Suppose that the process $X=\{X_n,n\in \mathbb{N}\}$ is observed sequentially. There are two random moments of time ${\theta }_1$ and ${\theta }_2$, independent of X, and X is a Markov process given ${\theta }_1$ and ${\theta }_2$. The transition probabilities of X change for the first time at time ${\theta }_1$ and for the second time at time ${\theta }_2$. Our objective is to find a strategy which immediately detects the distribution changes with maximal probability based on observation of X. The corresponding problem of double optimal stopping is constructed. The optimal strategy is found...

We offer the quantitative estimation of stability of risk-sensitive cost optimization in the problem of optimal stopping of Markov chain on a Borel space $X$. It is supposed that the transition probability $p(·|x)$, $x\in X$ is approximated by the transition probability $\tilde{p}(·|x)$, $x\in X$, and that the stopping rule ${\tilde{f}}_{*}$ , which is optimal for the process with the transition probability $\tilde{p}$ is applied to the process with the transition probability $p$. We give an upper bound (expressed in term of the total variation distance: ${sup}_{x\in X}\parallel p(·|x)-\tilde{p}(·|x)\parallel )$ for...

We establish some error estimates for the approximation of an optimal stopping problem along the paths of the Black–Scholes model. This approximation is based on a tree method. Moreover, we give a global approximation result for the related obstacle problem.

We establish some error estimates for the approximation of an optimal stopping problem along the paths of the Black–Scholes model. This approximation is based on a tree method. Moreover, we give a global approximation result for the related obstacle problem.

In this paper we solve an optimal stopping problem for processed indexed by N U{∞} with respect to a certain class of stopping times.

Let $(X_t,t\ge 0)$ be a Lévy process started at $0$, with Lévy measure $\nu$. We consider the first passage time $T_x$ of $(X_t,t\ge 0)$ to level $x\>0$, and $K_x:=X_{T_x}-𝑥$ the overshoot and $L_x:=x-X_{T_{{𝑥}^{-}}}$ the undershoot. We first prove that the Laplace transform of the random triple $(T_x,K_x,L_x)$ satisfies some kind of integral equation. Second, assuming that $\nu$ admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution as $x\to \infty$, where $\widetilde{T_x}$ denotes a suitable renormalization of $T_x$.

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévy measure ν. We consider the first passage time Tx of (Xt, t ≥ 0) to level x > 0, and Kx := XTx - x the overshoot and Lx := x- XTx- the undershoot. We first prove that the Laplace transform of the random triple (Tx,Kx,Lx) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution as x → ∞, where $\widetilde{T_x}$ denotes a suitable renormalization of Tx.