Martingale Solutions Research Articles

Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise

We study in this article the stochastic Zakharov-Kuznetsov equation driven by a multiplicative noise.We establish, in space dimensions two and three the global existence of martingale solutions, and in space dimension two the global pathwise uniqueness and the existence of pathwise solutions.New methods are employed to deal with a special type of boundary conditions and to verify the pathwise uniqueness of martingale solutions with a lack of regularity, where both difficulties arise due to the partly hyperbolic feature of the model.

On Martingale Problems with Continuous-Time Mixing and Values of Zero-Sum Games without the Isaacs Condition

We consider a zero-sum stochastic differential game over elementary mixed feedback strategies. These are strategies based only on the knowledge of the past state, randomized continuously in time from a sampling distribution which is kept constant in between some stopping rules. Once both players choose such strategies, the state equation admits a unique solution in the sense of the martingale problem of Stroock and Varadhan. We show that the game defined over martingale solutions has a value, which is the unique continuous viscosity solution of the randomized Isaacs equation.

AN INVARIANCE PRINCIPLE FOR THE TAGGED PARTICLE PROCESS IN CONTINUUM WITH SINGULAR INTERACTION POTENTIAL

We consider the dynamics of a tagged particle in an infinite particle environment moving according to a stochastic gradient dynamics. For singular interaction potentials this tagged particle dynamics was constructed first in Ref. 7, using closures of pre-Dirichlet forms which were already proposed in Refs. 13 and 24. The environment dynamics and the coupled dynamics of the tagged particle and the environment were constructed separately. Here we continue the analysis of these processes: Proving an essential m-dissipativity result for the generator of the coupled dynamics from Ref. 7, we show that this dynamics does not only contain the environment dynamics (as one component), but is, given the latter, the only possible choice for being the coupled process. Moreover, we identify the uniform motion of the environment as the reversed motion of the tagged particle. (Since the dynamics are constructed as martingale solutions on configuration space, this is not immediate.) Furthermore, we prove ergodicity of the environment dynamics, whenever the underlying reference measure is a pure phase of the system. Finally, we show that these considerations are sufficient to apply Ref. 4 for proving an invariance principle for the tagged particle process. We remark that such an invariance principle was studied before in Ref. 13 for smooth potentials, and shown by abstract Dirichlet form methods in Ref. 24 for singular potentials. Our results apply for a general class of Ruelle measures corresponding to potentials possibly having infinite range, a non-integrable singularity at 0 and a nontrivial negative part, and fulfill merely a weak differentiability condition on ℝd\{0}.

A convergent finite-element-based discretization of the stochastic Landau-Lifshitz-Gilbert equation

We propose a convergent finite-element-based discretization of the stochastic Landau–Lifshitz–Gilbert equation. The main difficulties in the convergence analysis for this nonlinear stochastic partial differential equation are to properly address the pointwise sphere condition and, in the fully discrete scheme, the Stratonovich noise. Approximations of the scheme proposed here satisfy a sphere constraint at nodal points of the spatial discretization and have finite energies, and their increments may be controlled uniformly with respect to discretization parameters. Sequences of corresponding continuified processes may then be generated which construct weak martingale solutions of the limiting equations.

ON THE EVOLUTIONARY STABILITY OF RATIONAL EXPECTATIONS

Evolutionary game theory provides a fresh perspective on the prospect that agents with heterogeneous expectations might eventually come to agree on a single expectation corresponding to the efficient markets hypothesis. We establish conditions under which agreement on a unique forecast is stable, but also show that persistent heterogeneous expectations can arise if those conditions do not hold. The critical element is the degree of curvature in the payoff weighting functions agents use to value forecasting performance. We illustrate our results in the context of an asset pricing model where a martingale solution competes with the fundamental solution for agents' attention.

