A Variational Inequality Related To Perpetual Convertible Bonds

In this paper, we consider the following probl em:The problem （1）-（2） is related to a perpetual convertible bond pricing problem.Here, x is the whole value of the company’s assets. f（x） is the total value of thecompany’s convertible bonds. x f（x） is the company’s stock price. δ is the stockdividend rate. σ is the volatility of the company’s total assets. r is the risk-free interestrate, where r> δ. c is the bond dividend rate. γ is the percentage of shares of thecompany total assets after all convertible bonds having been converted to the stocks,where0<γ <1. a is the upper bound of the company’s total assets.（reference to [1]）.Though the problem （1）-（2） is a one-dimensional problem, it has considerabledifculty by theoretical studies. Since operator N is a second-order nonlinear operatorand is degenerate at x=0, it has no explicit solution. In the reference[1], the authorproved the existence of the solution of the problem （1）-（2） by use of stochastic analysis,but he didn’t prove the uniqueness of the solution and didn’t estimate the bounds ofthe free boundary.Having used punishment method in the free boundary problem theories, we provedthe existence and uniqueness of the solution(C（[0, a]）∩W^2,∞（（η,a））, η∈（0, a）) ofthe problem （1）-（2） through appropriate approximation arguments and the delicateestimations. Moreover, we have got the upper and lower bounds of the free boundary.The method in this paper can be extended to study with convertible bonds pricingproblem which has fnite maturity（time-related）.