VISCOSITY SOLUTIONS OF HJB EQUATIONS ARISING FROM THE VALUATION OF EUROPEAN PASSPORT OPTIONS

The passport option is a call option on the balance of a trading account. The option holder retains the gain from trading, while the issuer is liable for the net loss. In this article, the mathematical foundation for pricing the European passport option is established. The pricing equation which is a fully nonlinear equation is derived using the dynamic programming principle. The comparison principle, uniqueness and convexity preserving of the viscosity solutions of related H J13 equation are proved. A relationship between the passport and lookback options is discussed.

The Time-Consistent Optimal Reinsurance Strategy of Insurance Group under the CEV Model

The article introduces proportional reinsurance contracts under the mean-variance criterion,studying the time-consistence investment portfolio problem considering the interests of both insurance companies and reinsurance companies.The insurance claims process follows a jump-diffusion model,assuming that the risk asset prices of insurance companies and reinsurance companies follow CEV models different from each other.In the framework of game theory,the time-consistent equilibrium reinsurance strategy is obtained by solving the extended HJB equation analytically.Finally,numerical examples are used to illustrate the impact of model parameters on equilibrium strategies and provide economic explanations.The results indicate that the decision weights of insurance companies and reinsurance companies do have a significant impact on both the reinsurance ratio and the equilibrium reinsurance strategy.

Adaptive Multi-Step Evaluation Design With Stability Guarantee for Discrete-Time Optimal Learning Control

This paper is concerned with a novel integrated multi-step heuristic dynamic programming(MsHDP)algorithm for solving optimal control problems.It is shown that,initialized by the zero cost function,MsHDP can converge to the optimal solution of the Hamilton-Jacobi-Bellman(HJB)equation.Then,the stability of the system is analyzed using control policies generated by MsHDP.Also,a general stability criterion is designed to determine the admissibility of the current control policy.That is,the criterion is applicable not only to traditional value iteration and policy iteration but also to MsHDP.Further,based on the convergence and the stability criterion,the integrated MsHDP algorithm using immature control policies is developed to accelerate learning efficiency greatly.Besides,actor-critic is utilized to implement the integrated MsHDP scheme,where neural networks are used to evaluate and improve the iterative policy as the parameter architecture.Finally,two simulation examples are given to demonstrate that the learning effectiveness of the integrated MsHDP scheme surpasses those of other fixed or integrated methods.

OPTIMAL CONTROL OF MARKOVIAN SWITCHING SYSTEMS WITH APPLICATIONS TO PORTFOLIO DECISIONS UNDER INFLATION

This article is concerned with a class of control systems with Markovian switching, in which an It5 formula for Markov-modulated processes is derived. Moreover, an optimal control law satisfying the generalized Hamilton-Jacobi-Bellman （HJB） equation with Markovian switching is characterized. Then, through the generalized HJB equation, we study an optimal consumption and portfolio problem with the financial markets of Markovian switching and inflation. Thus, we deduce the optimal policies and show that a modified Mutual Fund Theorem consisting of three funds holds. Finally, for the CRRA utility function, we explicitly give the optimal consumption and portfolio policies. Numerical examples are included to illustrate the obtained results.

THE OPTIMAL STRATEGY FOR INSURANCE COMPANY UNDER THE INFLUENCE OF TERMINAL VALUE

This paper considers a model of an insurance company which is allowed to invest a risky asset and to purchase proportional reinsurance. The objective is to find the policy which maximizes the expected total discounted dividend pay-out until the time of bankruptcy and the terminal value of the company under liquidity constraint. We find the solution of this problem via solving the problem with zero terminal value. We also analyze the influence of terminal value on the optimal policy.

OPTIMAL PORTFOLIO ON TRACKING THE EXPECTED WEALTH PROCESS WITH LIQUIDITY CONSTRAINTS

In this article, the authors consider the optimal portfolio on tracking the expected wealth process with liquidity constraints. The constrained optimal portfolio is first formulated as minimizing the cumulate variance between the wealth process and the expected wealth process. Then, the dynamic programming methodology is applied to reduce the whole problem to solving the Hamilton-Jacobi--Bellman equation coupled with the liquidity constraint, and the method of Lagrange multiplier is applied to handle the constraint. Finally, a numerical method is proposed to solve the constrained HJB equation and the constrained optimal strategy. Especially, the explicit solution to this optimal problem is derived when there is no liquidity constraint.

