An ODE approach for the expected discounted penalty at ruin in jump-diffusion model
The model under consideration is a jump-diffusion model, where the surplus process takes the form \(X_t=x+ct+\sigma W_t-Z_t\). Here, \(x\) is the initial capital, \(c>0\) is the rate of premiums income, \(\sigma>0\), \(W_t\) is a standard Brownian motion, and \(Z_t\) is a compound Poisson process (independent from \(W_t\)) which represents the aggregate claims on \([0,t]\). The authors study the properties of the expected discount penalty \(\Phi(x)=E_x[\exp(-rt)g(X_\tau)]\), where \(g\) is a penalty function, \(\tau\) is a ruin time and \(X_\tau\) is the negative surplus when ruin occurs, \(r>0\) is the risk-free rate. Under the assumption that the jump distribution of \(X\) is a two-sided phase-type distribution, the authors derive the integro-differential equation for \(\Phi\), then, by a Fourier transform argument, transform the integro-differential equation into a homogeneous ordinary differential equation and obtain the general form for the expected discount penalty \(\Phi\). In the case, if only downward jumps for \(X_t\) are allowed, they get an explicit formula in terms of the penalty function and jump distribution. On the other hand, if the downward jump distribution is a mixture of exponential distributions ( and upward jumps are determined by a general Lévy measure), closed-form solutions for \(\Phi\) are also obtained. To show possible finance and insurance applications of the results, a number of examples are discussed. In particular, in the setup of Leland's model with jumps, the optimal endogenous default and firm values are determined in closed-form formulas.
scientific article; zbMATH DE number 5520641
scientific article