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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
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.
Optimal stopping with random intervention times
 ADV. IN APPL. PROBAB
"... We consider a class of optimal stopping problems where the ability to stop depends an exogenous Poisson signal process  one can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an und ..."
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Cited by 5 (0 self)
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We consider a class of optimal stopping problems where the ability to stop depends an exogenous Poisson signal process  one can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous time structure. Depending on whether stopping is allowed at t = 0, the value function exhibits dierent properties across the optimal exercise boundary. Indeed, the value function is only C 0 across the optimal boundary when stopping is allowed at t = 0 and C 2
Dynamic System Evolution and Markov Chain Approximation
 Discrete Dynamics in NS, Gordon & Breach
, 1998
"... In this paper computational aspects of the mathematical modelling of dynamic system evolution have been considered as a problem in information theory. The construction of such models is treated as a decision making process with limited available information. The solution of the problem is associated ..."
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Cited by 4 (4 self)
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In this paper computational aspects of the mathematical modelling of dynamic system evolution have been considered as a problem in information theory. The construction of such models is treated as a decision making process with limited available information. The solution of the problem is associated with a computational model based on heuristics of a Markov Chain in a discrete spacetime of events. A stable approximation of the chain has been derived and the limiting cases are discussed. An intrinsic interconnection of constructive, sequential, and evolutionary approaches in related optimization problems provides new challenges for future work. Key words: decision making with limited information, optimal control theory, hyperbolicity of dynamic rules, generalized dynamic systems, Markov Chain approximation. 1 Introduction Many mathematical problems in information theory and optimal control related to dynamic system studies can be formulated in the following generic form. A decision...
ON THE CONVERGENCE FROM DISCRETE TO CONTINUOUS TIME IN AN OPTIMAL STOPPING PROBLEM 1
, 2005
"... We consider the problem of optimal stopping for a onedimensional diffusion process. Two classes of admissible stopping times are considered. The first class consists of all nonanticipating stopping times that take values in [0, ∞], while the second class further restricts the set of allowed values ..."
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Cited by 3 (0 self)
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We consider the problem of optimal stopping for a onedimensional diffusion process. Two classes of admissible stopping times are considered. The first class consists of all nonanticipating stopping times that take values in [0, ∞], while the second class further restricts the set of allowed values to the discrete grid {nh:n = 0,1,2,...,∞} for some parameter h> 0. The value functions for the two problems are denoted by V (x) and V h (x), respectively. We identify the rate of convergence of V h (x) to V (x) and the rate of convergence of the stopping regions, and provide simple formulas for the rate coefficients. 1. Introduction. One
Existence of optimal controls for singular control problems with state constraints
 Ann. Appl. Prob
, 2006
"... We establish the existence of an optimal control for a general class of singular control problems with state constraints. The proof uses weak convergence arguments and a time rescaling technique. The existence of optimal controls for Brownian control problems [14], associated with a broad family of ..."
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Cited by 3 (2 self)
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We establish the existence of an optimal control for a general class of singular control problems with state constraints. The proof uses weak convergence arguments and a time rescaling technique. The existence of optimal controls for Brownian control problems [14], associated with a broad family of stochastic networks, follows as a consequence.
Solving Singular Control from Optimal Switching
, 2008
"... This report summarizes some of our recent work (Guo and Tomecek (2008b,a)) on a new theoretical connection between singular control of finite variation and optimal switching problems. This correspondence not only provides a novel method for analyzing multidimensional singular control problems, but ..."
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This report summarizes some of our recent work (Guo and Tomecek (2008b,a)) on a new theoretical connection between singular control of finite variation and optimal switching problems. This correspondence not only provides a novel method for analyzing multidimensional singular control problems, but also builds links among singular controls, Dynkin games, and sequential optimal stopping problems. 1
List of papers................................................................................................................................... vii
"... Denmark Denne afhandling er, i forbindelse med de nedenfor anførte, tidligere offentliggjorte afhandlinger, af Det naturvidenskabelige Fakultet ved Aarhus Universitet antaget til forsvar for den naturvidenskabelige doktorgrad. Forsvarshandlingen finder sted fredag den 12. april 2002 kl. 13.15 i Audi ..."
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Denmark Denne afhandling er, i forbindelse med de nedenfor anførte, tidligere offentliggjorte afhandlinger, af Det naturvidenskabelige Fakultet ved Aarhus Universitet antaget til forsvar for den naturvidenskabelige doktorgrad. Forsvarshandlingen finder sted fredag den 12. april 2002 kl. 13.15 i Auditorium F p˚a Institut for Matematiske Fag, Aarhus Universitet.