALES WITH SPATIAL PARAMETERS

Our analysis of a certain stochastic difference equation driven by a martingale k?M(x,k) that depends on a spatial parameter x∈R ^d requires some regularity properties of the underlying martingale be satisfied. Because of their independent interest, we present these regularity properties in this article. We study first the continuity and Lipschitz continuity properties under corresponding conditions on the quadratic covariation of the martingale. We follow this with differentiability and integrability properties. Our analysis of the stochastic difference equation requires a discrete-time version of Itô's formula. The discrete-time Itô formula we have derived involves a martingale transform term. The purpose of the final section is to introduce linear and nonlinear martingale transforms and analyze their properties.     

Martingale solutions of a stochastic wave equation with reflection

We discuss an initial boundary value problem for a one-dimensional stochastic wave equation with reflection. For stochastic parabolic equations with reflection, there are some well-known results. However, there seems to be no existence result for a stochastic wave equation with reflection. Even for a deterministic wave equation, the problem has not been completely resolved. Our goal is to establish the existence of a martingale solution for this problem.     

Regularity property of solution to two-parameter stochastic volterra equation with non-lipschitz coefficients

This article proves the existence and uniqueness of solution to two-parameter stochastic Volterra equation with non-Lipschitz coefficients and driven by Brownian sheet, where the main tool is Bihari's inequality in the plane. Moreover, we also discuss the time regularity property of the solution by Kolmogorov's continuity criterion.     

A stochastic calculus for continuous N-parameter strong martingales

Let M be a 4N-integrable, real-valued continuous N-parameter strong martingale. Burkholder's inequalities prove to be an adequate tool to control the quadratic oscillations of M and the integral processes associated with it (i.e. multiple 1-stochastic integrals with respect to M and its quadratic variation) such that a 1-stochastic calculus for M can be designed. As the main results of this calculus, several Ito-type formulas are established: one in terms of the integral processes associated with M, another one in terms of the so-called ‘variations’, i.e. stochastic measures which arise as the limits of straightforward and simple approximations by Taylor's formula; finally, a third one which is derived from the first by iterated application of a stochastic version of Green's formula and which may be the strong martingale form of a prototype for general martingales.     

Energy methods for stochastic differential equations

This article is devoted to the existence of strong solutions to stochastic differential equations (SDEs). Compared with Ito's theory, we relax the assumptions on the volatility term and replace the global Lipschitz continuity condition with a local Lipschitz continuity condition and a Hoelder continuity condition. In particular, our general SDE covers the Cox–Ingersoll–Ross SDE as a special case. We note that the general weak existence theory presumably extends to our general SDE (although the explicit time dependence of the drift term and the volatility term might require some extra considerations). However, avoiding weak existence theory we prove the existence of a strong solution directly using a priori estimates (the so-called energy estimates) derived from the SDE. The benefit of this approach is that the argument only requires some basic knowledge about stochastic and functional analysis. Moreover, the underlying principle has developed to become one of the cornerstones of the modern theory of partial differential equations (PDEs). In this sense, the general goal of this article is not just to establish the existence of a strong solution to the SDE under consideration but rather to introduce a new principle in the context of SDEs that has already proven to be successful in the context of PDEs.     

Random variables and integral logic

We study model theory of random variables using finitary integral logic. We prove definability of some probability concepts such as having F(u) as distribution function, independence and martingale property. We then deduce Kolmogorov's existence theorem from the compactness theorem.     

Translated Brownian Motions and Associated Wick Products

Abstract

The concept of Wick product is strongly related to the underlying Brownian motion we have fixed on the probability space. Via the Girsanov's theorem we construct a family of new Brownian motions, obtained as translations of the original one, and to each of them we associate a Wick product. This produces a family of Wick products, named γ-Wick products, parameterized by the performed translations. We aim to describe this family of products. We also define a new family of stochastic integrals, which are related in a natural way to the γ-Wick products.     

Invariant measure for the stochastic Ginzburg Landau equation

The existence of martingale solutions and stationary solutions of stochastic Ginzburg-Landau equations under general hypothesizes on the dimension, the non linear term and the added noise is investigated. With a few more assumptions, it is established that the transition semi-group is well defined and that the stationary martingale solution yields the existence of an invariant measure. Moreover this invariant measure is shown to be unique.     

An approximation theorem of wong-zakai type for nonlinear stochastic partial differential equations

We present an extension of the Wong-Zakai approximation theorem for nonlinear 984 given by the Wiener process and a martingale. By approximating these disturbances we obtain in the limit equation the Ito correction term for the infinite dimensional case. Such form of the correction term connected with the Wiener process was proved in the author's papers [21] and [22], where the approximation theorem for semilinear stochastic evolution equations in Hilbert spaces was studied. Our model here is similar as the one considered by Pardoux [17]     

Closed Form Approximations for Spread Options

Abstract

Market mechanisms are increasingly being used as a tool for allocating somewhat scarce but unpriced rights and resources, and the European Emission Trading Scheme is an example. By means of dynamic optimization in the contest of firms covered by such environmental regulations, this article generates endogenously the price dynamics of emission permits under asymmetric information, allowing inter-temporal banking and borrowing. In the market, there are a finite number of firms and each firm's pollution emission follows an exogenously given stochastic process. We prove the discounted permit price is a martingale with respect to the relevant filtration. The model is solved numerically. Finally, a closed-form pricing formula for European-style options is derived.     

Set-Indexed Itô Calculus Along Paths

Abstract

Set-indexed stochastic analysis and set-indexed stochastic calculus are faced here with a new approach of dimension's reduction. We introduce a new tool (main flow) in order to deal with one-parameter calculus in set-indexed framework. We prove an Itô formula for any Brownian functional where the Brownian component is not a martingale on the whole set of indices but induces such a martingale. As first extensions, we provide definitions of bracket and local time in set-indexed context.     

Martingale solutions of nematic liquid crystals driven by pure jump noise in the Marcus canonical form

In this work we consider a stochastic evolution equation which describes the system governing the nematic liquid crystals driven by a pure jump noise in the Marcus canonical form. The existence of a martingale solution is proved for both 2D and 3D cases. The construction of the solution relies on a modified Faedo–Galerkin method based on the Littlewood–Paley-decomposition, compactness method and the Jakubowski version of the Skorokhod representation theorem for non-metric spaces. We prove that in the 2-D case the martingale solution is pathwise unique and hence deduce the existence of a strong solution.     

Diffusion-Approximations for Navier-Stokes Equation in Space-Time Gaussian Velocity Field

Abstract

By approximation methods, the existence of solutions for the Navier-Stokes equation with rapidly oscillating drift term for dimention 2 or 3 is proved. Also by the martingale method, stochastic Navier-Stokes equation is considered.     

Burkholder-Gundy-Davis inequality in martingale Hardy spaces with variable exponent

In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.