\documentclass{abstract}
\begin{document}

\title{A game-theoretical model of debt and bankruptcy}
\author{Khai T. Nguyen}
\institute{Department of Mathematics, Penn State University}
\email{ktn2@psu.edu}
\coauthor{Alberto Bressan} % comment out if not needed

\mymaketitle

\section{Abstract}
We study a game-theoretical model of debt and bankruptcy where the average interest rate $\alpha$ payed on debt for the borrower, depending on the bankruptcy risk $\rho$, is charged by a pool of risk-neutral lenders. It is assume that lenders will lose their investment if bankruptcy occurs. We consider the Nash equilibrium feedback solution for a simple model of the optimal control problem for the borrower:
\[
\mathrm{Minimize}\qquad E\Big[\int_0^{T_b}e^{-rt}L(u(t))dt+B\cdot e^{-rT_b}\Big]
\] 
subject to the evolution equation of the total debt $x$
\[
\dot{x}(t)~=~\alpha(x(t))\cdot x(t)-u(t)
\]
where the payment rate $u$ is the control variable, $L(u)$ is the cost to the borrower for implementing the control $u$, $B$ is the bankruptcy cost to the borrower and the random variable $T_b$ denotes the bankruptcy time. 
\end{document}
