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Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The probabilities of the type P[sup0≦t ≦ TW(t) − f(t) ≧ 0] have been extensively studied when f(t) is a deterministic function. This paper discusses the probabilities of the type P{sup0≦t ≦ TW(t) − [f(t) + X(t)] ≧ 0} when X(t) is a stochastic process. By taking compound Poisson processes as X(t), the paper gives procedures for finding such probabilities.

Let be a process with stationary, independent increments and no negative jumps. Let be this same process modified by a reflecting barrier at zero (a storage process). Assuming that – and denote by ψ(s) the exponent function of X. A simple formula is derived for the Laplace transform of as a function of W(0). Using the fact that the distribution of M is the unique stationary distribution of the Markov process W, this yields an elementary proof that the Laplace transform of M is µs/ψ(s). If it follows that These surprisingly simple formulas were originally obtained by Zolotarev using analytical methods.

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Stochastic barriers for the Wiener process

The supremum distribution of a Lévy process with no negative jumps

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2 results

Stochastic barriers for the Wiener process

The supremum distribution of a Lévy process with no negative jumps