A stochastic portfolio optimization model with complete memory

In this article, we consider a portfolio optimization problem of the Merton’s type with complete memory over a finite time horizon. The problem is formulated as a stochastic control problem on a finite time horizon and the state evolves according to a process governed by a stochastic process with memory. The goal is to choose investment and consumption controls such that the total expected discounted utility is maximized. Under certain conditions, we derive the explicit solutions for the associated Hamilton–Jacobi–Bellman (HJB) equations in a finite-dimensional space for exponential, logarithmic, and power utility functions. For those utility functions, verification results are established to ensure that the solutions are equal to the value functions, and the optimal controls are also derived.     

Omega diffusion risk model with surplus-dependent tax and capital injections

In this paper, we propose and study an Omega risk model with a constant bankruptcy function, surplus-dependent tax payments and capital injections in a time-homogeneous diffusion setting. The surplus value process is both refracted (paying tax) at its running maximum and reflected (injecting capital) at a lower constant boundary. The new model incorporates practical features from the Omega risk model (Albrecher et al., 2011), the risk model with tax (Albrecher and Hipp, 2007), and the risk model with capital injections (Albrecher and Ivanovs, 2014). The study of this new risk model is closely related to the Azéma–Yor process, which is a process refracted by its running maximum. We explicitly characterize the Laplace transform of the occupation time of an Azéma–Yor process below a constant level until the first passage time of another Azéma–Yor process or until an independent exponential time. We also consider the case when the process has a lower reflecting boundary. This result unifies and extends recent results of Li and Zhou (2013) and Zhang (2015). We explicitly characterize the Laplace transform of the time of bankruptcy in the Omega risk model with tax and capital injections up to eigen-functions, and determine the expected present value of tax payments until default. We also discuss a further extension to occupation functionals through stochastic time-change, which handles the case of a non-constant bankruptcy function. Finally we present examples using a Brownian motion with drift, and discuss the pricing of quantile options written on the Azéma–Yor process.     

Optimum Constrained Portfolio Rules in a Diffusion Market

A portfolio selection model is derived for diffusions where inequality constraints are imposed on portfolio security weights. Using the method of stochastic dynamic programming Hamilton–Jacobi–Bellman (HJB) equations are obtained for the problem of maximizing the expected utility of terminal wealth over a finite time horizon. Optimal portfolio weights are given in feedback form in terms of the solution of the HJB equations and its partial derivatives. An analysis of the no‐constraining (NC) region of a portfolio is also conducted.     

Optimal investment strategy for asset-liability management under the Heston model

ABSTRACT

This paper focuses on an asset-liability management problem for an investor who can invest in a risk-free asset and a risky asset whose price process is governed by the Heston model. The objective of the investor is to find an optimal investment strategy to maximize the expected exponential utility of the surplus process. By using the stochastic control method and variable change techniques, we obtain a closed-form solution of the corresponding Hamilton–Jacobi–Bellman equation. We also develop a verification theorem without the usual Lipschitz assumptions which can ensure that this closed-form solution is indeed the value function and then derive the optimal investment strategy explicitly. Finally, we provide numerical examples to show how the main parameters of the model affect the optimal investment strategy.     

Indifference pricing of a life insurance portfolio with risky asset driven by a shot-noise process

In this paper, we investigate the pricing problem for a portfolio of life insurance contracts where the life contingent payments are equity-linked depending on the performance of a risky stock or index. The shot-noise effects are incorporated in the modeling of stock prices, implying that sudden jumps in the stock price are allowed, but their effects may gradually decline over time. The contracts are priced using the principle of equivalent utility. Under the assumption of exponential utility, we find the optimal investment strategy and show that the indifference premium solves a non-linear partial integro-differential equation (PIDE). The Feynman–Kač form solutions are derived for two special cases of the PIDE. We further discuss the problem for the asymptotic shot-noise process, and find the probabilistic representation of the indifference premium. We also provide some numerical examples and analyze parameter sensitivities for the results obtained in this paper.     

Optimal control of stochastic functional differential equations with a bounded memory

This paper treats a finite time horizon optimal control problem in which the controlled state dynamics are governed by a general system of stochastic functional differential equations with a bounded memory. An infinite dimensional Hamilton–Jacobi–Bellman (HJB) equation is derived using a Bellman-type dynamic programming principle. It is shown that the value function is the unique viscosity solution of the HJB equation.     

