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We consider an insurance company whose surplus is represented by the classical Cramer-Lundberg process. The company can invest its surplus in a risk-free asset and in a risky asset, governed by the Black-Scholes equation. There is a constraint that the insurance company can only invest in the risky asset at a limited leveraging level; more precisely, when purchasing, the ratio of the investment amount in the risky asset to the surplus level is no more thana; and when short-selling, the proportion of the proceeds from the short-selling to the surplus level is no more thanb. The objective is to find an optimal investment policy that minimizes the probability of ruin. The minimal ruin probability as a function of the initial surplus is characterized by a classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We study the optimal control policy and its properties. The interrelation between the parameters of the model plays a crucial role in the qualitative behavior of the optimal policy. For example, for some ratios betweenaandb, quite unusual and at first ostensibly counterintuitive policies may appear, like short-selling a stock with a higher rate of return to earn lower interest, or borrowing at a higher rate to invest in a stock with lower rate of return. This is in sharp contrast with the unrestricted case, first studied in Hipp and Plum, or with the case of no short-selling and no borrowing studied in Azcue and Muler. [ABSTRACT FROM AUTHOR]

In this paper, first we study a stochastic volatility market model for which an explicit candidate solution to the problem of maximizing the utility function of terminal wealth is obtained. Applying this result, we present a complete solution for the Heston model, which is a particular case of the general model. A verification result and a martingale representation of the solution are provided for the Heston model. Finally, the same techniques are used to study a stochastic interest rate model and a necessary and sufficient condition for exploding growth is presented. [ABSTRACT FROM PUBLISHER]

An insurance company has a large numberNof potential customers characterized by i.i.d. r.v.'sgiving the arrival rates of claims. Customers are risk averse, and a customer accepts an offered premiumpaccording to hisA-value. The modelling further involves a discount rated>rof customers, whereris the risk-free interest rate. Based on calculations of the customers' present values of the alternative strategies of insuring and not insuring, the portfolio sizeis derived, and also the rate of claims from the insured customers is given. Furthermore, the value ofpwhich is optimal for minimizing the ruin probability is derived in a diffusion approximation to the Cramér–Lundberg risk process with an added liability rateLof the company. The solution involves the LambertWfunction. Similar discussion is given for extensions involving customers having only partial information on theirAand stochastic discount rates. [ABSTRACT FROM PUBLISHER]

We use the expectation of the range of an arithmetic Brownian motion and the method of moments on the daily high, low, opening, and closing prices to estimate the volatility of the stock price. This novel theoretical approach results in an estimator that is genuinely range-based on daily opening, high, low and closing data, unlike current estimators in the literature. The daily price jump at the opening is considered to be the result of the unobserved evolution of an after-hours virtual trading day. In comparison to an existing drift-independent estimator, we find that our estimator is actually more efficient when using a smaller number of data points, while for a larger number of points the efficiency of our estimator stays above 99% of the existing one. A toy example that uses this method to take advantage of mispricing opportunities in the options market illustrates potential applications of this method to algorithmic trading. [ABSTRACT FROM AUTHOR]

The paper examines random dynamical systems related to the classical von Neumann and Gale models of economic growth. Such systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. A central role in the theory of von Neumann-Gale dynamics is played by a special class of paths called rapid (they maximise properly defined growth rates). Up to now the theory lacked quite satisfactory results on the existence of such paths. This work provides a general existence theorem holding under assumptions analogous to the standard deterministic ones. The result solves a problem that remained open for more than three decades. [ABSTRACT FROM AUTHOR]

We consider a multistock market model. The stock price process satisfies a stochastic differential equation where both the drift and the volatility are driven by a discrete-time Markov chain of finite states. Not only the underlying Brownian motion but also the Markov chain in the stochastic differential equation are assumed to be unobservable. Investors can observe the stock price process only. The main result of this paper is that we derive the approximation of the optimal trading strategy and the corresponding optimal expected utility function from the terminal wealth for the CRRA utility function. [ABSTRACT FROM AUTHOR]

