On the problem of stochastic integral representations of functionals of the Brownian motion. I. (English. Russian original) Zbl 1057.60057

Shiryaev, A. N.; Yor, M.

Theory Probab. Appl. 48, No. 2, 304-313 (2003); translation from Teor. Veroyatn. Primen. 48, No. 2, 375-385 (2003).

The stochastic integral representation is well known, i.e. every quadratic integrable functional of the Brownian motion can be represented as the sum of its mean value and a stochastic integral w.r.t. the Brownian motion. Nevertheless, finding explicit representations is uneasy and not much studied. This lucid paper considers explicit representations of functionals of the “maximal” type: \(S_T\) and \(S_{T_{-a}}\), where \(S_t = \max_{s \leq t} B_t\), \(T\) is constant and \(T_{-a}\) is the first hitting time of \(a\). Among other results the representation of \(I(T_a < T)\) is also given.
It is a pleasure to read the paper, which only relies on the Itô formula and elementary calculations. More technical tools like Malliavin calculus and the Clark-Ocone formula are not used. The paper is the first in a series of two. The second paper considers the stochastic integral representation of \(S_{g_T}\) and \(S_{\theta_T}\). Here \(g_T\) is the time of the last zero of the Brownian motion on \([0,T]\), and \(\theta_T\) is the time when the Brownian motion achieves its maximal value on \([0,T]\). Both \(g_T\) and \(\theta_T\) are the non-Markov times.

Reviewer: Bo Markussen (Kopenhagen)

MSC:

60H05	Stochastic integrals
60J65	Brownian motion

Keywords:

Brownian motion; stochastic integral representation; Itô formula; Markov time