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A two-disorder detection problem

Krzysztof Szajowski (1997)

Applicationes Mathematicae

Suppose that the process X = { X n , n } is observed sequentially. There are two random moments of time θ 1 and θ 2 , independent of X, and X is a Markov process given θ 1 and θ 2 . The transition probabilities of X change for the first time at time θ 1 and for the second time at time θ 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...

About stability of risk-seeking optimal stopping

Raúl Montes-de-Oca, Elena Zaitseva (2014)

Kybernetika

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 X is approximated by the transition probability p ˜ ( · | x ) , x X , and that the stopping rule f ˜ * , which is optimal for the process with the transition probability 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 X p ( · | x ) - p ˜ ( · | x ) ) for...

Approximation of the Snell envelope and american options prices in dimension one

Vlad Bally, Bruno Saussereau (2002)

ESAIM: Probability and Statistics

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.

Approximation of the Snell Envelope and American Options Prices in dimension one

Vlad Bally, Bruno Saussereau (2010)

ESAIM: Probability and Statistics

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.

Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Bernard Roynette, Pierre Vallois, Agnès Volpi (2008)

ESAIM: Probability and Statistics

Let ( X t , t 0 ) be a Lévy process started at 0 , with Lévy measure ν . We consider the first passage time T x of ( X t , t 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 ν admits exponential moments, we show that ( T x ˜ , K x , L x ) converges in distribution as x , where T x ˜ denotes a suitable renormalization of T x .

Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Bernard Roynette, Pierre Vallois, Agnès Volpi (2007)

ESAIM: Probability and Statistics

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 ( T x ˜ , K x , L x ) converges in distribution as x → ∞, where T x ˜ denotes a suitable renormalization of Tx.


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