This paper deals with the dividend optimization problem for a financial or an insurance entity which can control its business activities, simultaneously reducing the risk and potential profits. It also controls the timing and the amount of dividends paid out to the shareholders. The objective of the corporation is to maximize the expected total discounted dividends paid out until the time of bankruptcy. Due to the presence of a fixed transaction cost, the resulting mathematical problem becomes a mixed classical-impulse stochastic control problem. The analytical part of the solution to this problem is reduced to quasivariational inequalities for a second-order nonlinear differential equation. We solve this problem explicitly and construct the value function together with the optimal policy. We also compute the expected time between dividend payments under the optimal policy. [ABSTRACT FROM AUTHOR]

This paper represents a model for risk management in a firm which exercises control of its risk as well as potential profit by choosing different business activities among those available to it. Furthermore, the firm has an option of investing its reserve in a financial market consisting of a risk-free asset (bond) and a risky asset (stock). The example we consider is that of a large corporation such as an insurance company, whose liquid assets in the absence of control and investments fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, while drift is associated with potential profit. At each moment of time there is an option to reduce risk exposure, simultaneously reducing the potential profit, like using proportional reinsurance with another carrier for an insurance company. The company invests its reserve in a financial market, which is described by a classical Black–Scholes model. The management of the company also controls the dividend pay-outs to shareholders. The objective is to find a policy, consisting of investment strategy, risk control and dividend distribution scheme, which maximizes the expected total discounted dividends paid out until the time of bankruptcy. We apply the theory of controlled diffusions to solve the problem and show that there is a level u 1 >0 such that the optimal action is to distribute all the reserve in excess of u 1 as dividends. Furthermore, there exists a constant x 0 , with x 0 < u 1 , such that the risk exposure monotonically increases on (0, x 0 ) from zero to the maximum possible. The optimal choice of investments depends on the market price of risk , where r 0 , r 1 denotes the mean rate of return of bond and stock respectively and s P denotes the volatility of the stock price. We get the following results. (1) m p =0: invest everything in bond.(2) m p is large: invest everything in stock.(3) m p is small: there exists x 0 < x 1 < u 1 , such that the optimal fraction of the reserve invested in stock is constant when the current reserve x is less than x 0 and it is an increasing function of x on [ x 0 , x 1 ] with all the reserve to be invested in stock whenever the reserve level exceeds x 1. [ABSTRACT FROM AUTHOR]

The paper generalizes and refines the Fundamental Theorem of Asset Pricing of in the following two respects: (a) the result is extended to a model with general portfolio constraints, and (b) versions of the no-arbitrage criterion based on the bang-bang principle in control theory are developed. [ABSTRACT FROM AUTHOR]

We consider a large insurance company whose surplus (reserve) is modeled by a Brownian motion. The company invests its surplus in stock market assets which may or may not contain an element of risk. To minimize the insurance risk there is a possibility to reinsure a part or the whole insurance portfolio. We consider the case of proportional reinsurance. There is a transaction cost, which manifests itself in the fact that the safety loading of the reinsurer is larger than that of the cedent. Stochastic optimal control theory is used to determine the optimal reinsurance policy which minimizes the ruin probability of the cedent. [ABSTRACT FROM AUTHOR]

Investigates a model of corporation which face constant liability payments which can chose a production/business policy from an available set of control policies with different expected profits and risks. Identification of constraints on business activity; Analysis of value function in great detail; Expression of the value function as well as the optimal policies.

We consider a zero-sum game of optimal stopping in which each of the opponents has the right to stop a one-dimensional diffusion process. There are two types of costs. The first is accumulated continuously at the rate H(X[sub t]), where X[sub t] is the current position of the process. The second is a cost associated with the stopping of the process. It is given by the function f[sub 1](x) for the first player and the function f[sub 2](x) for the second player, where x is the position of the process when the stopping option is exercised. We study the solution of the free boundary problem associated with this game via Dirichlet forms on the appropriate functional space. Integrating the value function of the game, we get a solution to another free boundary problem which yields the optimal return function for a singular stochastic control problem. [ABSTRACT FROM AUTHOR]

