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18
On the optimal stopping problem for onedimensional diffusions, 2002. Working Paper (http://www.stat.columbia.edu/ ˜ik/DAYKAR.pdf
"... A new characterization of excessive functions for arbitrary one–dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the ..."
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Cited by 52 (3 self)
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A new characterization of excessive functions for arbitrary one–dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the optimal stopping problem as “the smallest nonnegative concave majorant of the reward function ” and allows us to generalize results of Dynkin and Yushkevich for standard Brownian motion. Moreover, we show how to reduce the discounted optimal stopping problems for an arbitrary diffusion process to an undiscounted optimal stopping problem for standard Brownian motion. The concavity of the value functions also leads to conclusions about their smoothness, thanks to the properties of concave functions. One is thus led to a new perspective and new facts about the principle of smooth–fit in the context of optimal stopping. The results are illustrated in detail on a number of non–trivial, concrete optimal stopping problems, both old and new.
Option pricing with Markovmodulated dynamics, Working paper
, 2004
"... Abstract. Markovmodulated models for equity prices have recently been extensively studied in the literature. In this paper, we apply some old results on the Wiener–Hopf factorization of Markov processes to a range of optionpricing problems for such models. The first example is the perpetual Americ ..."
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Abstract. Markovmodulated models for equity prices have recently been extensively studied in the literature. In this paper, we apply some old results on the Wiener–Hopf factorization of Markov processes to a range of optionpricing problems for such models. The first example is the perpetual American put, where the exact (numerical) solution is obtained without discretizing any PDE. We then show how the methodology of Rogers and Stapleton [Finance Stoch., 2 (1997), pp. 3–17] can be used to tackle finitehorizon problems and illustrate the methodology by pricing European, American, single barrier, and double barrier options under Markovmodulated dynamics.
Optimal stopping of linear diffusions with random discounting. Working paper, http://www.princeton.edu/ sdayanik/papers/additive.pdf
, 2003
"... Abstract. We propose a new solution method for optimal stopping problems for linear diffusions with random discounting. First, we extend the class of excessive functions for general diffusions and show that they are essentially concave. Then we use the new characterization of excessive functions to ..."
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Cited by 8 (3 self)
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Abstract. We propose a new solution method for optimal stopping problems for linear diffusions with random discounting. First, we extend the class of excessive functions for general diffusions and show that they are essentially concave. Then we use the new characterization of excessive functions to show that optimal stopping problems for linear diffusions discounted with respect to a continuous additive functional, recently studied by Beibel and Lerche [Theory Probab. Appl., 45(4):547–557, 2001], can be reduced to an undiscounted optimal stopping problem for standard Brownian motion. The latter problem can be solved essentially by inspection. The necessary and sufficient conditions for the existence of an optimal stopping rule are proved when the reward function is continuous. A proof of the smooth–fit principle is also provided. The results are illustrated on examples. 1.
Discounted optimal stopping for maxima in diffusion models with finite horizon
 Electron. J. Probab
, 2006
"... We present a solution to some discounted optimal stopping problem for the maximum of a geometric Brownian motion on a finite time interval. The method of proof is based on reducing the initial optimal stopping problem with the continuation region determined by an increasing continuous boundary sur ..."
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Cited by 8 (1 self)
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We present a solution to some discounted optimal stopping problem for the maximum of a geometric Brownian motion on a finite time interval. The method of proof is based on reducing the initial optimal stopping problem with the continuation region determined by an increasing continuous boundary surface to a parabolic freeboundary problem. Using the changeofvariable formula with local time on surfaces we show that the optimal boundary can be characterized as a unique solution of a nonlinear integral equation. The result can be interpreted as pricing American fixedstrike lookback option in a diffusion model with finite time horizon. 1.
On the convergence from discrete to continuous time in an optimal stopping problem
 Ann. Appl. Probab
, 2005
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Discounted optimal stopping for maxima of some jumpdiffusion processes
 J. Appl. Probab
, 2007
"... We present closed form solutions to some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integrodifferential freeboundary prob ..."
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We present closed form solutions to some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integrodifferential freeboundary problems where the normal reflection and smooth fit may break down and the latter then be replaced by the continuous fit. We show that under certain relationships on the parameters of the model the optimal stopping boundary can be uniquely determined as a component of solution of a twodimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jumpdiffusion model. 1.
The optimal Timing of Investment Decisions
 University of London
, 2006
"... discretionary stopping problem with applications to ..."
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discretionary stopping problem with applications to
EulerMaruyama approximations in meanreverting stochastic volatility model under regimeswitching
 Journal of Applied Mathematics and Stochastic Analysis. Volume 2006, Article ID 80967
, 2006
"... Stochastic differential equations (SDEs) under regimeswitching have recently been developed to model various financial quantities. In general, SDEs under regimeswitching have no explicit solutions, so numerical methods for approximations have become one of the powerful techniques in the valuation ..."
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Cited by 3 (2 self)
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Stochastic differential equations (SDEs) under regimeswitching have recently been developed to model various financial quantities. In general, SDEs under regimeswitching have no explicit solutions, so numerical methods for approximations have become one of the powerful techniques in the valuation of financial quantities. In this paper, we will concentrate on the EulerMaruyama (EM) scheme for the typical hybrid meanreverting θprocess. To overcome the mathematical difficulties arising from the regimeswitching as well as the nonLipschitz coefficients, several new techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems. Copyright © 2006 Xuerong Mao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
Stochastic Processes and their Applications 120 (2010) 1033–1059 www.elsevier.com/locate/spa pi optionsI
, 2010
"... We consider a discretionary stopping problem that arises in the context of pricing a class of perpetual Americantype call options, which include the perpetual American, Russian and lookbackAmerican call options as special cases. We solve this genuinely twodimensional optimal stopping problem by m ..."
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We consider a discretionary stopping problem that arises in the context of pricing a class of perpetual Americantype call options, which include the perpetual American, Russian and lookbackAmerican call options as special cases. We solve this genuinely twodimensional optimal stopping problem by means of an explicit construction of its value function. In particular, we fully characterise the freeboundary that provides the optimal strategy, and which involves the analysis of a highly nonlinear ordinary differential equation (ODE). In accordance with other optimal stopping problems involving a running maximum process that have been studied in the literature, it turns out that the associated variational inequality has an uncountable set of solutions that satisfy the socalled principle of smooth fit.
time uncertainty
, 2007
"... This paper studies optimal stopping problems for general diffusion processes with an uncertain time horizon and with different levels of information based on which the decision to stop or to continue is made. First, corresponding value functions are compared and related explicitly to their counterp ..."
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This paper studies optimal stopping problems for general diffusion processes with an uncertain time horizon and with different levels of information based on which the decision to stop or to continue is made. First, corresponding value functions are compared and related explicitly to their counterparts without the time uncertainty. Second, to analyze optimal stopping strategies, characterization results regarding connectivity of the “stopping ” and “continuation ” regions are derived. In particular, sufficient conditions for the wellknown “threshold type ” optimal stopping rule are given, based on which explicit solutions are derived for problems with standard American call/put type payoff functions.