On moments of the integrated exponential Brownian motion

Francesco Caravelli; Toufik Mansour; Lorenzo Sindoni; Simone Severini

doi:10.1140/epjp/i2016-16245-9

EPJ

On moments of the integrated exponential Brownian motion

Francesco Caravelli¹^,2^,3^*, Toufik Mansour⁴, Lorenzo Sindoni¹ and Simone Severini³

¹ Invenia Labs, 27 Parkside Place, CB1 1HQ, Cambridge, UK
² London Institute for Mathematical Sciences, 35a South Street, W1K 2XF, London, UK
³ University College London, Gower Street, WC1E 6BT, London, UK
⁴ University of Haifa, 31905, Haifa, Israel

Received: 28 May 2016
Accepted: 23 June 2016
Published online: 27 July 2016

Abstract

We present new exact expressions for a class of moments of the geometric Brownian motion in terms of determinants, obtained using a recurrence relation and combinatorial arguments for the case of a Itô's Wiener process. We then apply the obtained exact formulas to computing averages of the solution of the logistic stochastic differential equation via a series expansion, and compare the results to the solution obtained via Monte Carlo.