Optimal dividend and stopping problems for two-dimensional compound poisson risk model

This work generalizes the connection between the optimal singular stochastic control problem and the optimal stopping problem for regime-switching processes. Via the optimal singular stochastic control, the optimal stopping time and the continuation region are characterized. Moreover, we prove the existence of optimal singular stochastic control for a finite horizon singular control problem with the cost function containing the terminal cost. We prove it directly by the compactification method, which is based on an elaborate application of the properties of probability measures over the càdlàg space. Such a problem was left open in Haussmann and Suo [SIAM J. Control Optim., 33 (1995), pp. 916–936] and [SIAM J. Control Optim., 33 (1995), pp. 937–959]). In addition, our compactification method can remove the convexity condition on the coefficients used in Dufour and Miller [SIAM J. Control Optim., 43 (2004), pp. 708–730].

A singular stochastic control problem with state constraints in two-dimensions is studied. We show that the value function is $C^1$ and its directional derivatives are the value functions of certain optimal stopping problems. Guided by the optimal stopping problem, we then introduce the associated no-action region and the free boundary and show that, under appropriate conditions, an optimally controlled process is a Brownian motion in the no-action region with reflection at the free boundary. This proves a conjecture of Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996) 2133--2171] on the form of an optimal control for this class of singular control problems. An important issue in our analysis is that the running cost is Lipschitz but not $C^1$. This lack of smoothness is one of the key obstacles in establishing regularity of the free boundary and of the value function. We show that the free boundary is Lipschitz and that the value function is $C^2$ in the interior of the no-action region. We then use a verification argument applied to a suitable $C^2$ approximation of the value function to establish optimality of the conjectured control.

Finite Horizon Portfolio Selection with Durable Goods

Robbins' problem of optimal stopping is that of minimising the expected rank of an observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding the value of the stopped variable under the rule that yields the minimal expected rank, by embedding the problem in a much more general context of selection problems with the nonanticipation constraint lifted, and with the payoff growing like a power function of the rank.

We consider the optimal stopping problem, based on the compound binomial model with capital injection controlled by optimal dividend strategy. First, the value function V(x) is established. Second, via the Bellman equation satisfied by the value function V(x) which maximizes the discounted value of define between dividend payment and the penalized discounted capital injection, we find the optimal stopping time τ*. Furthermore an optimal stopping model with capital injection in order to maximize the shareholders’ interests is obtained.

In sequential analysis and starting with the seminal treatment of the optimality of the SPRT due to Wald and Wolfowitz [Wald, A.; Wolfowitz, J. Bayes solutions of sequential decision problems. Ann. Math. Statist. 1950, 21, 82–99] various problems of optimal stopping with linear costs of observation arise. Here, we shall describe a new method for solving such problems. For a payoff g(X t ) − ct we propose a linear representation of the form where M t is a local martingale, and the function h and the local martingale depend on a parameter λ. In the case of a diffusion process X t we shall show that, for a proper choice of λ, the boundary points of the optimal stopping region can be obtained from those points of the state space where the maxima of h are located. This method is inspired by a method of Beibel and Lerche [Beibel, M.; Lerche, H.R. A new look at optimal stopping problems related to mathematical finance. Statistica Sinica 1997, 7, 93–108; Beibel, M.; Lerche, H.R. Optimal stopping of regular diffusions under random discounting. Theory Probab. Appl. 2001, 45 (4), 547–557] who, for optimal stopping problems with discounted payoff, used a multiplicative representation e −rt g(X t ) = h(X t )M t with a suitable martingale.

The connection between the optimal stopping problems for inhomogeneous standard Markov process and the corresponding homogeneous Markov process constructed in the extended state space is established. An excessive characterization of the value-function and the limit procedure for its construction in the problem of optimal stopping of an inhomogeneous standard Markov process is given. The form of ε-optimal (optimal) stopping times is also found.

We show that the value function of a singular stochastic control problem is equal to the integral of the value function of an associated optimal stopping problem. The connection is proved for a general class of diffusions using the method of viscosity solutions.

We consider the problem of tracking a Brownian motion by a process of bounded variation, in such a way as to minimize total expected cost of both ‘action' and ‘deviation from a target state 0'. The former is proportional to the amount of control exerted to date, while the latter is being measured by a function which can be viewed, for simplicity, as quadratic. We discuss the discounted, stationary and finite-horizon variants of the problem. The answer to all three questions takes the form of exerting control in a singular manner, in order not to exit from a certain region. Explicit solutions are found for the first and second questions, while the third is reduced to an appropriate optimal stopping problem. This reduction yields properties, as well as global upper and lower bounds, for the associated moving boundary. The pertinent Abelian and ergodic relationships for the corresponding value functions are also derived.

