Integrated Brownian Motion, Conditioned to be Positive

Abstract

We study the two-dimensional process of integrated Brownian motion and Brownian motion, where integrated Brownian motion is conditioned to be positive. The transition density of this process is derived from the asymptotic behavior of hitting times of the unconditioned process. Explicit expressions for the transition density in terms of confluent hypergeometric functions are derived, and it is shown how our results on the hitting time distributions imply previous results of Isozaki-Watanabe and Goldman. The conditioned process is characterized by a system of stochastic differential equations (SDEs) for which we prove an existence and unicity result. Some sample path properties are derived from the SDEs and it is shown that $t \to t^{9/10}$ is a “critical curve” for the conditioned process in the sense that the expected time that the integral part of the conditioned process spends below any curve $t \to t^{\alpha}$ is finite for $\alpha < 9 /10$ and infinite for $\alpha \geq 9/10$.