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29
The Russian Option: Reduced Regret
, 1993
"... this paper the value of the option (i.e. the supremum in (1.2)) will be found exactly, and in particular it will be shown that the maximum in (1.2) is finite if and only if r ? ¯ : (1.4) Assuming (1.4), an explicit formula is given for both the maximal expected present value and the optimal stopping ..."
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Cited by 36 (2 self)
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this paper the value of the option (i.e. the supremum in (1.2)) will be found exactly, and in particular it will be shown that the maximum in (1.2) is finite if and only if r ? ¯ : (1.4) Assuming (1.4), an explicit formula is given for both the maximal expected present value and the optimal stopping rule in (2.4), which is not a fixed time rule but depends heavily on the observed values of X t and S t . We call the financial option described above a "Russian option" for two reasons. First, this name serves to (facetiously) differentiate it from American and European options, which have been extensively studied in financial economics, especially with the new interest in market economics in Russia. Second, our solution of the stopping problem (1.2) is derived by the socalled principle of smooth fit, first enunciated by the great Russian mathematician, A. N. Kolmogorov, cf. [4, 5]. The Russian option is characterized by "reduced regret" because the owner is paid the maximum stock price up to the time of exercise and hence feels less remorse at not having exercised at the maximum. For purposes of comparison and to emphasize the mathematical nature of the contribution here, we conclude the paper by analyzing an optimal stopping problem for the Russian option based on Bachelier's (1900) original linear model of stock price fluctuations, X
Piecewiselinear diffusion processes
 Advances in Queueing
, 1995
"... Diffusion processes are often regarded as among the more abstruse stochastic processes, but diffusion processes are actually relatively elementary, and thus are natural first candidates to consider in queueing applications. To help demonstrate the advantages of diffusion processes, we show that ther ..."
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Cited by 28 (8 self)
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Diffusion processes are often regarded as among the more abstruse stochastic processes, but diffusion processes are actually relatively elementary, and thus are natural first candidates to consider in queueing applications. To help demonstrate the advantages of diffusion processes, we show that there is a large class of onedimensional diffusion processes for which it is possible to give convenient explicit expressions for the steadystate distribution, without writing down any partial differential equations or performing any numerical integration. We call these tractable diffusion processes piecewise linear; the drift function is piecewise linear, while the diffusion coefficient is piecewise constant. The explicit expressions for steadystate distributions in turn yield explicit expressions for longrun average costs in optimization problems, which can be analyzed with the aid of symbolic mathematics packages. Since diffusion processes have continuous sample paths, approximation is required when they are used to model discretevalued processes. We also discuss strategies for performing this approximation, and we investigate when this approximation is good for the steadystate distribution of birthanddeath processes. We show that the diffusion approximation tends to be good when the differences between the birth and death rates are small compared to the death rates.
Risk vs. ProfitPotential; A Model for Corporate Strategy
 J. Econ. Dynam. Control
, 1996
"... A firm whose net earnings are uncertain, and that is subject to the risk of bankruptcy, must choose between paying dividends and retaining earnings in a liquid reserve. Also, different operating strategies imply different combinations of expected return and variance. We model the firm's cash re ..."
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Cited by 27 (0 self)
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A firm whose net earnings are uncertain, and that is subject to the risk of bankruptcy, must choose between paying dividends and retaining earnings in a liquid reserve. Also, different operating strategies imply different combinations of expected return and variance. We model the firm's cash reserve as the difference between the cumulative net earnings and the cumulative dividends. The first is a diffusion (additive), whose drift/volatility pair is chosen dynamically from a finite set, A. The second is an arbitrary nondecreasing process, chosen by the firm. The firm's strategy must be nonclairvoyant. The firm is bankrupt at the first time, T , at which the cash reserve falls to zero (T may be infinite), and the firm's objective is to maximize the expected total discounted dividends from 0 to T , given an initial reserve, x; denote this maximum by V (x). We calculate V explicitly, as a function of the set A and the discount rate. The optimal policy has the form: (1) pay no dividends if ...
A Leavable BoundedVelocity Stochastic Control Problem
 Stochastic Process.Appl
, 2000
"... This paper studies boundedvelocity control of a Brownian motion when discretionary stopping, or `leaving', is allowed. The goal is to choose a control law and a stopping time in order to minimize the expected sum of a running and a termination cost, when both costs increase as a function of di ..."
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Cited by 9 (1 self)
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This paper studies boundedvelocity control of a Brownian motion when discretionary stopping, or `leaving', is allowed. The goal is to choose a control law and a stopping time in order to minimize the expected sum of a running and a termination cost, when both costs increase as a function of distance from the origin. There are two versions of this problem: the fullyobserved case, in which the control multiplies a known gain, and the partiallyobserved case, in which the gain is random and unknown. Without the extra feature of stopping, the fullyobserved problem originates with Benes (1974), who showed that the optimal control takes the `bangbang' form of pushing with maximum velocity toward the origin. We show here that this same control is optimal in the case of discretionary stopping; in the case of powerlaw costs, we solve the variational equation for the value function and explicitly determine the optimal stopping policy. We also discuss qualitative features of the solution for...
