Sto chastic Maximum Principle for Optimal Control of Forward-backward Sto chastic Pantograph Systems with Regime Switching

In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stochastic pantograph equation and modulated by a continuous-time finite-state Markov chain. By virtue of classical variational approach, duality method, and convex analysis, we obtain a stochastic maximum principle for the optimal control.     

Stochastic maximum principle for mean-field forward-backward stochastic control system with terminal state constraints

In this paper,we consider an optimal control problem with state constraints,where the control system is described by a mean-field forward-backward stochastic differential equation(MFFBSDE,for short)and the admissible control is mean-field type.Making full use of the backward stochastic differential equation theory,we transform the original control system into an equivalent backward form,i.e.,the equations in the control system are all backward.In addition,Ekeland’s variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control.We also study a stochastic linear quadratic control problem with state constraints.     

MAXIMUM PRINCIPLE FOR STOCHASTIC OPTIMAL CONTROL PROBLEM WITH DISTRIBUTED DELAYS

This paper is concerned with a Pontryagin's maximum principle for the stochastic optimal control problem with distributed delays given by integrals of not necessarily linear functions of state or control variables.By virtue of the duality method and the generalized anticipated backward stochastic differential equations,we establish a necessary maximum principle and a sufficient verification theorem.In particular,we deal with the controlled stochastic system where the distributed delays enter both the state and the control.To explain the theoretical results,we apply them to a dynamic advertising problem.     

Large deviation principle for diffusion processes under a sublinear expectation

We represent the exponential moment of the Brownian functionals under a nonlinear expectation according to the solution to a backward stochastic differential equation.As an application,we establish a large deviation principle of the Freidlin and Wentzell type under the corresponding nonlinear probability for diffusion processes with a small diffusion coefficient.     

A MAXIMUM PRINCIPLE APPROACH TO STOCHASTIC H_2/H_∞ CONTROL WITH RANDOM JUMPS

A necessary maximum principle is given for nonzero-sum stochastic differential games with random jumps.The result is applied to solve the H_2/H_∞ control problem of stochastic systems with random jumps.A necessary and sufficient condition for the existence of a unique solution to the H_2/H_∞ control problem is derived.The resulting solution is given by the solution of an uncontrolled forward backward stochastic differential equation with random jumps.     

A Type of General Forward-Backward Stochastic Differential Equations and Applications

The authors discuss one type of general forward-backward stochastic differential equations (FBSDEs) with It?o’s stochastic delayed equations as the forward equations and anticipated backward stochastic differential equations as the backward equations. The existence and uniqueness results of the general FBSDEs are obtained. In the framework of the general FBSDEs in this paper, the explicit form of the optimal control for linearquadratic stochastic optimal control problem with delay and the Nash equilibrium point for nonzero sum differential games problem with delay are obtained.     

Optimal variational principle for backward stochastic control systems associated with Lévy processes

The paper is concerned with optimal control of backward stochastic differential equation (BSDE) driven by Teugel’s martingales and an independent multi-dimensional Brownian motion,where Teugel’s martin- gales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see e.g.,Nualart and Schoutens’ paper in 2000).We derive the necessary and sufficient conditions for the existence of the op- timal control by means of convex variation methods and duality techniques.As an application,the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria (or backward linear-quadratic problem,or BLQ problem for short) is discussed and characterized by a stochastic Hamilton system.     

Necessary Maximum Principle of Stochastic Optimal Control with Delay and Jump Diffusion

In this paper, we have studied the necessary maximum principle of stochastic optimal control problem with delay and jump diffusion.     

Infinite Horizon Forward-Backward Doubly Stochastic Differential Equations and Related SPDEs

A type of infinite horizon forward-backward doubly stochastic differential equations is studied.Under some monotonicity assumptions,the existence and uniqueness results for measurable solutions are established by means of homotopy method.A probabilistic interpretation for solutions to a class of stochastic partial differential equations combined with algebra equations is given.A significant feature of this result is that the forward component of the FBDSDEs is coupled with the backward variable.     

STABILITY OF THE SOLUTIONS TO STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper,we obtain the stability of solutions to stochastic functional differential equations with infinite delay at phase space BC((-∞,0];Rd),under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition by means of the corollary of Bihari inequality.     