Optimal dividend and capital injection problem with a random time horizon and a ruin penalty in the dual model

In the dual risk model, we consider the optimal dividend and capital injection problem, which involves a random time horizon and a ruin penalty. Both fixed and proportional costs from the transactions of capital injection are considered. The objective is to maximize the total value of the expected discounted dividends, and the penalized discounted both capital injections and ruin penalty during the horizon, which is described by the minimum of the time of ruin and an exponential random variable. The explicit solutions for optimal strategy and value function are obtained, when the income jumps follow a hyper-exponential distribution.Besides, some numerical examples are presented to illustrate our results.

THEORY AND ALGORITHM OF OPTIMAL CONTROL SOLUTION TO DYNAMIC SYSTEM PARAMETERS IDENTIFICATION(Ⅱ)──STOCHASTIC SYSTEM PARAMETERS IDENTIFICATION AND APPLICATION EXAMPLE

Based on the contents Of part (Ⅰ) and stochastic optimal control theory, the concept of optimal control solution to parameters identification of stochastic dynamic system is discussed at first. For the completeness of the theory developed in this paper and part (Ⅰ), then the procedure of establishing HamiltonJacobi-Bellman (HJB) equations of parameters identification problem is presented.And then, parameters identification algorithm of stochastic dynamic system is introduced. At last, an application example-local nonlinear parameters identification of dynamic system is presented.

带跳一扩散风险过程保险人的最优控制（英文）

In this paper, the optimal XL-reinsurance of an insurer with jump-diffusion risk process is studied. With the assumptions that the risk process is a compound Possion process perturbed by a standard Brownian motion and the reinsurance premium is calculated according to the variance principle, the implicit expression of the priority and corresponding value function when the utility function is exponential are obtained. At last, the value function is argued, the properties of the priority about parameters are discussed and numerical results of the priority for various claim-size distributions are shown.

Optimal Synchronization Control of Heterogeneous Asymmetric Input-Constrained Unknown Nonlinear MASs via Reinforcement Learning

The asymmetric input-constrained optimal synchronization problem of heterogeneous unknown nonlinear multiagent systems(MASs)is considered in the paper.Intuitively,a state-space transformation is performed such that satisfaction of symmetric input constraints for the transformed system guarantees satisfaction of asymmetric input constraints for the original system.Then,considering that the leader’s information is not available to every follower,a novel distributed observer is designed to estimate the leader’s state using only exchange of information among neighboring followers.After that,a network of augmented systems is constructed by combining observers and followers dynamics.A nonquadratic cost function is then leveraged for each augmented system(agent)for which its optimization satisfies input constraints and its corresponding constrained Hamilton-Jacobi-Bellman(HJB)equation is solved in a data-based fashion.More specifically,a data-based off-policy reinforcement learning(RL)algorithm is presented to learn the solution to the constrained HJB equation without requiring the complete knowledge of the agents’dynamics.Convergence of the improved RL algorithm to the solution to the constrained HJB equation is also demonstrated.Finally,the correctness and validity of the theoretical results are demonstrated by a simulation example.

Optimal consumption–investment under partial information in conditionally log-Gaussian models

Certain Merton type consumption−investment problems under partial information are reduced to the ones of full information and within the framework of a complete market model.Then,specializing to conditionally log−Gaussian diffusion models,concrete analysis about the optimal values and optimal strategies is performed by using analytical tools like Feynman−Kac formula,or HJB equations.The explicit solutions to the related forward-backward equations are also given.

Homotopy Analysis Method for Portfolio Optimization Problem Under the 3/2 Model

This paper considers the Merton portfolio optimization problem for an investor that aims at maximizing the expected power utility of the terminal wealth and intermediate consumption.Applying the homotopy analysis method,an analytical solution for value function as well as optimal strategy under the 3/2 model is derived,respectively.Compared with the existing explicit solutions for Merton problem under the 3/2 model,the formulas provide certain parameters with less requirement since the homotopy analysis method does not depend on the existence of small parameters in the equation.Finally,numerical examples are examined with the approach,and the proposed solution provides more accurate approximation as the number of terms in infinite series increases.

Tensor Decomposition and High-Performance Computing for Solving High-Dimensional Stochastic Control System Numerically

The paper presents a numerical method for solving a class of high-dimensional stochastic control systems based on tensor decomposition and parallel computing.The HJB solution provides a globally optimal controller to the associated dynamical system.Variable substitution is used to simplify the nonlinear HJB equation.The curse of dimensionality is avoided by representing the HJB equation using separated representation.Alternating least squares(ALS)is used to reduced the separation rank.The experiment is conducted and the numerical solution is obtained.A high-performance algorithm is designed to reduce the separation rank in the parallel environment,solving the high-dimensional HJB equation with high efficiency.

