The problem of disorder seeks to determine a stopping time which is as close as possible to the unknown time of "disorder" when the observed process changes its probability characteristics. We give a partial answer to this question for some special cases of Lévy processes and present a complete solution of the Bayesian and variational problem for a compound Poisson process with exponential jumps. The method of proof is based on reducing the Bayesian problem to an integro-differential free-boundary problem where, in some cases, the smooth-fit principle breaks down and is replaced by the principle of continuous fit.1. Introduction. Assume that at time t = 0 we begin to observe a continuously updated process X = (X t ) t≥0 whose probability characteristics change at some unknown time θ, called the time of disorder, which cannot be observed directly. Throughout this paper the random time θ can take the value 0 with probability π; under the condition that θ > 0, it is exponentially distributed with parameter λ > 0. The disorder problem or the problem of quickest disorder detection is to decide by observing the process X the time instant at which we should give an alarm to indicate the occurrence of disorder. This time instant should be as close as possible to the time θ in the sense that both the probability of false alarm and the expectation of the time interval between the occurrence of disorder and the alarm (when the latter is given correctly) should be minimal.The problem of detecting a change in drift of a Wiener process was formulated and solved by Shiryaev [12,13,14,15] (see also [16] and [17], Chapter IV and page 208, for historical notes and references). Some particular
We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process with linear and exponential penalty costs for a detection delay. The optimal times of alarms are found as the first times at which the weighted likelihood ratios hit stochastic boundaries depending on the current observations. The proof is based on the reduction of the initial problems into appropriate three-dimensional optimal stopping problems and the analysis of the associated parabolic-type free-boundary problems. We provide closed form estimates for the value functions and the boundaries, under certain nontrivial relations between the coefficients of the observable diffusion.2010 Mathematics Subject Classification: Primary 60G40, 62M20, 34K10.Secondary 62C10, 62L15, 60J60.
Statistics and Decisions (2005) 23(1) (15-31)
AbstractA convertible (callable) bond is a security that the holder can convert into a specified number of underlying shares. In addition, the issuer can recall the bond, paying some compensation, or force the holder to convert it immediately. We give an explicit solution to the corresponding optimal stopping game in the context of a reduced form model driven by a Brownian motion and a compound Poisson process with exponential jumps. It turns out that the occurrence of jumps leads to optimal stopping strategies whose structure differs from the results for continuous models.
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 integro-differential 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 two-dimensional 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 jump-diffusion model.
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