A Large Deviation Principle of Retarded Ornstein-Uhlenbeck Processes Driven by Lévy Noise

In this article, we develop a large deviation principle (LDP) for a class of retarded Ornstein-Uhlenbeck processes driven by Lévy processes. We first present a LDP result for time delay systems driven by cylindrical Wiener processes based on the large deviations of Gaussian processes. By using a contraction technique and passing on a finite-dimensional approximation, an LDP is obtained for stochastic time delay evolution equations driven by additive Lévy noise, whose solutions are generally not Lévy processes any more.     

Random attractors for stochastic differential equations driven by two-sided Lévy processes

Abstract

In this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor.     

Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields

We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations (SPDE) with jumps. This is a type of equations, which appear as adjoint equations in the maximum principle approach to optimal control of systems described by SPDE driven by Lévy processes.     

Simulation of stochastic integrals with respect to Lévy processes of type G

We study the simulation of stochastic processes defined as stochastic integrals with respect to type G Lévy processes for the case where it is not possible to simulate the type G process exactly. The type G Lévy process as well as the stochastic integral can on compact intervals be represented as an infinite series. In a practical simulation we must truncate this representation. We examine the approximation of the remaining terms with a simpler process to get an approximation of the stochastic integral. We also show that a stochastic time change representation can be used to obtain an approximation of stochastic integrals with respect to type G Lévy processes provided that the integrator and the integrand are independent.     

Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations

We prove a comparison principle for unbounded semicontinuous viscosity sub- and supersolutions of non-linear degenerate parabolic integro-partial differential equations coming from applications in mathematical finance in which geometric Lévy processes act as the underlying stochastic processes for the assets dynamics. As a consequence of the “geometric form” of these processes, the comparison principle holds without assigning spatial boundary data. We present applications of our result to (i) backward stochastic differential equations (BSDEs) and (ii) pricing of European and American derivatives via BSDEs. Regarding (i), we extend previous results on BSDEs in a Lévy setting and the connection to semilinear integro-partial differential equations.     

Fractional Lévy processes on Gel’fand triple and stochastic integration

In this paper, we investigate the long-range dependence of fractional Lévy processes on Gel’fand triple and construct stochastic integral with respect to fractional Lévy processes for a class of deterministic integrands.      

A Maximum Principle for Stochastic Control with Partial Information

Abstract

We study the problem of optimal control of a jump diffusion, that is, a process which is the solution of a stochastic differential equation driven by Lévy processes. It is required that the control process is adapted to a given subfiltration of the filtration generated by the underlying Lévy processes. We prove two maximum principles (one sufficient and one necessary) for this type of partial information control. The results are applied to a partial information mean-variance portfolio selection problem in finance.     

Selfdecomposable Fields

In the present paper, we study selfdecomposability of random fields, as defined directly rather than in terms of finite-dimensional distributions. The main tools in our analysis are the master Lévy measure and the associated Lévy-Itô representation. We give the dilation criterion for selfdecomposability analogous to the classical one. Next, we give necessary and sufficient conditions (in terms of the kernel function) for a Volterra field driven by a Lévy basis to be selfdecomposable. In this context, we also study the so-called Urbanik classes of random fields. We follow this with the study of existence and selfdecomposability of integrated Volterra fields. Finally, we introduce infinitely divisible field-valued Lévy processes, give the Lévy-Itô representation associated with them and study stochastic integration with respect to such processes. We provide examples in the form of Lévy semistationary processes with a Gamma kernel and Ornstein–Uhlenbeck processes.     

On a Class of Lévy Stochastic Networks

We consider a Lévy stochastic network as a regulated multidimensional Lévy process. The reflection direction is constant on each boundary of the positive orthant and the corresponding reflection matrix corresponds to a single-class network. We use the representation of the Lévy process and Itô's formula to arrive at some equations for the steady-state process; the latter is shown to exist, under natural stability conditions. We specialize first to the class of Lévy processes with non-negative jumps and then add the assumption of self-similarity. We show that the stationary distribution of the network corresponding the the latter process does not has product form (except in trivial cases). Finally, we derive asymptotic bounds for two-dimensional Lévy stochastic network.