We consider the problem of finding an optimal financing mix of retained earnings and external equity for maximizing the value of a corporation in a stochastic environment. We formulate the problem as a singular stochastic control for a diffusion process. We show that the value function satisfies a free-boundary problem. We characterize the value function and show that the optimal policy can be characterized in terms of two threshold parameters. With asset level below the lower threshold, optimal policy is to finance the firm’s growth by retaining all earnings and raising the required external equity financing. With asset level above the higher threshold, optimal policy is to pay all retained earnings as dividends and to bring in no new equity. Between the two thresholds, the optimal policy is to retain all earnings but not raise any external equity. We obtain an explicit solution for the value function when there is no brokerage commission in floating external equity. We provide economic interpretations of the results obtained in the paper. [ABSTRACT FROM AUTHOR]

This paper examines set-valued random dynamical systems defined by convex homogeneous stochastic operators. The operators under consideration transform elements of a cone contained in a space of random vectors into subsets of the cone. We study rapid paths of such dynamical systems, i.e. those paths which maximize (appropriately defined) growth rates at every time period. Questions of existence, uniqueness and asymptotic behavior of infinite rapid trajectories are considered. The study is motivated by problems related to stochastic models of economic growth. [ABSTRACT FROM AUTHOR]

We consider a problem of risk control and dividend optimization for a financial corporation facing a constant liability payment. More specifically we investigate the case of excess-of-loss reinsurance for an insurance company. In this scheme the insurance company diverts a part of its premium stream to another company, the reinsurer, in exchange for an obligation to pick up that amount of each claim which exceeds a certain level a . The objective of the insurer is to maximize the expected present value of total future dividend pay-outs. We consider cases when there is restriction on the rate of dividend pay-outs and when there is no restriction. In both cases we describe explicitly the optimal return function as well as the optimal policy. [ABSTRACT FROM AUTHOR]

Reports the optimal risk control for a large corporation in the presence of returns on investments. Presentation of a model of financial valuation of a firm controlling the dividend payment stream and its risk; Elements of risk; Association of the value of the company with the expected present value of the net dividend distributions.

We study the optimal investment policy for an investor who has available one bank account and n risky assets modeled by log-normal diffusions. The objective is to maximize the long-run average growth of wealth for a logarithmic utility function in the presence of proportional transaction costs. This problem is formulated as an ergodic singular stochastic control problem and interpreted as the limit of a discounted control problem for vanishing discount factor. The variational inequalities for the discounted control problem and the limiting ergodic problem are established in the viscosity sense. The ergodic variational inequality is solved by using a numerical algorithm based on policy iterations and multigrid methods. A numerical example is displayed for two risky assets. [ABSTRACT FROM AUTHOR]

The paper deals with the model of a firm which has a possibility to choose among a variety of production/business policies with different risk and profit potential. The objective is to find the policy which maximizes the expected total discounted dividend pay-out until the time of bankruptcy. The bankruptcy is defined as the time when the liquid assets of the company vanish. A typical example of such a corporation would be an insurance company whose different business activities correspond to choosing different levels of reinsurance. The main novelty of this model is in introduction of terminal value of the company at the time of the bankruptcy. This could be the value of non liquid assets (such as real estate or the rights to conduct business or the trade name), which at the time of bankruptcy are subject to sale with proceeds distributed among shareholders. We model the dynamics of the corporate liquid assets as a diffusion process with controllable drift and diffusion coefficients. Diffusion coefficient corresponds to risk, while drift represents potential profit. In our model the potential profit proportional to the risk. The dividend distribution is modeled by an increasing functional, which is also controllable. We show how to obtain solution for this problem starting with the solution to the problem with zero terminal value. [ABSTRACT FROM AUTHOR]