The stochastic economic lot scheduling problem: heavy traffic analysis of dynamic cyclic policies
, 2000
"... We consider two queueing control problems that are stochastic versions of the economic lot scheduling problem: A single server processes N customer classes, and completed units enter a finished goods inventory that services exogenous customer demand. Unsatisfied demand is backordered, and each class ..."
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Cited by 9 (2 self)
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We consider two queueing control problems that are stochastic versions of the economic lot scheduling problem: A single server processes N customer classes, and completed units enter a finished goods inventory that services exogenous customer demand. Unsatisfied demand is backordered, and each class has its own general service time distribution, renewal demand process, and holding and backordering cost rates. In the first problem, a setup cost is incurred when the server switches class, and the setup cost is replaced by a setup time in the second problem. In both problems we employ a longrun average cost criterion and restrict ourselves to a class of dynamic cyclic policies, where idle periods and lot sizes are statedependent, but the N classes must be served in a fixed sequence. Motivated by existing heavy traffic limit theorems, we make a time scale decomposition assumption that allows us to approximate these scheduling problems by diffusion control problems. Our analysis of the approximating setup cost problem yields a closedform dynamic lotsizing policy and a computational procedure for an idling threshold. We derive structural results and an algorithmic procedure for the setup time problem. A computational study compares the proposed policy and several alternative policies to the numerically computed optimal policy.
Inside Information And Stock Fluctuations
, 1999
"... A model of an incomplete market with the incorporation of a new notion of "inside information" is posed. The usual assumption that the stock price is Markovian is modified by adjoining a hidden Markov process to the BlackScholes exponential Brownian motion model for stock fluctuations. ..."
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Cited by 8 (4 self)
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A model of an incomplete market with the incorporation of a new notion of "inside information" is posed. The usual assumption that the stock price is Markovian is modified by adjoining a hidden Markov process to the BlackScholes exponential Brownian motion model for stock fluctuations. The drift and volatility parameters take different values when the hidden Markov process is in different states. For example, it is 0 when there is no subset of the market which has or which believes it has, extra information. However, the hidden process is in state 1 when information is not equally shared by all, and then the behavior of the members in the subset causes increased fluctuations in the stock price. This model
Drift rate control of a Brownian processing system
 Annals of Applied Probability
, 2005
"... A system manager dynamically controls a diffusion process Z that lives in a finite interval [0,b]. Control takes the form of a negative drift rate θ that is chosen from a fixed set A of available values. The controlled process evolves according to the differential relationship dZ = dX − θ(Z)dt + dL ..."
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Cited by 7 (0 self)
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A system manager dynamically controls a diffusion process Z that lives in a finite interval [0,b]. Control takes the form of a negative drift rate θ that is chosen from a fixed set A of available values. The controlled process evolves according to the differential relationship dZ = dX − θ(Z)dt + dL − dU, where X is a (0,σ) Brownian motion, and L and U are increasing processes that enforce a lower reflecting barrier at Z = 0 and an upper reflecting barrier at Z = b, respectively. The cumulative cost process increases according to the differential relationship dξ = c(θ(Z))dt+pdU, where c(·) is a nondecreasing cost of control and p> 0 is a penalty rate associated with displacement at the upper boundary. The objective is to minimize longrun average cost. This problem is solved explicitly, which allows one to also solve the following, essentially equivalent formulation: minimize the longrun average cost of control subject to an upper bound constraint on the average rate at which U increases. The two
The relaxed stochastic maximum principles in singular optimal control of diffusions
 SIAM J. Cont. and Opt
"... E l e c t r o n ..."
Connections between singular control and optimal switching
 SIAM J. on Control and Optimization
, 2008
"... Abstract. This paper builds a new theoretical connection between singular control of finite variation and optimal switching problems. This correspondence provides a novel method for solving highdimensional singular control problems and enables us to extend the theory of reversible investment: Suffi ..."
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Cited by 6 (4 self)
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Abstract. This paper builds a new theoretical connection between singular control of finite variation and optimal switching problems. This correspondence provides a novel method for solving highdimensional singular control problems and enables us to extend the theory of reversible investment: Sufficient conditions are derived for the existence of optimal controls and for the regularity of value functions. Consequently, our regularity result links singular controls and Dynkin games through sequential optimal stopping problems.
Irreversible Investment
, 1998
"... This paper proposes, solves and characterizes a model of sequential irreversible investment by a firm facing uncertainty in technology, demand and price of capital. The solution can be found in closed form if simple (but not totally unrealistic) functional forms are assumed, and can be given an ..."
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Cited by 6 (0 self)
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This paper proposes, solves and characterizes a model of sequential irreversible investment by a firm facing uncertainty in technology, demand and price of capital. The solution can be found in closed form if simple (but not totally unrealistic) functional forms are assumed, and can be given an optimal stopping interpretation. The marginal revenue product of capital that induces additional investment is higher, under irreversibility, than the conventionally measured user cost of capital. In ergodic steady state, however, the former quantity is on average lower than the latter