on singular control problems with state constraints and regime switching this paper investigates a singular stochastic control problem for a multi dimensional regime switching diffusion process confined in an unbounded domain the objective is to maximize the total expected discounted rewards from exerting the singular control such a formulation stems from application areas such as optimal harvesting multiple species and optimal dividends payments schemes in random environments with the aid of weak dynamic programming principle and an exponential transformation we characterize the value function to be the unique constrained viscosity solution of a certain system of coupled nonlinear quasi variational inequalities several examples are analyzed in detail to demonstrate the main results

the stochastic maximum principle in singular optimal control with recursive utilities abstract in this paper we consider stochastic recursive optimal control problem in which the control variable has two components with the first absolutely continuous and the second singular the control domain of the first component needs not to be convex by using a spike variation on the absolutely continuous part of the control and a convex perturbation on the singular one respectively we obtain a stochastic maximum principle of the optimal control also we give the relationship of the backward variational equation the adjoint equation and forward variational equation

optimal impulse control for a multidimensional cash management system with generalized cost functions abstract we consider the optimal control of a multidimensional cash management system where the cash balances fluctuate as a homogeneous diffusion process in r n we formulate the model as an impulse control problem on an unbounded domain with unbounded cost functions under general assumptions we characterize the value function as a weak solution of a quasi variational inequality in a weighted sobolev space and we show the existence of an optimal policy moreover we prove the local uniform convergence of a finite element scheme to compute numerically the value function and the optimal cost we compute the solution of the model in two dimensions with linear and distance cost functions showing what are the shapes of the optimal policies in these two simple cases finally our third numerical experiment computes the solution in the realistic case of the cash concentration of two bank accounts made by a centralized treasury

stochastic maximum principle for optimal control problems of forward backward systems involving impulse controls we consider a stochastic optimal control problem of a forward backward system in which the control variable consists of two components the continuous control and the impulse control the domain of the control is assumed to be convex necessary optimality conditions of the pontryagin maximum principle type are obtained for this stochastic optimal control problem we also give additional conditions under which the necessary optimality conditions turn out to be sufficient

on an investment consumption model with transaction costs this paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and n risky assets whose prices are log normal diffusions we suppose that transactions between the assets incur a cost proportional to the size of the transaction the problem is to maximize the total utility of consumption dynamic programming leads to a variational inequality for the value function existence and uniqueness of a viscosity solution are proved the variational inequality is solved by using a numerical algorithm based on policies iterations and multigrid methods numerical results are displayed for n 1 and n 2

optimal consumption and portfolio with both fixed and proportional transaction costs we consider a market model with one risk free and one risky asset in which the dynamics of the risky asset are governed by a geometric brownian motion in this market we consider an investor who consumes from the bank account and who has the opportunity at any time to transfer funds between the two assets we suppose that these transfers involve a fixed transaction cost k 0 independent of the size of the transaction plus a cost proportional to the size of the transaction the objective is to maximize the cumulative expected utility of consumption over a planning horizon we formulate this problem as a combined stochastic control impulse control problem which in turn leads to a nonlinear quasi variational hamilton jacobi bellman inequality qvhjbi we prove that the value function is the unique viscosity solution of this qvhjbi finally numerical results are presented

on optimal harvesting problems in random environments this paper investigates the optimal harvesting strategy for a single species living in random environments whose population growth is given by a regime switching diffusion harvesting acts as a stochastic control on the size of the population the objective is to find a harvesting strategy which maximizes the expected total discounted income from harvesting up to the time of extinction of the species the income rate is allowed to be state and environment dependent this is a singular stochastic control problem with both the extinction time and the optimal harvesting policy depending on the initial condition one aspect of receiving payments up to the random time of extinction is that small changes in the initial population size may significantly alter the extinction time when using the same harvesting policy consequently one no longer obtains continuity of the value function using standard arguments for either regular or singular control problems having a fixed time horizon this paper introduces a new sufficient condition under which the continuity of the value function for the regime switching model is established further it is shown that the value function is a viscosity solution of a coupled system of quasi variational inequalities the paper also establishes a verification theorem and based on this theorem an varepsilon optimal harvesting strategy is constructed under certain conditions on the model two examples are analyzed in detail