Abstract. We consider a model of a financial corporation which has to find an optimal policy balancing its risk and expected profits. The example treated in this paper is related to an insurance company with the risk control method known in the industry as excess-of-loss reinsurance. Under this scheme the insurance company divert part of its premium stream to another company in exchange of an obligation to pick up that amount of each claim which exceeds a certain level a. This reduces the risk but it also reduces the potential profit. The objective is to make a dynamic choice of a and find the dividend distribution policy, which maximizes the cumulative expected discounted dividend pay-outs. We use diffusion approximation for this optimal control problem, where two situations are considered: (a) The rate of dividend pay-out are unrestricted and in this case mathematically the problem becomes a mixed singular-regular control problem for diffusion processes. Its analytical part is related to a free boundary (Stephan) problem for a linear second order differential equation. The optimal policy prescribes to reinsure using a certain retention level (depending on the reserve) and pay no dividends when the reserve is below some critical level x~ and to pay out everything that exceeds x[sub 1]. Reinsurance will stop at a level x[sub 0] less than or equal to x[sub 1] depending on the claim size distribution. (b) The rate of dividend pay-out is bounded by some positive constant M < Infinity, in... [ABSTRACT FROM AUTHOR]

Abstract. The current paper presents a short survey of stochastic models of risk control and dividend optimization techniques for a financial corporation. While being close to consumption/investment models of Mathematical Finance, dividend optimization models possess special features which do not allow them to be treated as a particular case of consumption/investment models. In a typical model of this sort, in the absence of control, the reserve (surplus) process, which represents the liquid assets of the company, is governed by a Brownian motion with constant drift and diffusion coefficient. This is a limiting case of the classical Cramer-Lundberg model in which the reserve is a compound Poisson process, amended by a linear term, representing a constant influx of the insurance premiums. Risk control action corresponds to reinsuring part of the claims the cedent is required to pay simultaneously diverting part of the premiums to a reinsurance company. This translates into controlling the drift and the diffusion coefficient of the approximating process. The dividend distribution policy consists of choosing the times and the amounts of dividends to be paid out to shareholders. Mathematically, the cumulative dividend process is described by an increasing functional which may or may not be continuous with respect to time. The objective in the models presented here is maximization of the dividend pay-outs. We will discuss models with different types of conditions imposed upon a company and different types of reinsurances available, such as proportional, noncheap, proportional in a presence of a constant debt liability, excess-of-loss. We will show that in most cases the optimal dividend [ABSTRACT FROM AUTHOR]

The paper represents a model for financial valuation of a firm which has control of the dividend payment stream and its risk as well as potential profit by choosing different business activities among those available to it. This model extends the classical Miller-Modigliani theory of firm valuation to the situation of controllable business activities in a stochastic environment. We associate the value of the company with the expected present value of the net dividend distributions (under the optimal policy). The example we consider is a large corporation, such as an insurance company, whose liquid assets in the absence of control fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, and drift is understood as potential profit. At each moment of time there is an option to reduce risk exposure while simultaneously reducing the potential profit-for example, by using proportional reinsurance with another carrier for an insurance company. Management of a company controls the dividends paid out to the shareholders, and the objective is to find a policy that maximizes the expected total discounted dividends paid out until the time of bankruptcy. Two cases are considered: one in which the rate of dividend payout is bounded by some positive constant M, and one in which there is no restriction on the rate of dividend payout. We use recently developed techniques of mathematical finance to obtain an easy understandable closed form solution. We show that there are two levels u0 and u1 with u0 ≤ u1. As a function of currently available reserve, the risk exposure monotonically increases on (0, u0) from 0 to the maximum possible. When the reserve exceeds u1 the dividends are paid at the maximal rate in the first case and in the second case every excess above u 1 is distributed as dividend. We also show that for M small enough u0 = u1 and the optimal risk exposure is always less than the maximal. [ABSTRACT FROM AUTHOR]

We apply regenerative theory to derive certain relations between steady state probabilities of a Markov chain. These relations are then used to develop a numerical algorithm to find these probabilities. The algorithm is a modification of the Gauss-Jordan method, in which all elements used in numerical computations are nonnegative; as a consequence, the algorithm is numerically stable. [ABSTRACT FROM AUTHOR]

The article presents an asymptomatic optimal hierarchical approach to show capacity and production decisions in stochastic manufacturing systems. One of the most important methods of dealing with optimization of large, complex systems is that of hierarchical decomposition. The idea is to reduce the overall complex problem into manageable approximate problems or sub problems, to solve these problems, and to construct a solution of the original problem from solutions of these simpler problems. Most manufacturing firms are large, complex systems characterized by several decision subsystems, such as finance, personnel, marketing, and operations. Furthermore, they may have a number of plants and warehouses and produce a large number of different products using a wide variety of machines and equipment. These events could be deterministic or stochastic. It shows that under reasonable assumptions, the strategic level management can base the capacity decision on aggregated information from the shop floor and the operational level management.

