The integral option in a model with jumps
Let \(J_t=Y_1+Y_2+\cdots+Y_{N_t}\) \((t\geq 0)\) be a jump process where \((N_t)\) is a Poisson process with intensity \(\lambda>0\), and \(Y_1,Y_2,\dots\) are independent random variables being exponentially distributed (parameter 1). Let \((Y_n)\) and \((N_t)\) be independent of each other. Define the process \((S_t)\) by \[ S_t=s\exp (((r-\delta-\lambda\theta)/(1-\theta))t+\theta J_t),\;t\geq 0, \] where \(0\leq\delta <r\) and \(\theta<1\) \((\theta\neq 0)\) are constants. The process \((S_t)\) may be interpreted as the risk process of an insurance company where \(r\) and \(\delta\) are, respectively, the riskless interest rate and the dividend rate payed by the company. The main purpose of the present paper is to derive a solution of the optimal stopping problem \[ V_*=\sup_\tau E\left[ e^{-r\tau}\left(\int_0^\tau S_udu+x\right)\right]\tag{*} \] (the supremum being taken over all finite stopping times \(\tau\) w.r.t. the filtration generated by \((S_t))\). Here, \(x\geq 0\) is a given constant. The value in (*) coincides with an arbitrage-free price of a perpetual integral option. Using arguments in the paper \textit{D. O. Kramkov} and \textit{E. Mordecki} [Theory Probab. Appl. 39, No. 1, 162--172 (1994); translation from Teor. Veroyatn. Primen. 39, No. 1, 201--211 (1994; Zbl 0836.90012)], the authors solve (*) by reduction to an optimal stopping problem for Shiryaev's diffusion process; see, e.g., \textit{A. M. Shiryaev} [Optimal stopping rules. Applications of Mathematics. 8. New York etc.: Springer (1978; Zbl 0391.60002), Chapter IV, Section 4].
scientific article; zbMATH DE number 5366042
scientific article