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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
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