In this paper, we consider the following probl em:The problem ï¼ˆ1ï¼‰-ï¼ˆ2ï¼‰ is related to a perpetual convertible bond pricing problem.Here, x is the whole value of the companyâ€™s assets. fï¼ˆxï¼‰ is the total value of thecompanyâ€™s convertible bonds. x fï¼ˆxï¼‰ is the companyâ€™s stock price. Î´ is the stockdividend rate. Ïƒ is the volatility of the companyâ€™s total assets. r is the risk-free interestrate, where r> Î´. c is the bond dividend rate. Î³ is the percentage of shares of thecompany total assets after all convertible bonds having been converted to the stocks,where0<Î³ <1. a is the upper bound of the companyâ€™s total assets.ï¼ˆreference to [1]ï¼‰.Though the problem ï¼ˆ1ï¼‰-ï¼ˆ2ï¼‰ 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 ï¼ˆ1ï¼‰-ï¼ˆ2ï¼‰ by use of stochastic analysis,but he didnâ€™t prove the uniqueness of the solution and didnâ€™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(Cï¼ˆ[0, a]ï¼‰âˆ©W^{2,âˆž}ï¼ˆï¼ˆÎ·,aï¼‰ï¼‰, Î·âˆˆï¼ˆ0, aï¼‰) ofthe problem ï¼ˆ1ï¼‰-ï¼ˆ2ï¼‰ 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 maturityï¼ˆtime-relatedï¼‰. |