Infinite sums of the geometric Brownian motion and generalizations
Dan Pirjol: JP Morgan
Location: Hill 705
Date & time: Tuesday, 07 February 2017 at 11:45AM - 11:45AM
The infinite sum of a geometric Brownian motion (gBM) sampled on a sequence of uniformly spaced times appears in problems of actuarial science and theoretical probability. For example this appears when considering the present value of a perpetuity: a fixed recurring payment made in perpetuity from an initial deposit of stock, assumed to follow a geometric Brownian motion. The talk studies the distributional properties of the infinite sum of the gBM. This can be characterized as the stationary distribution of a linear stochastic recursion. Tail asymptotics are derived, and the distribution is found numerically by solving an integral equation. Similar results are obtained for the sum of the gBM with a geometrically distributed stopping time. The results can be generalized further by replacing the gBM with an exponential Levy process.