Figure 5

The free boundary when \(\pmb{c(1-\gamma)-rP\leq0}\) with different volatility. Plot of the optimal bankruptcy boundary \(h(\tau)\) as the function of time τ when \(c(1-\gamma)-rP\leq0\). The parameter values used in the calculations are \(T=1\), \(N=2{,}000\), \(r=0.3\), \(\beta=0.02\), \(\delta=0.01\), \(c=0.04\), \(\gamma=0.2\), \(P=2\); \(h1(\tau)\) and \(h2(\tau)\) are the free boundaries when \(\sigma_{1}=0.3\) and \(\sigma_{2}=0.7\), respectively. The numerical result (see Figure 5) shows that the optimal bankruptcy boundary is decreasing not only with τ, which coincides with Theorem 4.1, but also with volatility σ.