We investigate in detail works of Peskir [13] and Meilijson [9] and develop a link between them. We consider the following optimal stopping problem: maximize V# = E 0 c(Bs )ds over all stopping times with E 0 c(Bs )ds < where S = (S t ) t#0 is the supremum process associated with real valued Brownian motion B, # is non-decreasing and c is continuous. From work of Peskir [13] we deduce that this problem has a unique solution if and only if the differential equation admits a maximal solution g# (s) such that g# (s) s for all s 0. The stopping time which yields the highest payoff can be written as ## = g# (S t )}. The problem is actually solved in a general case of a real-valued, time homogeneous diffusion X = (X t : t 0) instead of B. We then proceed to solve the problem for more general functions # and c. Explicit formulae for payoff are given. We apply

Let Z = (Zt)t≥0 be a regular diffusion process started at 0, let ℓ be an independent random variable with a strictly increasing and continuous distribution function F, and let τℓ = inf { t ≥ 0 | Zt = ℓ} be the first entry time of Z at the level ℓ. We show that the quickest detection problem inf P(τ <τℓ) + c E(τ −τℓ)

We investigate in detail works of Peskir [13] and Meilijson [9] and develop a link between them. [ We consider the following optimal stopping problem: maximize Vτ = E φ(Sτ) − ∫ τ 0 c(Bs)ds over all stopping times with E ∫ τ 0 c(Bs)ds < ∞, where S = (St)t≥0 is the supremum process associated with real valued Brownian motion B, φ ∈ C 1 is non-decreasing and c is continuous. From work of Peskir [13] we deduce that this problem has a unique solution if and only if the differential equation g ′ (s) = φ ′ (s) 2c(g(s))(s − g(s)) admits a maximal solution g∗(s) such that g∗(s) ≤ s for all s ≥ 0. The stopping time which yields the highest payoff can be written as τ ∗ = inf{t ≥ 0: Bt ≤ g∗(St)}. The problem is actually solved in a general case of a real-valued, time homogeneous diffusion X = (Xt: t ≥ 0) instead of B. We then proceed to solve the problem for more general functions φ and c. Explicit formulae for payoff are given. We apply the results to solve the so-called optimal Skorokhod embedding problem. We give also a sample of applications to various inequalities dealing with terminal value and supremum of a process.

develop a link between [ them. We consider the following optimal stopping problem: maximize Vτ = E φ(Sτ) − ∫ τ 0 c(Bs)ds over all stopping times with E ∫ τ c(Bs)ds < 0 ∞, where S = (St)t≥0 is the maximum process associated with real valued Brownian motion B, φ ∈ C 1 is non-decreasing and c ≥ 0 is continuous. From work of Peskir [15] we deduce that this problem has a unique solution if and only if the differential equation g ′ (s) = φ ′ (s) 2c(g(s))(s − g(s)) admits a maximal solution g∗(s) such that g∗(s) < s for all s ≥ 0. The stopping time which yields the highest payoff can be written as τ ∗ = inf{t ≥ 0: Bt ≤ g∗(St)}. The problem is actually solved in a general case of a real-valued, time homogeneous diffusion X = (Xt: t ≥ 0) instead of B. We then proceed to solve the problem for more general functions φ and c. Explicit formulae for payoff are given. We apply the results to solve the so-called optimal Skorokhod embedding problem. We give also a sample of applications to various inequalities dealing with terminal value and maximum of a process.