BV functions in a Gelfand triple for differentiable measure and its applications

Abstract In this paper, we introduce a definition of BV functions for (non-Gaussian) differentiable measure in a Gelfand triple which is an extension of the definition of BV functions in [Ann. Probab. 40 (2012), 1759–1794], using Dirichlet form theory. By this definition, we can analyze the reflected stochastic quantization problem associated with a self-adjoint operator A and a cylindrical Wiener process on a convex set Γ in a Banach space E. We prove the existence of a martingale solution of this problem if Γ is a regular convex set.

Stochastic stability of the Ekman spiral

Consider the stochastic Navier-Stokes-Coriolis equations in T2 × (0, b) subject to Dirichlet boundary conditions as well as the Ekman spiral which is a stationary solution to the deterministic equations. It is proved that the stochastic Navier-Stokes-Coriolis equation admits a weak martingale solution. Moreover, as an stochastic analogue of the existing deterministic stability results for the Ekman spiral, stochastic stability of the Ekman spiral is proved by considering stationary martingale solutions.

Existence of a martingale solution of the stochastic Navier–Stokes equations in unbounded 2D and 3D domains

Stochastic Navier–Stokes equations in 2D and 3D possibly unbounded domains driven by a multiplicative Gaussian noise are considered. The noise term depends on the unknown velocity and its spatial derivatives. The existence of a martingale solution is proved. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. Moreover, some compactness and tightness criteria in nonmetric spaces are proved. Compactness results are based on a certain generalization of the classical Dubinsky Theorem.

Martingale Solution to Equations for Differential Type Fluids of Grade Two Driven by Random Force of Lévy Type

In this article we study a system of nonlinear non-parabolic stochastic evolution equations driven by Levy noise type. This system describes the motion of second grade fluids driven by random force. Global existence of a martingale solution is proved under general conditions on the noise. Since the coefficient of the noise does not satisfy a Lipschitz property, we could not prove any pathwise uniqueness result. We note that this is the first work dealing with a stochastic model for non-Newtonian fluids excited by external forces of Levy noise type.

Stochastic Navier–Stokes Equations Driven by Lévy Noise in Unbounded 3D Domains

Martingale solutions of the stochastic Navier–Stokes equations in 2D and 3D possibly unbounded domains, driven by the Lévy noise consisting of the compensated time homogeneous Poisson random measure and the Wiener process are considered. Using the classical Faedo–Galerkin approximation and the compactness method we prove existence of a martingale solution. We prove also the compactness and tightness criteria in a certain space contained in some spaces of càdlàg functions, weakly càdlàg functions and some Fréchet spaces. Moreover, we use a version of the Skorokhod Embedding Theorem for nonmetric spaces.

A New Term Risk Dynamic for Q Jumps for Measuring Systemic Risk

We propose a model which enables the measurement of term risk in markets which are sensitive to systemic risk. With its origins in the spectralisation of the AR(1) process (using the Wiener-Khintchine theorem, and a P ~ Q transform), a Q jump martingale solution is found which is unique and independent of the wiener process. The model is tested, in differential equation form, on the risk premia generated in the yield curve, the credit spread of risky bonds, and the term risk in the implied volatility skew (forward variance). An excellent agreement, in both graphical and regression forms for the scale and patterns of term risk premia, is displayed. Because these measures also typify systemic risk characteristics (with their traded risk versions seen in the CDS and forward VIX markets), the model also defines a useful connection between systemic (bank distress) risk with the Q jump systematic risk.

Weak Solutions of a Stochastic Landau-Lifshitz-Gilbert Equation

The Landau-Lifshitz-Gilbert equation perturbed by a multiplicative space- dependent noise is considered for a ferromagnet filling a bounded three-dimensional do- main. We show the existence of weak martingale solutions taking values in a sphere S 2 . The regularity of weak solutions is also discussed. Some of the regularity results are new even for the deterministic Landau-Lifshitz-Gilbert equation.