a general method for comparing probability assessors a probability assessor or forecaster is a person who assigns subjective probabilities to events which will eventually occur or not occur in this paper a general method for comparing forecasters after a finite number of trials is introduced the general method is proven to include calculating all proper scoring rules as special cases it also includes comparison of forecasters in all simple two decision problems as special cases the relationship between the general method and calibration is also explored the general method is also translated into a method for deciding who will give better forecasts in the future

the well calibrated bayesian abstract suppose that a forecaster sequentially assigns probabilities to events he is well calibrated if for example of those events to which he assigns a probability 30 percent the long run proportion that actually occurs turns out to be 30 percent we prove a theorem to the effect that a coherent bayesian expects to be well calibrated and consider its destructive implications for the theory of coherence

optimal switching with constraints and utility maximization of an indivisible market this work focuses on optimal switching with constraints our motivation stems from utility maximization of an indivisible market the dynamic programming approach is used the value function is characterized as the unique viscosity solution of a quasi variational inequality the unbounded domain introduces new challenges by studying the sample paths of the diffusion at the boundary a sufficient condition for the continuity of the value function is provided yielding the desired characterization not only are the results of this work applicable to the utility maximization problem but also they can be used for general optimal switching problems with finite regimes

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

error estimates for approximate solutions to bellman equations associated with controlled jump diffusions we derive error estimates for approximate viscosity solutions of bellman equations associated to controlled jump diffusion processes which are fully nonlinear integro partial differential equations two main results are obtained i error bounds for a class of monotone approximation schemes which under some assumptions includes finite difference schemes and ii bounds on the error induced when the original levy measure is replaced by a finite measure with compact support an approximation process that is commonly used when designing numerical schemes for integro partial differential equations our proofs use and extend techniques introduced by krylov and barles jakobsen

on a characterization of algebraic number fields with class number less than three abstract a well known theorem of l carlitz states that classical algebraic number fields have class number less than or equal to two if and only if any two factorizations of an algebraic integer into irreducible elements contains the same number of factors we show that the same result holds using a somewhat weaker factorization condition this leads to a characterization of dedekind domains with class number less than or equal to three

optimal mixed impulse equity insurance control problem with reinsurance we investigate an optimal financing and dividend control problem of an insurance company facing fixed and proportional transaction costs the goal of the company is to maximize the expected present value of future dividends after deduction of the equity issuance until the time of bankruptcy we formulate the problem as a mixed classical impulse control and discuss the problem using the hjb dynamic programming approach a viscosity solution is considered and its uniqueness is established we also give results for the regularity and structure of the value function and the optimal policy of the control problem

finite horizon optimal stopping of time discontinuous functionals with applications to impulse control with delay we study finite horizon optimal stopping problems for continuous time feller markov processes the functional depends on time state and external parameters and may exhibit discontinuities with respect to the time variable both left and right hand discontinuities are considered we investigate the dependence of the value function on the parameters on the initial state of the process and on the stopping horizon we construct varepsilon optimal stopping times and provide conditions under which an optimal stopping time exists we demonstrate how to approximate this optimal stopping time by solutions to discrete time problems our results are applied to the study of impulse control problems with finite time horizon decision lag and execution delay

inventory control for spectrally positive levy demand processes a new approach to solve the continuous time stochastic inventory problem using the fluctuation theory of levy processes is developed this approach involves the recent developments of the scale function that is capable of expressing many fluctuation identities of spectrally one sided levy processes for the case with a fixed cost and a general spectrally positive levy demand process we show the optimality of an s s policy the optimal policy and the value function are concisely expressed via the scale function numerical examples under a levy process in the family with jumps of infinite activity are provided to confirm the analytical results furthermore the case with no fixed ordering costs is studied

combining probability forecasts summary linear pooling is by far the most popular method for combining probability forecasts however any non trivial weighted average of two or more distinct calibrated probability forecasts is necessarily uncalibrated and lacks sharpness in view of this linear pooling requires recalibration even in the ideal case in which the individual forecasts are calibrated towards this end we propose a beta transformed linear opinion pool for the aggregation of probability forecasts from distinct calibrated or uncalibrated sources the method fits an optimal non linearly recalibrated forecast combination by compositing a beta transform and the traditional linear opinion pool the technique is illustrated in a simulation example and in a case study on statistical and national weather service probability of precipitation forecasts

a numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit gmwb in this paper we outline an impulse stochastic control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit gmwb assuming the policyholder is allowed to withdraw funds continuously we develop a numerical scheme for solving the hamilton jacobi bellman hjb variational inequality corresponding to the impulse control problem we prove the convergence of our scheme to the viscosity solution of the continuous withdrawal problem provided a strong comparison result holds the scheme can be easily generalized to price discrete withdrawal contracts numerical experiments are conducted which show a region where the optimal control appears to be non unique

impulse control of multidimensional jump diffusions in finite time horizon this paper analyzes a class of impulse control problems for multidimensional jump diffusions in the finite time horizon following the basic mathematical setup from stroock and varadhan multidimensional diffusion processes springer verlag heidelberg 2006 this paper first establishes rigorously an appropriate form of the dynamic programming principle it then shows that the value function is a viscosity solution for the associated hamilton jacobi bellman equation involving integro differential operators finally under additional assumptions that the jumps are of infinite activity but are of finite variation and that the diffusion is uniformly elliptic it proves that the value function is the unique viscosity solution and has w loc 2 1 p regularity for 1 p infty

the stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls this paper is concerned with a pontryagin s maximum principle for stochastic optimal control problems of delay systems with random coefficients involving both continuous and impulse controls this kind of control problems is motivated by some interesting phenomena arising from economics and finance we establish a necessary maximum principle and a sufficient verification theorem by virtue of the duality and the convex analysis to explain the theoretical results we apply them to a production and consumption choice problem

optimal impulse control of portfolios an investor has the opportunity of holding shares in n risky assets and one nonrisky asset at every time in a fixed interval t t the risky assets are governed by a stochastic differential equation at random instants of his choice he may intervene in order to rebalance his portfolio and consume a nonnegative amount of money fixed and variable transactions costs are incurred upon intervention at time t all remaining wealth is consumed the solution to the problem of maximizing total utility of consumption is given by way of quasi variational inequalities for the value function with probability one the investor only intervenes finitely many times indication of the solution of the quasi variational inequalities in the case of one risky asset with log normal prices is given together with a description of a discretization procedure

impulse control of multidimensional jump diffusions this paper studies regularity properties of the value function for an infinite horizon discounted cost impulse control problem where the underlying controlled process is a multidimensional jump diffusion with possibly infinite activity jumps surprisingly despite these jumps we obtain the same degree of regularity as for the diffusion case at least when the jump satisfies certain integrability conditions

hamilton jacobi bellman quasi variational inequality arising in an environmental problem and its numerical discretization abstract a hamilton jacobi bellman quasi variational inequality hjbqvi for a river environmental restoration problem with wise use of sediment is formulated and its mathematical properties are analyzed a finite difference scheme with a penalization technique is then established for solving the hjbqvi the scheme is free from any iterative solvers and is unconditionally stable and convergent in the viscosity sense under certain conditions a demonstrative application example of the hjbqvi is finally presented

smooth fit principle for impulse control of multidimensional diffusion processes value functions of impulse control problems are known to satisfy quasi variational inequalities qvis a bensoussan and j l lions impulse control and quasivariational inequalities heyden son philadelphia 1984 translation of controle impulsionnel et inequations quasi variationnelles gauthier villars paris 1982 this paper proves the smooth fit c 1 property of the value function for multidimensional controlled diffusions using a viscosity solution approach we show by examples how to exploit this regularity property to derive explicitly optimal policy and value functions

portfolio selection with transaction costs in this paper optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log normal diffusion this problem was solved by merton and others when transactions between bank and stock are costless here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted it is shown that the optimal buying and selling policies are the local times of the two dimensional process of bank and stock holdings at the boundaries of a wedge shaped region which is determined by the solution of a nonlinear free boundary problem an algorithm for solving the free boundary problem is given

evaluating probability forecasts probability forecasts of events are routinely used in climate predictions in forecasting default probabilities on bank loans or in estimating the probability of a patient s positive response to treatment scoring rules have long been used to assess the efficacy of the forecast probabilities after observing the occurrence or nonoccurrence of the predicted events we develop herein a statistical theory for scoring rules and propose an alternative approach to the evaluation of probability forecasts this approach uses loss functions relating the predicted to the actual probabilities of the events and applies martingale theory to exploit the temporal structure between the forecast and the subsequent occurrence or nonoccurrence of the event