The disorder problem for compound Poisson processes with exponential jumps

Abstract

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

Optimal time to change premiums

“…solved the simple Poisson disorder problem for exponential penalty for delay, and Bayraktar et al (2005) solved the standard Poisson disorder problem. These results were recently extended by Dayanik and Sezer (2006) (using the results developed in ) and Gapeev (2005) for compound Poisson procesesses. On the other hand solved the simple Poisson disorder problem when the post disorder rate is a random variable and Bayraktar and Sezer (2006) solved this problem for the case with a Phase-type disorder distribution.…”

“…Second, in the general problem if we set Λ to be a constant, then we obtain a version of the main problem in which the rate change and change of the distribution of the claim sizes occur simultaneously. This case was analyzed by Dayanik and Sezer (2006) and Gapeev (2005). Finally, the more general problem represents a situation in which the insurance company has only some apriori information about the post disorder rate λ 1 , but the company can not pin λ 1 down to a constant because it might only have very few claims after the regime change occurs.…”

Parameter Estimation: The Proper Way to Use Bayesian Posterior Processes with Brownian Noise

“…I managed to show that the Bayesian posterior distribution process of a disorder discrete process that is close in distribution to a disorder Brownian motion, has a similar structure to the posterior 20 As in Section 6.3, the function Vτ (q 1 ,q 2 ) (π) can be expressed explicitly through the parameters of the problem, but since it has no fundamental contribution, this expression is omitted. 21 This model was generalized in the context of Brownian motion by, e.g., Vellekoop and Clark (2001) [25], Gapeev and Peskir (2006) [13], Dayanik (2010) [9], Sezer (2010) [23], and in the context of other processes different from the Brownian motion, e.g., Peskir and Shiryaev (2002) [21], Gapeev (2005) [11], and Bayraktar, Dayanik, and Karatzas (2006) [1]. distribution process in our paper.…”

On the Laplace transforms of the first exit times in one-dimensional non-affine jump–diffusion models

“…Optimal stopping problems for some mean-reverting and diverting jump-diffusion processes were initiated by Davis [4], Peskir and Shiryaev [25]- [26], Dayanik and Sezer [5]- [6], and [10]- [11] among others, with the aim to detect the change points in the associated discontinuous observable processes. Discounted optimal stopping problems for certain payoff functions depending on the current values of geometric compound Poisson processes with multi-exponential jumps and their various extensions were considered by Mordecki [20]- [21], Kou [16], and Kou and Wang [18] among others, with the aim of computing rational values for the perpetual American options.…”