The Azéma-Yor Embedding in Brownian Motion with Drift

Abstract

Let B = (B t ) t0 be standard Brownian motion started at zero, let > 0 be given and fixed, and let be a probability measure on IR having a strictly positive density F 0 . Then there exists a stopping time 3 of B such that In addition, it is proved that 3 is pointwise the smallest possible stopping time satisfying (B 3 + 3 ) which generates stochastically the largest possible maximum of the process (B t +t) t0 up to the time of stopping. This minimax property characterizes 3 uniquely. The result recovers the Azéma-Yor solution of the Skorokhod embedding problem [1] by passing to the limit when # 0 . The condition on the existence of a strictly positive density is imposed for simplicity, and more general cases can be treated similarly. The line of arguments used in the proof can be extended to treat the case of more general nonrecurrent diffusions.

The Skorokhod embedding problem and its offspring

“…This optimality property characterizes the Azéma-Yor solution. Furthermore, it characterizes some of its generalizations -for example in the case of Brownian motion with drift, see Peskir [93]. It also implies an optimality property of the reversed solution presented in Section 5.3.…”

“…It was done by Hall back in 1969 as we stressed in Section 3.4, however his work doesn't seem to be well known. Consequently, the subject was treated again in a similar manner by Grandits and Falkner [46] and Peskir [93]. They show that for a probability measure µ there exists a stopping time T such that B…”

An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift

“…Especially, this class of processes includes Brownian motions with non-linear drift. The SEP for Brownian motion with linear drift was first solved in the technical report [Hal68] and 30 years later again in [GF00] and [Pes00]. Techniques developed in these works can be extended to timehomogeneous diffusions, as done in [PP01], and can be seen as generalization of the Azéma-Yor solution.…”

Processes That Can Be Embedded in a Geometric Brownian Motion

“…embedding of a single random variable in a geometric Brownian motion). Different solutions to this problem (in fact, to the one for a Brownian motion with drift) were proposed in [20], [19], [33], [3], [5], and [4]. In Section 2.1, we suggest an alternative construction, which is, in our view, of interest in its own right.…”