We consider a queue in which the server is available for primary customers for a random number of busy cycles, after which he leaves for a random amount of time as soon as the system becomes empty. Under appropriate normalization we establish a heavy traffic limit which turns out to be a regenerative generalized elastic screen process (RGESP) with random jumps or linear parts of the trajectories. These jumps or linear parts occur when the local time at zero accumulates to a certain random level. As a main tool we first establish a heavy traffic limit theorem for the renewal counting process associated with the busy cycles in the underlying queueing system. In particular we show that this limit is proportional to the local time at zero of a reflected Brownian motion. We compare the stationary distribution for the RGESP with the corresponding one for the M/G/1 queue, where for both explicit expressions are obtained, and show that the results are consistent. [ABSTRACT FROM AUTHOR]

We consider the problem of production control in a single machine, single product, unreliable manufacturing system facing a constant demand d. The goal is to minimize the expected average (per unit time) inventory/backlog costs. Under heavy traffic condition, i.e., when the average production capacity is close to demand, the problem is approximated by a singular stochastic control problem. The approximate problem can be solved explicitly. The solution is then interpreted in terms of the actual manufacturing system and a control policy for this system is derived. We prove that the resulting policy is nearly optimal under the heavy traffic condition. This policy is characterized by a critical level z0. That is, produce at maximal rate r when inventory is less than z0 and at the demand rate d when the inventory is equal to z0. The quality of the approximate control is also discussed by comparing it to known results, when they are available. [ABSTRACT FROM AUTHOR]

This paper presents an asymptotic analysis of a hierarchical manufacturing system with machines subject to breakdown and repair. The rate of change in machine states is much larger than the rate of fluctuation in demand and the rate of discounting of costs, and this gives rise to a limiting problem in which the stochastic machine availability is replaced by the equilibrium mean availability. The value function for the original problem converges to the value function of the limiting problem. Moreover, the control for the original problem can be constructed from the optimal controls of the limiting problem in a way which guarantees asymptotic optimality of the value function. The limiting problem is computationally more tractable and sometimes has a closed form solution. [ABSTRACT FROM AUTHOR]

Focuses on a study which introduced a continuous time model in which a rate of return or a risky investment fluctuates like a Brownian motion. Details of the mathematical formulation of the problem; Heuristic deduction of the optimality equation; Construction of control limit policies; Proof of sufficiency of the optimality equation; Solution for the optimality equation.

A controller continuously monitors a storage system, such as an inventory or bank account, whose content Z = {Z[subt],t > 0} fluctuates as a (μ,&theta²) Brownian motion in the absence of control. Holding costs are incurred continuously at rate h(Z[subt]). At any time, the controller may instantaneously increase the content of the system, incurring a proportional cost of r times the size of the increase, or decrease the content at a cost of l times the size of the decrease. We consider the case where h is convex on a finite interval [α,Β] and h = ∞ outside this interval. The objective is to minimize the expected discounted sum of holding costs and control costs over an infinite planning horizon. It is shown that there exists an optimal control limit policy, characterized by two parameters a and b (α 0. Put another way, the optimal control limit policy imposes on Z a lower reflecting barrier at a and an upper reflecting barrier at b. The optimality of a particular control limit policy is proved directly, with heavy reliance on the change of variable formula for semimartingales. We do not give a full-blown algorithm for construction of the optimal control limits, but a computational scheme could easily be developed from our constructive proof of existence. [ABSTRACT FROM AUTHOR]