Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.     

A Note On the Existence of Consistent Estimatein the Simple Regression Model

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A Dichotomy for the Gelfand–Kirillov Dimensions of Simple Modules over Simple Differential Rings

The Gelfand–Kirillov dimension has gained importance since its introduction as a tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand–Kirillov dimension of simple modules over certain simple rings of differential operators. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. McConnell (J. Algebra 76(2), 489–493, 1982) concerning this dimension for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand–Kirillov dimension of an induced module.     

Permanence of Stochastic Lotka–Volterra Systems

This paper proposes a new definition of permanence for stochastic population models, which overcomes some limitations and deficiency of the existing ones. Then, we explore the permanence of two-dimensional stochastic Lotka–Volterra systems in a general setting, which models several different interactions between two species such as cooperation, competition, and predation. Sharp sufficient criteria are established with the help of the Lyapunov direct method and some new techniques. This study reveals that the stochastic noises play an essential role in the permanence and characterize the systems being permanent or not.     

A Signed Generalization of the Bernoulli–Laplace Diffusion Model

We bound the rate of convergence to stationarity for a signed generalization of the Bernoulli–Laplace diffusion model; this signed generalization is a Markov chain on the homogeneous space ( ₂S _n )/(S _r ×S _n–r ). Specifically, for r not too far from n/2, we determine that, to first order in n, n log n steps are both necessary and sufficient for total variation distance to become small. Moreover, for r not too far from n/2, we show that our signed generalization also exhibits the cutoff phenomenon.     

Stochastic continuity equations–A general uniqueness result

We showexistence and uniqueness of L^∞-solutions to stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition. The main issue of uniqueness is proved reeling on stochastic characteristic method.     

On the Zakharov–L’vov Stochastic Model for Wave Turbulence

Doklady Mathematics - In this paper we discuss a number of rigorous results in the stochastic model for wave turbulence due to Zakharov–L’vov. Namely, we consider the damped/driven...     

Stochastic Evolution of Augmented Born–Infeld Equations

This paper compares the results of applying a recently developed method of stochastic uncertainty quantification designed for fluid dynamics to the Born–Infeld model of nonlinear electromagnetism. The similarities in the results are striking. Namely, the introduction of Stratonovich cylindrical noise into each of their Hamiltonian formulations introduces stochastic Lie transport into their dynamics in the same form for both theories. Moreover, the resulting stochastic partial differential equations retain their unperturbed form, except for an additional term representing induced Lie transport by the set of divergence-free vector fields associated with the spatial correlations of the cylindrical noise. The explanation for this remarkable similarity lies in the method of construction of the Hamiltonian for the Stratonovich stochastic contribution to the motion in both cases, which is done via pairing spatial correlation eigenvectors for cylindrical noise with the momentum map for the deterministic motion. This momentum map is responsible for the well-known analogy between hydrodynamics and electromagnetism. The momentum map for the Maxwell and Born–Infeld theories of electromagnetism treated here is the 1-form density known as the Poynting vector. Two appendices treat the Hamiltonian structures underlying these results.

     

A Reaction–Diffusion Model of Vector-Borne Disease with Periodic Delays

Journal of Nonlinear Science - A vector-borne disease is caused by a range of pathogens and transmitted to hosts through vectors. To investigate the multiple effects of the spatial heterogeneity,...     

Stochastic goal programming: A mean–variance approach

We propose a stochastic goal programming (GP) model leading to a structure of mean–variance minimisation. The solution to the stochastic problem is obtained from a linkage between the standard expected utility theory and a strictly linear, weighted GP model under uncertainty. The approach essentially consists in specifying the expected utility equation corresponding to every goal. Arrow's absolute risk aversion coefficients play their role in the calculation process. Once the model is defined and justified, an illustrative example is developed.     

Asymptotic Normality of LS Estimate in Simple Linear EV Regression Model

Though EV model is theoretically more appropriate for applications in which measurement errors exist, people are still more inclined to use the ordinary regression models and the traditional LS method owing to the difficulties of statistical inference and computation. So it is meaningful to study the performance of LS estimate in EV model. In this article we obtain general conditions guaranteeing the asymptotic normality of the estimates of regression coefficients in the linear EV model. It is noticeable that the result is in some way different from the corresponding result in the ordinary regression model.     

Simple nonstandard proofs of Daniell–Kolmogorov-type theorems

This brief note employs a proof technique essentially due to Lindstr?m, based on the Loeb measure construction, to obtain short and simple proofs of theorems of the Daniell?CKolmogorov type.     

A Notion of Stochastic Input-to-State Stability and Its Application to Stability of Cascaded Stochastic Nonlinear Systems

In this paper, the property of practical input-to-state stability and its application to stability of cascaded nonlinear systems are investigated in the stochastic framework. Firstly, the notion of （practical） stochastic input-to-state stability with respect to a stochastic input is introduced, and then by the method of changing supply functions, （a） an （practical） SISS-Lyapunov function for the overall system is obtained from the corresponding Lyapunov functions for cascaded （practical） SISS subsystems.     

Stochastic Ordering of Infinite Geometric Galton–Watson Trees

We consider Galton–Watson trees with Geom\((p)\) offspring distribution. We let \(T_{\infty }(p)\) denote such a tree conditioned on being infinite. We prove that for any \(1/2\le p_1 <p_2 \le 1\), there exists a coupling between \(T_{\infty }(p_1)\) and \(T_{\infty }(p_2)\) such that \({\mathbb {P}}(T_{\infty }(p_1) \subseteq T_{\infty }(p_2))=1\).     

A Note on the Rate of Convergence of the Euler–Maruyama Method for Stochastic Differential Equations

Stochastic processes are approximated by wavelet operators. For the stochastic process and the scale function satisfying some conditions, a new method is given to estimate the degree of approximation.     

A Note on Stochastic Burgers’ System of Equations

Abstract

We consider a stochastic version of a system of two equations formulated by Burgers [Burgers, J.M. Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Verh. Kon. Nerderl. Akad. Weten-Schappen Amsterdam, Afdeel Natuurkunde, 1939, 17 (2), 1–53] with the aim to describe the laminar and turbulent motions of a fluid in a channel. The existence and uniqueness theorem for a global solution is established. The paper generalizes the result from the paper by Da Prato and Ga¸tarek [Da Prato, G.; Ga¸tarek, D. Stochastic Burgers equation with correlated noise. Stochastics Stochastics Rep. 1995, 52, 29–41] dealing with the equation describing only the turbulent motion.