On the dimension reduction in the quickest detection problem for diffusion processes with exponential penalty for the delay

The problem of the quickest detection of a change in the drift of a time-homogeneous diffusion process is considered under the assumption that the detection delay is exponentially penalized. In this framework, the past literature has shown that a two-or three-dimensional optimal stopping problem needs to be faced. In this note, we show how a change of measure significantly simplifies the setting by reducing the dimension of the optimal stopping problem to one or two, respectively. We illustrate this result in the well known Brownian motion case analyzed by Beibel [4] and when a Bessel process is observed, generalizing therefore the results for the linear penalty case obtained by Johnson and Peskir [13]

A note on some sequential for the equilibrium value of a Vasicek process

We apply the Shiryaev's sequential procedure to the Vasicek mode

The disorder problem for diffusion processes with the ε-linear and expected total miss criteria

We study the disorder problem for a time-homogeneous diffusion process. The aim is to determine an efficient detection strategy of the disorder time θ, at which the process changes its drift. We focus on the ϵ-linear and the expected total miss criteria, where, unlike the well known linear penalty criterion, the expected penalty for an early/wrong detection of θ is expressed as the frequency of false alarms launched at least ϵ units of time before θ and as the expected advance in the detection of θ, respectively. We show that the original optimal stopping problems can be reduced to a unifying optimal stopping problem; then, we derive the associated free-boundary problem and we provide sufficient conditions for the existence and uniqueness of its solution

The disorder problem for purely jump L\ue9vy processes with completely monotone jumps

We study the problem of detecting as quickly as possible the disorder time at which a purely jump L\ue9vy process changes its probabilistic features. Assuming that its jumps are completely monotone, the monitored process is approximated by a sequence of hyperexponential processes. Then, the solution to the disorder problem for a hyperexponential process is used to approximate the one of the original problem. The efficiency of the proposed approximation scheme is investigated for some popular L\ue9vy processes, such as the gamma, inverse Gaussian, variance-gamma and CGMY processes

Dynamic optimality in optimal variance stopping problems

In an optimal variance stopping (O.V.S.) problem one seeks to determine the stopping time that maximizes the variance of an observed process. As originally shown by Pedersen (2011), the variance criterion leads to optimal stopping boundaries that depend explicitly on the initial point of the process. Then, following the lines of Pedersen and Peskir (2016), we introduce the concept of dynamic optimality for an O.V.S. problem, a type of optimality that disregards the starting point of the process. We examine when an O.V.S. problem admits a dynamically optimal stopping time and we illustrate our findings through several examples

On the martingale and free-boundary approaches in sequential detection problems with exponential penalty for delay

We study the connection between the martingale and free-boundary approaches in sequential detection problems for the drift of a Brownian motion, under the assumption of exponential penalty for the delay. By means of the solution of a suitable free-boundary problem, we show that the reward process can be decomposed into the product between a gain function of the boundary point and a positive martingale inside the continuation region

A collocation method for the sequential testing of a gamma process

We study the Bayesian problem of sequential testing of two simple hypotheses about the parameter \u3b1 > 0 of a L\ue9vy gamma process. The initial optimal stopping problem is reduced to a free-boundary problem where, at the unknown boundary points separating the stopping and continuation set, the principles of the smooth and/or continuous fit hold and the unknown value function satisfies on the continuation set a linear integro-differential equation. Due to the form of the L\ue9vy measure of a gamma process, determining the solution of this equation and the boundaries is not an easy task. Hence, instead of solving the problem analytically, we use a collocation technique: the value function is replaced by a truncated series of polynomials with unknown coefficients that, together with the boundary points, are determined by forcing the series to satisfy the boundary conditions and, at fixed points, the integro-differential equation. The proposed numerical technique is employed in well-understood problems to assess its efficiency