System Of Riccati Differential Equations Research Articles

A Stackelberg reinsurance-investment game under α-maxmin mean-variance criterion and stochastic volatility

ABSTRACT This paper investigates a Stackelberg game between an insurer and a reinsurer under the α-maxmin mean-variance criterion. The insurer can purchase per-loss reinsurance from the reinsurer. With the insurer's feedback reinsurance strategy, the reinsurer optimizes the reinsurance premium in the Stackelberg game. The financial market consists of cash and stock with Heston's stochastic volatility. Both the insurer and reinsurer maximize their respective α-maxmin mean-variance preferences in the market. The criterion is time-inconsistent and we derive the equilibrium strategies by the extended Hamilton-Jacobi-Bellman equations. Similar to the non-robust case in [Li, D. & Young, V. R. (2022). Stackelberg differential game for reinsurance: mean-variance framework and random horizon. Insurance: Mathematics and Economics 102, 42–55.], excess-of-loss reinsurance is the optimal form of reinsurance strategy for the insurer. The equilibrium investment strategy is determined by a system of Riccati differential equations. Besides, the equations determining the equilibrium reinsurance strategy and reinsurance premium rate are given semi-explicitly, which is simplified to an algebraic equation in a specific example. Numerical examples illustrate that the game between the insurer and reinsurer makes the insurance more radical when the agents become more ambiguity averse or risk averse. Furthermore, the level of ambiguity, ambiguity attitude, and risk attitude of the insurer (reinsurer) have similar effects on the equilibrium reinsurance strategy, reinsurance premium, and investment strategy.

The step-wise construction of solitary solutions to Riccati equations with diffusive coupling

<abstract><p>A novel scheme based on the generalized differential operator and computer algebra was used to construct solitary solutions to a system of Riccati differential equations with diffusive coupling. The presented approach yields necessary and sufficient existence conditions of solitary solutions with respect to the system parameters. The proposed stepwise approach enabled the derivation of the explicit analytic solution, which could not be derived using direct balancing techniques due to the complexity of algebraic relationships. Computational experiments were used to demonstrate the efficacy of proposed scheme.</p></abstract>

DISCRETE-CONTINUOUS MODEL OF SALES MANAGEMENT IN REAL TIME

The subject of the research is the development of a strategy for dynamic price management when selling products over a fixed time interval. We consider the case when the demand for the seller's products can be represented as a combination of two random processes: 1) Poisson flow of potential consumers; 2) the purchase of goods by an individual consumer, the probability of which is inversely related to the price of the product. Consumers need at most one unit of the good and have independent equally distributed estimates of its consumer value. Such demand structure allows to formalize the choice of the optimal pricing strategy as an optimal control problem. Employing dynamic programming methods to solving this problem yields a system of Riccati differential equations. The optimal solution is obtained in the closedloop form as a function of the time to expiration of the product value and unsold inventory levels. Examples of a practical solution to the optimal pricing problem are given for special cases when it is possible to find an analytical solution. For the general case, it is shown how to find the optimal prices using numerical methods. Calculations show that optimal prices are decreasing functions of time and inventory levels. The combination of these factors, together with the random nature of the product sales, leads to rather complex observed price trajectories, examples of which were obtained using computer simulations. In particular, in many cases, the implementation of the proposed strategy results in cyclical price behavior, the prevalence of which in retail is well documented. The problem of optimizing the expected income of the seller when using constant prices was also solved. Comparison of the expected income of the seller under static and dynamic prices indicates a significant advantage of the latter. The economic effect of using dynamic pricing is most significant near the expiration of the product value.

Infinite dimensional affine processes

The goal of this article is to investigate infinite dimensional affine diffusion processes on the canonical state space. This includes a derivation of the corresponding system of Riccati differential equations and an existence proof for such processes, which has been missing in the literature so far. For the existence proof, we will regard affine processes as solutions to infinite dimensional stochastic differential equations with values in Hilbert spaces. This requires a suitable version of the Yamada–Watanabe theorem, which we will provide in this paper. Several examples of infinite dimensional affine processes accompany our results.

AN ANALYTICAL SCHEME FOR THE ANALYSIS OF MULTI-HUMP SOLITONS

An analytical framework for the analysis of multi-hump solitons is proposed in this paper. Multi-hump solitons are defined by imposing special symmetry conditions on the classical soliton expression. Such soliton solutions have a wide range of potential applications in the field of optical communications. The proposed algebras of soliton solutions enable a new look at the propagation dynamics of complex nonlinear wave phenomena. The efficiency of the presented analytical scheme is demonstrated using a system of Riccati differential equations with diffusive and multiplicative coupling.

A general closed-form solution for the overall response of piezoelectric composites with periodic and random particulate microstructures

In this paper, we make use of a new iterative homogenization technique in finite electroelastostatics (Lopez-Pamies, 2014) to derive a closed-form solution for the overall response of two-phase piezoelectric composites with particulate microstructures. The calculations amount to solving a system of Riccati differential equations for the effective elastic, dielectric, and piezoelectric tensors of the composites, where the volume fraction of the inclusions plays the role of independent variable. The solution is valid for any choice of piezoelectric behaviors for the underlying matrix and inclusions, and any choice of the one- and two-point correlation functions describing the microstructure. In addition to discussing the key theoretical and practical features of the solution, its descriptive and predictive capabilities are illustrated via comparisons with a broad range of experimental data and full-field simulations (available from the literature) for composites with periodic and random distributions of inclusions of various shapes.

Game-theoretic coupled riccati equations associated to controlled linear differential systems with jump markov perturbations

In this paper we investigate several properties of the stabilizing solution of a class of systems of Riccati type differential equations with indefinite sign associated to controlled systems described by differential equations with Markovian jumping. We show that the existence of a bounded on R + and stabilizing solution for this class of systems of Riccati type differential equations is equivalent to the solvability of a control-theoretic problem, namely disturbance attenuation problem. If the coefficients of the considered system are theta;-periodic functions then the stabilizing solution is also theta;-periodic and if the coefficients are asymptotic almost periodic functions, then the stabilizing solution is also asymptotic almost periodic and its almost periodic component is a stabilizing solution for a system of Riccati type differential equations defined on the whole real axis. One proves also that the existence of a stabilizing and bounded on R + solution of a system of Riccati differential equations with indefinite sign is equivalent to the existence of a solution to a corresponding system of matrix inequalities. Finally, a minimality property of the stabilizing solution is derived.

Coupled matrix Riccati equations in minimal cost variance control problems

We present an algorithm for the solution of a nontrivial coupled system of algebraic Riccati equations appearing in risk sensitive control problems. Moreover, we use comparison methods to derive non-blowup conditions for the solutions of a corresponding terminal value problem for coupled systems of Riccati differential equations.