On the 3-D stochastic magnetohydrodynamic-[formula omitted] model

We consider the stochastic three dimensional magnetohydrodynamic-α model (MHD-α) which arises in the modeling of turbulent flows of fluids and magnetofluids. We introduce a suitable notion of weak martingale solution and prove its existence. We also discuss the relation of the stochastic 3D MHD-α model to the stochastic 3D magnetohydrodynamic equations by proving a convergence theorem, that is, as the length scale α tends to zero, a subsequence of weak martingale solutions of the stochastic 3D MHD-α model converges to a certain weak martingale solution of the stochastic 3D magnetohydrodynamic equations. Finally, we prove the existence and uniqueness of the probabilistic strong solution of the 3D MHD-α under strong assumptions on the external forces.

STOCHASTIC QUASI-GEOSTROPHIC EQUATION

In this note we study the 2D stochastic quasi-geostrophic equation in 𝕋2 for general parameter α ∈ (0, 1) and multiplicative noise. We prove the existence of martingale solutions and pathwise uniqueness under some condition in the general case, i.e. for all α ∈ (0, 1). In the subcritical case α > 1/2, we prove existence and uniqueness of (probabilistically) strong solutions and construct a Markov family of solutions. In particular, it is uniquely ergodic for α > 2/3 provided the noise is non-degenerate. In this case, the convergence to the (unique) invariant measure is exponentially fast. In the general case, we prove the existence of Markov selections.

Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise

In this paper, we establish the solvability of martingale solutions for the stochastic Navier-Stokes equations with Ito-Levy noise in bounded and unbounded domains in $ \mathbb{R} ^d$,$d=2,3.$ The tightness criteria for the laws of a sequence of semimartingales is obtained from a theorem of Rebolledo as formulated by Metivier for the Lusin space valued processes. The existence of martingale solutions (in the sense of Stroock and Varadhan) relies on a generalization of Minty-Browder technique to stochastic case obtained from the local monotonicity of the drift term.

Stochastic evolution equations driven by Lévy processes

Existence of weak (martingale) solutions and pathwise uniqueness are established for stochastic evolution equations driven by Levy processes.

Local martingale and pathwise solutions for an abstract fluids model

We establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system. The primary application of this abstract framework is to infer the local existence of strong, pathwise solutions to the 3D primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise. Instead of developing our results, specifically for the 3D primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations (atmosphere, oceans, coupled atmosphere–ocean, on the sphere, on the β -plane approximation etc. and the incompressible Navier–Stokes equations). In applications, all the details are given for the β -plane approximation of the equations of the ocean.

Martingale solutions and Markov selection of stochastic 3D Navier–Stokes equations with jump

In this paper, we study the existence of martingale solutions of stochastic 3D Navier–Stokes equations with jump, and following Flandoli and Romito (2008) [7] and Goldys et al. (2009) [8], we prove the existence of Markov selections for the martingale solutions.

N/V-LIMIT FOR LANGEVIN DYNAMICS IN CONTINUUM

We construct an infinite particle/infinite volume Langevin dynamics on the space of simple configurations in ℝd having velocities as marks. The construction is done via a limiting procedure using N-particle dynamics in cubes (-λ, λ]d with periodic boundary condition. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of N-particle systems in (-λ, λ]d with periodic boundary condition. After proving tightness of the laws of the finite particle dynamics, the identification of accumulation points as martingale solutions of the Langevin equation is based on a general study of properties of measures on configuration space fulfilling a uniform Ruelle bound (and their weak limits). Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for a wide class of repulsive interaction potentials ϕ (including, e.g., the Lennard–Jones potential) and all temperatures, densities and dimensions d ≥ 1.

Regular solutions for multiplicative stochastic Landau-Lifshitz-Gilbert equation and blow-up phenomena

The two-dimensional Landau-Lifshitz-Gilbert equation of motion for a classical magnetic moment perturbed by a multiplicative noise is considered. This equation is highly nonlinear in nature and, for this reason, many mathematical results in stochastic partial differential equations (SPDEs) cannot be applied. The aim of this work is to introduce the difference method to handle SPDEs and prove the existence of regular martingale solutions in dimension two. Some blow-up phenomena are presented, which are drastically different from the deterministic case. Finally, to yield correct thermal-equilibrium properties, Stratonovitch integral is used instead of Ito integral.