Optimal dividend strategies in discrete risk model with capital injections

In this paper we consider a doubly discrete model used in Dickson and Waters (biASTIN Bulletin 1991; 21 :199–221) to approximate the Cramér–Lundberg model. The company controls the amount of dividends paid out to the shareholders as well as the capital injections which make the company never ruin in order to maximize the cumulative expected discounted dividends minus the penalized discounted capital injections. We show that the optimal value function is the unique solution of a discrete Hamilton–Jacobi–Bellman equation by contraction mapping principle. Moreover, with capital injection, we reduce the optimal dividend strategy from band strategy in the discrete classical risk model without external capital injection into barrier strategy , which is consistent with the result in continuous time. We also give the equivalent condition when the optimal dividend barrier is equal to 0. Although there is no explicit solution to the value function and the optimal dividend barrier, we obtain the optimal dividend barrier and the approximating solution of the value function by Bellman's recursive algorithm. From the numerical calculations, we obtain some relevant economical insights. Copyright © 2010 John Wiley & Sons, Ltd.     

Explicit solutions of optimal consumption,investment and insurance problems with regime switching

We consider an investor who wants to select his optimal consumption, investment and insurance policies. Motivated by new insurance products, we allow not only the financial market but also the insurable loss to depend on the regime of the economy. The objective of the investor is to maximize his expected total discounted utility of consumption over an infinite time horizon. For the case of hyperbolic absolute risk aversion (HARA) utility functions, we obtain the first explicit solutions for simultaneous optimal consumption, investment, and insurance problems when there is regime switching. We determine that the optimal insurance contract is either no-insurance or deductible insurance, and calculate when it is optimal to buy insurance. The optimal policy depends strongly on the regime of the economy. Through an economic analysis, we calculate the advantage of buying insurance.     

Taylor expansions of the value function associated with a bilinear optimal control problem

A general bilinear optimal control problem subject to an infinite-dimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated formal differentiation of the Hamilton–Jacobi–Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-hand sides. They form the basis for defining a suboptimal feedback law. The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker–Planck equation is also provided.     

Non-standard power-law Lagrangians in classical and quantum dynamics

Recently, non-standard Lagrangians (NSL) have gained an increasing significance due to their wide implications in the theory of non-linear differential equations, complex dynamical systems and theoretical physics. In this work, we choose the power-law non-standard Lagrangian and we discuss some of its implications in classical and quantum theories.     

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

We consider a controlled system driven by a coupled forward–backward stochastic differential equation with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation, at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem. The existence of a strict control follows from the Filippov convexity condition.     

Mayer control problem with probabilistic uncertainty on initial positions

In this paper we introduce and study an optimal control problem in the Mayer's form in the space of probability measures on ${\mathbb{R}}^n$ endowed with the Wasserstein distance. Our aim is to study optimality conditions when the knowledge of the initial state and velocity is subject to some uncertainty, which are modeled by a probability measure on ${\mathbb{R}}^d$ and by a vector-valued measure on ${\mathbb{R}}^d$, respectively. We provide a characterization of the value function of such a problem as unique solution of an Hamilton–Jacobi–Bellman equation in the space of measures in a suitable viscosity sense. Some applications to a pursuit-evasion game with uncertainty in the state space is also discussed, proving the existence of a value for the game.     

Pricing a European Basket Option in the Presence of Proportional Transaction Costs

A crucial assumption in the Black–Scholes theory of options pricing is the no transaction costs assumption. However, following such a strategy in the presence of transaction costs would lead to immediate ruin. This paper presents a stochastic control approach to the pricing and hedging of a European basket option, dependent on primitive assets whose prices are modelled as lognormal diffusions, in the presence of costs proportional to the size of the transaction. Under certain assumptions on the individual preferences, it is able to reduce the dimensionality of the resulting control problem. This facilitates considerably the study of the value function and the characterisation of the optimal trading policy. For solution of the problem a perturbation analysis scheme is utilized to derive a non‐trivial, asymptotically optimal result. The findings reveal that this result can be expressed by means of a small correction to the corresponding solution of the frictionless Black–Scholes type problem, resembling a multi‐dimensional ‘bandwidth’ around the vanilla case, which, moreover, is readily tractable.     

