On the drawdowns and drawups in diffusion-type models with running maxima and minima

Abstract

This document is the author's final accepted version of the journal article. There may be differences between this version and the published version. You are advised to consult the publisher's version if you wish to cite from it.On the drawdowns and drawups in diffusion-type models with running maxima and minima Pavel V. Gapeev * Neofytos Rodosthenous † To appear in Journal of Mathematical Analysis and ApplicationsWe obtain closed-form expressions for the values of joint Laplace transforms of the running maximum and minimum of a diffusion-type process stopped at the first time at which the associated drawdown and drawup processes hit constant levels. It is assumed that the coefficients of the diffusion-type process are regular functions of the running values of the process itself, its maximum and minimum, as well as its maximum drawdown and maximum drawup processes. The proof is based on the solution to the equivalent boundary-value problems and application of the normal-reflection conditions for the value functions at the edges of the state space of the resulting five-dimensional Markov process. We show that the joint Laplace transforms represent linear combinations of solutions to the systems of first-order partial differential equations arising from the application of the normal-reflection conditions.

Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs

“…(ii) Let us now assume that λ i > λ 1−i = 0 holds, for any i = 0, 1 fixed. In this case, by using straightforward calculations from part (i) of Subsection 3.2, it can be shown that the candidate solution of the system in (5.3)-(5.9) takes the form: 18) for k = 1, 2, where the value b * 1−i is determined by the equation: 19) for i = 0, 1 (see [8]). Substituting the expression for W 1−i (x, s; b * 1−i ) from (5.17) into the equation of (5.3) for W i (x, s) and applying the conditions of (5.4)-(5.6), we obtain that the candidate value function admits the representation:…”

Optimal stopping problems for maxima and minima in models with asymmetric information

“…are continuous uniformly integrable martingales under the probability measure P, when α < 0 and α > 0, respectively. Note that the processes S and Q may change their values only at the times when X t = S t and X t = Q t , for t ≥ 0, respectively, and such times accumulated over the infinite horizon form the sets of the Lebesgue measure zero, so that the indicators in the expressions of ( 15)-( 16) and ( 17)- (18)…”

“…Discounted optimal stopping problems for the running maxima and minima of the initial continuous (diffusion-type) processes were initiated by Shepp and Shiryaev [44] and further developed by Pedersen [36], Guo and Shepp [24], Gapeev [14], Guo and Zervos [25], Peskir [39,40], Glover, Hulley, and Peskir [22], Gapeev and Rodosthenous [17][18][19], Rodosthenous and Zervos [43], Gapeev, Kort, and Lavrutich [20], and Gapeev and Al Motairi [15] among others. It was shown, by means of the maximality principle for solutions of optimal stopping stopping problems established by Peskir [37], which is equivalent to the superharmonic characterization of the value functions, that the optimal stopping boundaries are given by the appropriate extremal solutions of certain (systems of) first-order nonlinear ordinary differential equations.…”

A variation of the Azéma martingale and drawdown options

“…The distributional properties of and are also studied in Douady, Shiryaev, and Yor () and Magdon‐Ismail, Atiya, Pratap, and Abu‐Mostafa (). Furthermore, studies that involve both the drawdown and drawup processes have been done by Popisil, Vecer, and Hadjiliadis (), Zhang and Hadjiliadis () and Gapeev and Rodosthenous (). This stopping time is commonly referred to as drawdown time in the literature but we note the distinction between this and .…”