Optimal Contract for the Principal-Agent Under Knightian Uncertainty

Under the Knightian uncertainty,this paper constructs the optimal principal(he)-agent(she)contract model based on the principal’s expected profit and the agent’s expected utility function by using the sublinear expectation theory.The output process in the model is provided by the agent’s continuous efforts and the principal cannot directly observe the agent’s efforts.In the process of work,risk-averse agent will have the opportunity to make external choices.In order to promote the agent’s continuous efforts,the principal will continuously provide the agents with consumption according to the observable output process after the probation period.In this paper,the Hamilton–Jacobi–Bellman equation is deduced by using the optimality principle under sublinear expectation while the smoothness viscosity condition of the principal-agent optimal contract is given.Moreover,the continuation value of the agent is taken as the state variable to characterize the optimal expected profit of the principal,the agent’s effort and the consumption level under different degrees of Knightian uncertainty.Finally,the behavioral economics is used to analyze the simulation results.The research findings are that the increasing Knightian uncertainty incurs the decline of the principal’s maximum profit;within the probation period,the increasing Knightian uncertainty leads to the shortening of probation period and makes the agent give higher effort when she faces the outside option;what’s more,after the smooth completion of the probation period for the agent,the agent’s consumption level will rise and her effort level will drop as Knightian uncertainty increasing.

Domain Decomposition Method for a System of Quasivariational Inequalities

We propose a domain decomposition method for a system of quasivariational inequalities related to the HJB equation. The monotone convergence of the algorithm is also established.

The Stochastic Control Model for Use Conversion of Land

In this paper, the authors investigate the optimal conversion rate at which land use is irreversibly converted from biodiversity conservation to agricultural production. This problem is formulated as a stochastic control model, then transformed into a HJB equation involving free boundary. Since the state equation has singularity, it is difficult to directly derive the boundary value condition for the HJB equation. They provide a new method to overcome the difficulty via constructing another auxiliary stochastic control problem,and impose a proper boundary value condition. Moreover, they establish the existence and uniqueness of the viscosity solution of the HJB equation. Finally, they propose a stable numerical method for the HJB equation involving free boundary, and show some numerical results.

The Optimal Investment,Liability and Dividends in Insurance

In this paper,we build an optimal control model with the objective to maximize the expected value of the time discount utility by selecting optimal investment,liability and dividend strategies for insurance companies.We then use the techniques from Merton(J Econ Theory 3(4):373–413,1971)to solve our optimal control problem and deduce the optimal control solutions.Finally,we analyze the economic impacts on the optimal controls of the parameters in insurance market.

Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs

In this paper,we propose a new type of viscosity solutions for fully nonlinear path-dependent PDEs.By restricting the solution to a pseudo-Markovian structure defined below,we remove the uniform non-degeneracy condition needed in our earlier works(Ekren,I,Touzi,N,Zhang,J,Ann Probab,44:1212–1253,2016a;Ekren,I,Touzi,N,Zhang,J,Ann Probab,44:2507–2553,2016b)to establish the uniqueness result.We establish the comparison principle under natural and mild conditions.Moreover,we apply our results to two important classes of PPDEs:the stochastic HJB equations and the path-dependent Isaacs equations,induced from the stochastic optimization with random coefficients and the path-dependent zero-sum game problem,respectively.

Optimal Insurance-Package and Investment Problem for an Insurer

An insurance-package is a combination being tie-in at least two different categories of insurances with different underwriting-yield-rate. In this paper, the optimal insurance-package and investment problem is investigated by maximizing the insurer’s exponential utility of terminal wealth to find the optimal combination-share and investment strategy. Using the methods of stochastic analysis and stochastic optimal control, the Hamilton-Jacobi-Bellman(HJB) equations are established, the optimal strategy and the value function are obtained in closed form. By comparing with classical results, it shows that the insurance-package can enhance the utility of terminal wealth, meanwhile,reduce the insurer’s claim risk.

TIME OPTIMAL IMPLUSE CONTROL PROBLEMS

In this paper, a class of time optimal problem with impluse control is considered. Under certain conditions we prove that the optimal impluse control exists and its impluse number is finite. Moreover, it is proved that the minimum time function is locally Lipschitz continuous in its domain and is the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman system.