Optimal proportional reinsurance with common shock dependence

In this paper, we consider the optimal proportional reinsurance strategy in a risk model with multiple dependent classes of insurance business, which extends the work of Liang and Yuen (2014) to the case with the reinsurance premium calculated under the expected value principle and to the model with two or more classes of dependent risks. Under the criterion of maximizing the expected exponential utility, closed-form expressions for the optimal strategies and value function are derived not only for the compound Poisson risk model but also for the diffusion approximation risk model. In particular, we find that the optimal reinsurance strategies under the expected value premium principle are very different from those under the variance premium principle in the diffusion risk model. The former depends not only on the safety loading, time and interest rate, but also on the claim size distributions and the counting processes, while the latter depends only on the safety loading, time and interest rate. Finally, numerical examples are presented to show the impact of model parameters on the optimal strategies.     

On optimal dividends with exponential and linear penalty payments

We study the optimal dividend problem where the surplus process of an insurance company is modelled by a diffusion process. The insurer is not ruined when the surplus becomes negative, but penalty payments occur, depending on the level of the surplus. The penalty payments shall avoid that losses can rise above any number and can be seen as a preference measure or costs for negative capital. As examples, exponential and linear penalty payments are considered. It turns out that a barrier dividend strategy is optimal.     

Optimal investment strategies and risk-sharing arrangements for a hybrid pension plan

A continuous time stochastic model is used to study a hybrid pension plan, where both the contribution and benefit levels are adjusted depending on the performance of the plan, with risk sharing between different generations. The pension fund is invested in a risk-free asset and multiple risky assets. The objective is to seek an optimal investment strategy and optimal risk-sharing arrangements for plan trustees and participants so that this proposed hybrid pension system provides adequate and stable income to retirees while adjusting contributions effectively, as well as keeping its sustainability in the long run. These goals are achieved by minimizing the expected discount disutility of intermediate adjustment for both benefits and contributions and that of terminal wealth in finite time horizon. Using the stochastic optimal control approach, closed-form solutions are derived under quadratic loss function and exponential loss function. Numerical analysis is presented to illustrate the sensitivity of the optimal strategies to parameters of the financial market and how the optimal benefit changes with respect to different risk aversions. Through numerical analysis, we find that the optimal strategies do adjust the contributions and retirement benefits according to fund performance and model objectives so the intergenerational risk sharing seem effectively achieved for this collective hybrid pension plan.     

Ergodic control of diffusions with compound Poisson jumps under a general structural hypothesis

We study the ergodic control problem for a class of controlled jump diffusions driven by a compound Poisson process. This extends the results of Arapostathis et al. (2019) to running costs that are not near-monotone. This generality is needed in applications such as optimal scheduling of large-scale parallel server networks.We provide a full characterizations of optimality via the Hamilton–Jacobi–Bellman (HJB) equation, for which we additionally exhibit regularity of solutions under mild hypotheses. In addition, we show that optimal stationary Markov controls are a.s. pathwise optimal. Lastly, we show that one can fix a stable control outside a compact set and obtain near-optimal solutions by solving the HJB on a sufficiently large bounded domain. This is useful for constructing asymptotically optimal scheduling policies for multiclass parallel server networks.     

Risk sensitive control of pure jump processes on a general state space

We study stochastic control problem for pure jump processes on a general state space with risk sensitive discounted and ergodic cost criteria. For the discounted cost criterion we prove the existence and Hamilton–Jacobi–Bellman characterization of optimal α-discounted control for bounded cost function. For the ergodic cost criterion we assume a Lyapunov type stability assumption and a small cost condition. Under these assumptions we show the existence of the optimal risk-sensitive ergodic control.     

A Markov Process Modeling and Analysis of Indifference Pricing of Insurance Contracts for Home Reversion Plan for a Pair of Insureds

The intent of the home reversion plan for retired homeowners in United Kingdom is to help house-rich but cash-poor seniors by the way of releasing their home equity into cash so as to meet living expenses and continue to enjoy the right to live in their home until their death. We present multiple finite state Markov models to price the continuous annuities of insurance policies relevant to home reversion plan for a pair of insureds (meaning husband and wife). Our modeling assumes that the home value follows a geometric Brownian motion. By applying the principle of equivalent utility, we derive the partial differential equation system that the indifferent annuities satisfy under the exponential utility function, and find their explicit representations. Furthermore, we employ an explicit finite difference scheme to calculate the numerical solution and discuss the impacts of the risk aversion of insurer, the volatility of home value, and the interest rate on the annuities of home reversion plan for a couple. We observe that these results reflect our intuition.