On future drawdowns of Levy processes

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:127

Article

Baurdoux, E. J.¹  Palmowski, Zbigniew²  Pistorius, M. R.³ 

[1] London Sch Econ, Dept Stat, Houghton St, London WC2A 2AE, England

[2] Wroclaw Univ Sci & Technol, Fac Pure & Appl Math, Wyb Wyspianskiego 27, PL-50370 Wroclaw, Poland

[3] Imperial Coll London, Dept Math, South Kensington Campus, London SW7 2AZ, England

关键词: Reflected process;    Levy process;    Drawdown process;    Cramer-asymptotics;    Heavy-tailed distributions;    Queueing;    Workload process;   

DOI  :  10.1016/j.spa.2016.12.008

来源: Elsevier

【 摘 要 】

For a given Levy process X = (X-t)(t is an element of R+) and for fixed s is an element of R+ U {infinity} and t is an element of R+ we analyse the future drawdown extremes that are defined as follows: (D) over bar (*)(t, s) = sup(0 <= u <= t) inf(u <= w < t+s) (X-w, X-u), D-t, s(*) = inf(0 <= u <= t) inf(u <= w(*)(t,s) and D-t,s(*) are of interest in various areas of application, including financial mathematics and queueing theory. In the case that X has a strictly positive mean, we find the exact asymptotic decay as x -> infinity of the tail probabilities P((D) over bar (*)(t) < x) and P(<(D)over bar>(*)(t) < x) of <(D)over bar>(*)(t) = lim(s ->infinity)(D) over bar (*)(t,s) and (D) over bar (*)(t) = lim(s ->infinity)(D) over bar (*)(t,s) both when the jumps satisfy the Cramer assumption and in a heavy-tailed case. Furthermore, in the case that the jumps of the Levy process X are of single sign and X is not subordinator, we identify the one-dimensional distributions in terms of the scale function of X. By way of example, we derive explicit results for the Black Scholes Samuelson model. (C) 2016 Elsevier B.V. All rights reserved.

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