Asymptotics for stochastic reaction–diffusion equation driven by subordinate Brownian motion

We study the ergodicity of stochastic reaction–diffusion equation driven by subordinate Brownian motion. After establishing the strong Feller property and irreducibility of the system, we prove the tightness of the solution’s law. These properties imply that this stochastic system admits a unique invariant measure according to Doob’s and Krylov–Bogolyubov’s theories. Furthermore, we establish a large deviation principle for the occupation measure of this system by a hyper-exponential recurrence criterion. It is well known that S(P)DEs driven by $\alpha$-stable type noises do not satisfy Freidlin–Wentzell type large deviation, our result gives an example that strong dissipation overcomes heavy tailed noises to produce a Donsker–Varadhan type large deviation as time tends to infinity.     

Dual regularization and Pontryagin’s maximum principle in a problem of optimal boundary control for a parabolic equation with nondifferentiable functionals

A problem of optimal boundary control is considered for a divergent linear parabolic equation. Equality constraints of the problem are given by nondifferentiable functionals. A dual regularization algorithm stable to errors in initial data is constructed for solving the problem. Pontryagin’s maximum principle plays the key role in this algorithm.     

Kolmogorov equation and large-time behaviour for fractional brownian motion driven linear sde’s

We consider a stochastic process X _t ^x which solves an equation

Power law Pólya’s urn and fractional Brownian motion

We introduce a natural family of random walks $S_n$ on $\mathbb{Z }$ that scale to fractional Brownian motion. The increments $X_n := S_n - S_{n-1} \in \{\pm 1\}$ have the property that given $\{ X_k : k < n \}$ , the conditional law of $X_n$ is that of $X_{n - k_n}$ , where $k_n$ is sampled independently from a fixed law $\mu $ on the positive integers. When $\mu $ has a roughly power law decay (precisely, when $\mu $ lies in the domain of attraction of an $\alpha $ -stable subordinator, for $0<\alpha <1/2$ ) the walks scale to fractional Brownian motion with Hurst parameter $\alpha + 1/2$ . The walks are easy to simulate and their increments satisfy an FKG inequality. In a sense we describe, they are the natural “fractional” analogues of simple random walk on $\mathbb{Z }$ .     

Dynamic programming principle for stochastic control problems driven by general Lévy noise

We extend the proof of the dynamic programming principle (DPP) for standard stochastic optimal control problems driven by general Lévy noise. Under appropriate assumptions, it is shown that the DPP still holds when the state process fails to have any moments at all.     

Singleton sets random attractor for stochastic FitzHugh–Nagumo lattice equations driven by fractional Brownian motions

The paper is devoted to the study of the dynamical behavior of the solutions of stochastic FitzHugh–Nagumo lattice equations, driven by fractional Brownian motions, with Hurst parameter greater than 1/2. Under some usual dissipativity conditions, the system considered here features different dynamics from the same one perturbed by Brownian motion. In our case, the random dynamical system has a unique random equilibrium, which constitutes a singleton sets random attractor.     

The first variation and Pontryagin’s maximum principle in optimal control for partial differential equations

A modification of the classical needle variation, namely, the so-called two-parameter variation of controls is proposed. The first variation of a functional is understood as a repeated limit. It is shown that the modified needle variation can be effectively used to derive necessary optimality conditions for a rather wide class of optimal control problems involving partial differential equations with weak solutions. Specifically, the two-parameter variation is used to obtain necessary optimality conditions in the form of a maximum principle for the optimal control of divergent hyperbolic equations.     

Estimation of the lead–lag parameter between two stochastic processes driven by fractional Brownian motions

Statistical Inference for Stochastic Processes - In this paper, we consider the problem of estimating the lead–lag parameter between two stochastic processes driven by fractional Brownian...     

Fractional stochastic integration and Black–Scholes equation for fractional Brownian model with stochastic volatility

We modify the Hu-Øksendal and Elliot-van der Hoek approach to arbitrage-free financial markets driven by a fractional Brownian motion that is defined on a white noise space. We deduce and solve a Black–Scholes fractional equation for constant volatility and outline the corresponding equation with stochastic volatility. As an auxiliary result, we produce some simple conditions implying the existence of the Wick integral w.r.t. fractional noise.     

Invariant measure for the stochastic Cauchy problem driven by a cylindrical Lévy process

In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process, and show that these conditions are also necessary if the semigroup is stable, in which case the invariant measure is unique. For typical situations such as the heat equation, we significantly simplify these conditions without assuming any further restrictions on the driving cylindrical Lévy process and demonstrate their application in some examples.     

Maximum principle for anticipated recursive stochastic optimal control problem with delay and Lvy processes

In this paper,we study the stochastic maximum principle for optimal control problem of anticipated forward-backward system with delay and Lvy processes as the random disturbance. This control system can be described by the anticipated forward-backward stochastic differential equations with delay and L′evy processes(AFBSDEDLs),we first obtain the existence and uniqueness theorem of adapted solutions for AFBSDEDLs; combining the AFBSDEDLs' preliminary result with certain classical convex variational techniques,the corresponding maximum principle is proved.     

On certain minimax problems and Pontryagin’s maximum principle

This paper deals with minimax problems for nonlinear differential expressions involving a vector-valued function of a scalar variable under rather conventional structure conditions on the cost function. It is proved that an absolutely minimizing (i.e. globally and locally minimizing) function is continuously differentiable. A minimizing function is also continuously differentiable, provided a certain extra condition is satisfied. The variational method of V.G. Boltyanskii, developed within optimal control theory, is adapted and used in the proof. The case of higher order derivatives is also considered.     

The Cauchy problem for fractional conservation laws driven by Lévy noise

In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by Lévy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of a solution is established. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that the Lévy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. Furthermore, we establish a result on vanishing non-local regularization of scalar stochastic conservation laws.     

Handelman’s hierarchy for the maximum stable set problem

The maximum stable set problem is a well-known NP-hard problem in combinatorial optimization, which can be formulated as the maximization of a quadratic square-free polynomial over the (Boolean) hypercube. We investigate a hierarchy of linear programming relaxations for this problem, based on a result of Handelman showing that a positive polynomial over a polytope with non-empty interior can be represented as conic combination of products of the linear constraints defining the polytope. We relate the rank of Handelman’s hierarchy with structural properties of graphs. In particular we show a relation to fractional clique covers which we use to upper bound the Handelman rank for perfect graphs and determine its exact value in the vertex-transitive case. Moreover we show two upper bounds on the Handelman rank in terms of the (fractional) stability number of the graph and compute the Handelman rank for several classes of graphs including odd cycles and wheels and their complements. We also point out links to several other linear and semidefinite programming hierarchies.