The integral option in a model with jumps
type: Item
instance of: scholarly article
title: The integral option in a model with jumps
zbMATH Open document ID: 1221.60062
published in: Statistics \& Probability Letters
publication date: 2008-11-14T00:00:00Z
review text: 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].
reviewed by: Klaus Schürger
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 34K10
Mathematics Subject Classification ID: 91B70
Mathematics Subject Classification ID: 91G20
Mathematics Subject Classification ID: 60J75
zbMATH DE Number: 5366042
author: Pavel Gapeev
MaRDI profile type: Publication
full work available at URL: j.spl.2008.02.028
OpenAlex ID: W2053483706
cites work: Q3331506
cites work: Some remarks on first passage of Lévy processes, the American put and pasting principles
cites work: Russian and American put options under exponential phase-type Lévy models.
cites work: Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options
cites work: Q5820721
cites work: Q5653410
cites work: Perpetual convertible bonds in jump-diffusion models
cites work: An optimal stopping problem in a diffusion-type model with delay
cites work: Q4856610
cites work: Pricing derivatives of American and game type in incomplete markets
cites work: A Jump-Diffusion Model for Option Pricing
cites work: Q4845602
cites work: Optimal stopping for a diffusion with jumps
cites work: Optimal stopping and perpetual options for Lévy processes
cites work: Sequential testing problems for Poisson processes.
cites work: Q4550924
cites work: Q3378055
cites work: Q4226355
cites work: Q4845599
cites work: Q4172681
cites work: Q4220653
DOI: 10.1016/J.SPL.2008.02.028
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description: scientific article; zbMATH DE number 5366042