On three methods for analytic Laplace inversion in the framework of Brownian motion and their excursions (Q2850031)

The framework, in which the paper (under review) analyzes the problem undertaken, originates with Brownian motion and its properties. It describes three mutually connected issues. The first pertains to an analytic approach, which involves isolation of a function out of a convolution for a known Laplace transform of the convolution and of the complementary factor of the function present. The second that is considered involves the means for the valuation and creates a fence of a class of barrier options (the Parisian barrier options); whereas the third issue is to study the explicit structure of minimal-length excursion of the stochastic process. The functions, denoted by a certain symbol, studied here are defined on the positive real and depend on two complex parameters there in the symbol. The origin of the Laplace transform equation is due to Azémas martingales (see [\textit{J. Azéma} and \textit{M. Yor}, Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 88--130 (1989; Zbl 0743.60045), ibid. XXVI, Lect. Notes Math. 1526, 248--306 (1992; Zbl 0765.60038)]) by virtue of Brownian excursions and Parisian options. In Section 3, the endeavour is to connect with the second issue (mentioned above). Complete proofs of the results and applications is given through Sections 4--6.