Infinite horizon singular optimal control problems with control taking values in a closed cone U in Rn lead to a dynamic programming equation of the form: maxF2x,v, v',v'' ,F1x,v,v' =0, forall x∈Q, where Q , the state space of the control problem, is some nonempty connected open subset of Rn , and F1, F2 are continuous real-valued functions on QxR2 and QxR3 respectively, with the coercivity assumption for F 2, that is, the function r&rarrr;F2x,u,p,r is nonincreasing on R . A major concern is to determine how smooth the value function v is across the free boundary of the problem. Linear-convex deterministic and stochastic singular control problems in dimension one are considered. We present the analysis of the above equation together with smoothness of the value function across the free boundary. The interest here is the explicitness of the results and the fact that the smooth fit principle depends on the parameters of the control problem. In particular, the value function for the stochastic control problem is founded explicitly, and at the same time, optimal controls are identified using a verification theorem.

Residential demand response is one of the key enabling technologies which plays an important role in managing the load demand of prosumers. However, the load scheduling problem becomes quite challenging due to the involvement of dynamic parameters and renewable energy resources. This work has proposed a bi-level load scheduling mechanism with dynamic electricity pricing integrated with renewable energy and storage system to overcome this problem. The first level involves the formulation of load scheduling and optimization problems as optimal stopping problems with the objective of energy consumption and delay cost minimization. This problem involved the real-time electricity pricing signal, customers load scheduling priority, machine learning (ML) based forecasted load demand, and renewable & storage unit profiles, which is solved using mathematical programming with branch-and-cut & branch-and-bound algorithms. Since the first-level optimization problem is formulated as a stopping problem, the optimal time slots are obtained using a one-step lookahead rule to schedule the load with the ability to handle the uncertainties. The second level is used to further model the load scheduling problem through the dynamic electricity pricing signal. The cost minimization objective function is then solved using the genetic algorithm (GA), where the input parameters are obtained from the first-level optimization solution. Furthermore, the impact of load prioritization in terms of time factor and electricity price is also modeled to allow the end-users to control their load. Analytical and simulation results are conducted using solar-home electricity data, Ausgrid, AUS to validate the proposed model. Results show that the proposed model can handle uncertainties involved in the load scheduling process along with a cost-effective solution in terms of cost and discomfort reduction. Furthermore, the bi-level process ensures cost minimization with end-user satisfaction regarding the dynamic electricity price signal.

In this paper, we consider the optimal stopping problems on semi-Markov processes (sMPs) with finite horizon and aim to establish the existence and algorithm of optimal stopping times. The key method is the equivalence between optimal stopping problems on sMPs and a special class of semi-Markov decision processes (sMDPs). We first introduce the optimality equation and show the existence of the optimal policies of finite-horizon sMDPs with additional terminal costs. Based on the optimal stopping problems on sMPs, we give an explicit construction of sMDPs such that the optimal stopping times of sMPs are equivalent to the optimal policies of the constructed sMDPs. Then, using the results of sMDPs developed here, we not only prove the existence of the optimal stopping times of sMPs, but also provide an algorithm for computing the optimal stopping times of sMPs. Moreover, we show that the optimal and \(\varepsilon \)-optimal stopping time can be characterized by the hitting time of some special sets. Finally, we give an example to illustrate the effectiveness of our conclusions.

In this paper we study the optimal multiple stopping problem with weak regularity for the reward, where the reward is given by a set of random variables indexed by stopping times. When the reward family is upper semicontinuous in expectation along stopping times, we construct the optimal multiple stopping strategy using the auxiliary optimal single stopping problems. We also obtain the corresponding results when the reward is given by a progressively measurable process.

The main aim of this chapter is to present and analyze two methods for the pricing of multi-asset American – in practice Bermudan – options: the regression methods “a la Longstaff-Schwarz” and the quantization methods. The pricing of American options is a typical example of an optimal stopping problem. So, we start from such a general optimal stopping problem but for a diffusion model, we introduce the Snell envelope of a payoff or reward process and its value function in ou Markov framework. We give some rate results for various “levels” of time discretization of such Snell envelope. Then we consider a discrete time optimal stopping problem in a Markov framework. We introduce the backward dynamic programming principle and its variant based on optimal stopping times. We describe in detail the algorithmic aspects of both the regression and the quantization method. The Monte Carlo error (Central Limit Theorem) induced by the regression method is described. Finally, a theoretical analysis of the convergence rate of the quantization method is carried out.

For i.i.d. random variables in the domain of attraction of a max-stable distribution with discount and observation costs we determine asymptotic approximations of the optimal stopping values and asymptotically optimal stopping times. The results are based on Poisson approximation of related embedded planar point processes. The optimal stopping problem for the limiting Poisson point processes can be reduced to differential equations for the boundaries. In several cases we obtain numerical solutions of the differential equations. In some cases the analysis allows us to obtain explicit optimal stopping values. This approach thus leads to approximate solutions of the optimal stopping problem of discrete time sequences.