Switching games of stochastic differential systems

Version of RecordPublishe

Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems

In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This implies that DNNs can break the curse of dimensionality in numerical approximations and optimal control of PDEs and SPDEs. The main ingredient of our proof is to construct a suitable discrete-time system to effectively approximate the evolution of the underlying stochastic dynamics. Similar ideas can also be applied to obtain expression rates of DNNs for value functions induced by stiff systems with regime switching coefficients and driven by general L\'{e}vy noise.Comment: This revised version has been accepted for publication in Analysis and Application

Multidimensional indefinite stochastic Riccati equations and zero-sum linear-quadratic stochastic differential games with non-markovian regime switching

This paper is concerned with two-player zero-sum linear-quadratic stochastic differential games in a regime switching model. The controlled inhomogeneous system coefficients depend on the underlying noises, so it is a non-Markovian regime switching model. Based on a new kind of multidimensional indefinite stochastic Riccati equation (SRE) and a multidimensional linear backward stochastic differential equation (BSDE) with unbounded coefficients, we can provide optimal feedback control-strategy pairs for the two players in a closed-loop form. The main contribution of this paper, which is important in its own right from the BSDE theory point of view, is to prove the existence and uniqueness of the new kind of multidimensional indefinite SRE. Interestingly, the components of the solution can take positive, zero and negative values simultaneously. We also obtain the corresponding optimal feedback control-strategy pairs for homogeneous systems under closed convex cone control constraints. Finally, these results are applied to portfolio selection problems with different short-selling prohibition constraints in a regime switching market with random coefficients

Switching Diffusions: Applications To Ecological Models, And Numerical Methods For Games In Insurance

Recently, a class of dynamic systems called ``hybrid systems containing both continuous dynamics and discrete events has been adapted to treat a wide variety of situations arising in many real-world situations. Motivated by such development, this dissertation is devoted to the study of dynamical systems involving a Markov chain as the randomly switching process. The systems studied include hybrid competitive Lotka-Volterra ecosystems and non-zero-sum stochastic differential games between two insurance companies with regime-switching. The first part is concerned with competitive Lotka-Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. Our main objective is to reduce the computational complexity by using the two-time-scale formulation. Because the existence and uniqueness as well as continuity of solutions for Lotka-Volterra ecosystems with Markovian switching in which the switching takes place in a countable set are not available, such properties are studied first. The two-time scale feature is highlighted by introducing a small parameter into the generator of the Markov chain. When the small parameter goes to 0, there is a limit system or reduced system. It is established in this work that if the reduced system possesses certain properties such as permanence and extinction, etc., then the complex original system also has the same properties when the parameter is sufficiently small. These results are obtained by using the perturbed Lyapunov function methods. The second part develops an approximation procedure for a class of non-zero-sum stochastic differential games for investment and reinsurance between two insurance companies. Both proportional reinsurance and excess-of-loss reinsurance policies are considered. We develop numerical algorithms to obtain the approximation to the Nash equilibrium by adopting the Markov chain approximation methodology. We establish the convergence of the approximation sequences and the approximation to the value functions. Numerical examples are presented to illustrate the applicability of the algorithms

Differential Games Controllers That Confine a System to a Safe Region in the State Space, With Applications to Surge Tank Control

Surge tanks are units employed in chemical processing to regulate the flow of fluids between reactors. A notable feature of surge tank control is the need to constrain the magnitude of the Maximum Rate of Change (MROC) of the surge tank outflow, since excessive fluctuations in the rate of change of outflow can adversely affect down-stream processing (through disturbance of sediments, initiation of turbulence, etc.). Proportional + Integral controllers, traditionally employed in surge tank control, do not take direct account of the MROC. It is therefore of interest to explore alternative approaches. We show that the surge tank controller design problem naturally fits a differential games framework, proposed by Dupuis and McEneaney, for controlling a system to confine the state to a safe region of the state space. We show furthermore that the differential game arising in this way can be solved by decomposing it into a collection of (one player) optimal control problems. We discuss the implications of this decomposition technique, for the solution of other controller design problems possessing some features of the surge tank controller design problem