Existence and uniqueness of solutions to the inverse boundary crossing problem for diffusions

We study the inverse boundary crossing problem for diffusions. Given a diffusion process $X_t$, and a survival distribution $p$ on $[0,\infty)$, we demonstrate that there exists a boundary $b(t)$ such that $p(t)=\mathbb{P}[\tau >t]$, where $\tau$ is the first hitting time of $X_t$ to the boundary $b(t)$. The approach taken is analytic, based on solving a parabolic variational inequality to find $b$. Existence and uniqueness of the solution to this variational inequality were proven in earlier work. In this paper, we demonstrate that the resulting boundary $b$ does indeed have $p$ as its boundary crossing distribution. Since little is known regarding the regularity of $b$ arising from the variational inequality, this requires a detailed study of the problem of computing the boundary crossing distribution of $X_t$ to a rough boundary. Results regarding the formulation of this problem in terms of weak solutions to the corresponding Kolmogorov forward equation are presented.Comment: Published in at http://dx.doi.org/10.1214/10-AAP714 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Analysis of an Optimal Stopping Problem Arising from Hedge Fund Investing

The final publication is available at Elsevier via https://doi.org/10.1016/j.jmaa.2019.123559. © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We analyze the optimal withdrawal time for an investor in a hedge fund with a first-loss or shared-loss fee structure, given as the solution of an optimal stopping problem on the fund's assets with a piecewise linear payoff function. Assuming that the underlying follows a geometric Brownian motion, we present a complete solution of the problem in the infinite horizon case, showing that the continuation region is a finite interval, and that the smooth-fit condition may fail to hold at one of the endpoints. In the finite horizon case, we show the existence of a pair of optimal exercise boundaries and analyze their properties, including smoothness and convexity.NSERC, RGPIN-2017-04220

Toxic Assets and Financial Crises

This talk will begin with a brief introduction to risk-neutral pricing of financial derivatives—such as bonds, options, and credit default swaps—using the Black-Scholes-Merton math model. Using these ideas, Chadam will then explore areas of concern in the equity and bond markets. In particular, he will discuss insider trading in the recent sale of H. J. Heinz, and examine the topics of exchange-traded funds (ETFs) and emerging instabilities in bond markets. If time permits, Chadam will discuss how mathematical methods might be used to understand contagion in networks of financial institutions, the functioning of dark markets and high-frequency trading, and commodity markets and sustainability