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First passage times of a jump diffusion process. (English) Zbl 1037.60073
The jump diffusion process $X_t=\sigma W_t+\mu t+\sum_{i=1}^{N_t} Y_i, \quad X_0=0,$ is considered where $$W$$ is a standard Brownian motion, $$N$$ is a Poisson process with rate $$\lambda$$, $$\sigma$$ and $$\mu$$ are positive constants. The i.i.d. r.v.s $$(Y_i)_{i\geq 1}$$ have a double exponential distribution given by the density $f_Y(y)=p\cdot\eta_1 e^{-\eta_1 y}\mathbf{1}_{\{y\geq 0\}}+ q\cdot\eta_2 e^{-| \eta_2| y}\mathbf{1}_{\{y< 0\}},$ $$p,q\geq 0$$, $$p+q=1$$, $$\eta_1,\eta_2>0$$. The authors derive the closed-form formulae for the Laplace transform of the first passage time $$\tau_b=\inf\{t\geq 0: X_t\geq b\}$$, $$b>0$$, as well as for $$\mathbf{E}[e^{-\alpha \tau_b}\mathbf{1}_{\{X_{\tau_b}-b>y\}}]$$ and $$\mathbf{E}[e^{-\alpha \tau_b}\mathbf{1}_{\{X_{\tau_b}-b=y\}}]$$, $$y\geq 0$$. Connections with renewal-type equations are discussed. The Laplace transform of the joint law of $$X_t$$ and $$\max_{0\leq s\leq t}X_s$$ is obtained in terms of special functions. The Gaver-Stehfest algorithm for the numerical inversion of Laplace transforms is tested.

##### MSC:
 60J75 Jump processes (MSC2010) 44A10 Laplace transform 60J27 Continuous-time Markov processes on discrete state spaces 60G51 Processes with independent increments; Lévy processes
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##### References:
 [1] Abate, J. and Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10, 5–88. · Zbl 0749.60013 · doi:10.1007/BF01158520 [2] Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC. · Zbl 0543.33001 [3] Asmussen, S., Glynn, P. and Pitman, J. (1995). Discretization error in simulation of one-dimensional reflecting Brownian motion. Ann. Appl. Prob. 5, 875–896. · Zbl 0853.65147 · doi:10.1214/aoap/1177004597 [4] Bateman, H. (1953). Higher Transcendental Functions , Vol. 2. McGraw-Hill, New York. · Zbl 0051.34703 [5] Bateman, H. (1954). Tables of Integral Transforms , Vol. 1. McGraw-Hill, New York. · Zbl 0055.27503 [6] Bertoin, J. (1996). Lévy Processes . Cambridge University Press. · Zbl 0861.60003 [7] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766. · Zbl 0322.60068 · doi:10.2307/1426397 [8] Boyarchenko, S. and Levendorskiĭ, S. (2002). Barrier options and touch-and-out options under regular Lévy processes of exponential type. Ann. Appl. Prob. 12, 1261–1298. · Zbl 1015.60036 · doi:10.1214/aoap/1037125863 [9] Brémaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York. [10] Duffie, D. (1995). Dynamic Asset Pricing Theory , 2nd edn. Princeton University Press. · Zbl 1140.91041 [11] Gerber, H. and Landry, B. (1998). On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurance Math. Econom. 22, 263–276. · Zbl 0924.60075 · doi:10.1016/S0167-6687(98)00014-6 [12] Glasserman, P. and Kou, S. G. (2003). The term structure of simple forward rates with jump risk. To appear in Math. Finance . · Zbl 1087.91024 · doi:10.1111/1467-9965.00021 [13] Hull, J. C. (1999). Options, Futures, and Other Derivative Securities , 4th edn. Prentice-Hall, Englewood Cliffs, NJ. [14] Ikeda, N. and Watanabe, S. (1962). On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Kyoto Univ. 2, 79–95. · Zbl 0118.13401 [15] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin. · Zbl 0635.60021 · doi:10.1016/0304-4149(87)90050-0 [16] Karatzas, I. and Shreve, S. (1991). Brownian Motion and Stochastic Calculus. Springer, New York. · Zbl 0734.60060 [17] Karlin, S. and Taylor, H. (1975). A First Course in Stochastic Processes , 2nd edn. Academic Press, New York. · Zbl 0315.60016 [18] Kou, S. G. (2002). A jump diffusion model for option pricing. Manag. Sci. 48, 1086–1101. · Zbl 1216.91039 · doi:10.1287/mnsc.48.8.1086.166 [19] Kou, S. G. and Wang, H. (2001). Option pricing under a double exponential jump diffusion model. Preprint, Columbia University and Brown University. [20] Merton, R. C. (1976). Option pricing when the underlying stock returns are discontinuous. J. Financial Econom. 3, 115–144. · Zbl 1131.91344 · doi:10.1016/0304-405X(76)90022-2 [21] Pecherskii, E. A. and Rogozin, B. A. (1969). On joint distributions of random variables associated with fluctuations of a process with independent increment. Theory Prob. Appl. 15, 410–423. [22] Pitman, J. W. (1981). Lévy system and path decompositions. In Seminar on Stochastic Processes (Evanston, IL, 1981), Birkhäuser, Boston, MA, pp. 79–110. · Zbl 0518.60077 [23] Protter, P. (1990). Stochastic Integration and Differential Equations. A New Approach (Appl. Math. 21 ). Springer, Berlin. · Zbl 0694.60047 [24] Rogers, L. C. G. (2000). Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Prob. 37, 1173–1180. · Zbl 0981.60048 · doi:10.1239/jap/1014843099 [25] Rogozin, B. A. (1966). On the distribution of functionals related to boundary problems for processes with independent increments. Theory Prob. Appl. 11, 580–591. · Zbl 0178.52701 · doi:10.1137/1111062 [26] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. · Zbl 0973.60001 [27] Siegmund, D. (1985). Sequential Analysis. Springer, New York. · Zbl 0573.62071 [28] Weil, M. (1971). Conditionnement par rapport au passé strict. In Séminaire de Probabilités V, Université de Strasbourg, année universitaire 1969–1970 (Lecture Notes Math. 191 ). Springer, Berlin, pp. 362–372. [29] Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis (CBMS-NSF Regional Conf. Ser. Appl. Math. 39 ). SIAM, Philadelphia, PA. · Zbl 0487.62062
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