×

zbMATH — the first resource for mathematics

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

MSC:
60H05 Stochastic integrals
60J65 Brownian motion
PDF BibTeX XML Cite
Full Text: DOI