## Some remarks on first passage of Lévy processes, the American put and pasting principles

Venue: | Annals of Appl. Probability |

Citations: | 31 - 3 self |

### BibTeX

@ARTICLE{Alili_someremarks,

author = {L. Alili and A. E. Kyprianou},

title = {Some remarks on first passage of Lévy processes, the American put and pasting principles},

journal = {Annals of Appl. Probability},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Lévy process and the solution of Gerber and Shiu [Astin

### Citations

683 |
Lévy Processes and Infinitely Divisible Distributions
- Sato
- 1999
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Citation Context ...nding subordinator, with the help of (2), we come to rest at the Pecherskii–Rogozin identity ∫ ∞ e 0 −qx E[e −ατ+ x −β(X τ + x ( −x) 1 ]dx = q − β . 1 − Ψ+ α(−q) Ψ + α(−β) for any q > 0. See [30] and =-=[34]-=- for a comparison with existing proofs. )6 L. ALILI AND A. E. KYPRIANOU 3.2. Spectrally one-sided processes. Suppose that X is spectrally negative, but not a negative subordinator, with Laplace expon... |

417 |
Lévy Processes
- Bertoin
- 1996
(Show Context)
Citation Context ... important tool in the study of the fluctuations of Lévy processes is the Wiener–Hopf factorization which we now review for convenience. For a more detailed account, the reader is referred to [22] or =-=[7]-=-. Assume that r > 0. We have that Xer and Xer − Xer are independent where Xt = sup 0≤s≤t Xt and er is an independent exponentially distributed random variable with parameter r. As a consequence, for θ... |

279 |
Financial modelling with jump processes
- Cont, Tankov
- 2004
(Show Context)
Citation Context ...rom the original in pagination and typographic detail. 12 L. ALILI AND A. E. KYPRIANOU of an American put option in an incomplete Black–Scholes-type markets driven by Lévy processes (see [13, 35] or =-=[18]-=-). For this reason, we refer to (1) as the American put optimal stopping problem. In a number of numerical simulations and theoretical calculations for specific choices of Lévy processes, various auth... |

98 |
Optimal stopping rules
- Shiryaev
- 1978
(Show Context)
Citation Context ... extent, the conditions (i)–(iv) may be seen as a stochastic analogue of a free boundary value problem which one often sees as a way of characterizing the solution to an optimal stopping problem; see =-=[36]-=-. In this case it is also possible to write down a free boundary value problem (cf. [12]), although one must be a little careful about the sense in which the associated integro-differential operator i... |

89 |
Ruin Probabilities
- Asmussen
- 2000
(Show Context)
Citation Context ... F(s) = ∫ ∞ 0 e−sxf(x)dx = a(sI − T) −1t for s > 0. The latter can be extended to the complex plane except at a finite number of poles (the eigenvalues of T). For full details, we refer the reader to =-=[3]-=-. The process X enjoys the representation Xt = X (+) t − N(t) ∑ Uj, t ≥ 0, j=1 where {X (+) t :t ≥ 0} is a spectrally positive Lévy process, {Nt :t ≥ 0} is a Poisson process with rate λ and {Uj :j ≥ 1... |

73 |
Lévy Processes in Finance: Pricing Financial Derivatives
- Schoutens
- 2003
(Show Context)
Citation Context ...nt differs from the original in pagination and typographic detail. 12 L. ALILI AND A. E. KYPRIANOU of an American put option in an incomplete Black–Scholes-type markets driven by Lévy processes (see =-=[13, 35]-=- or [18]). For this reason, we refer to (1) as the American put optimal stopping problem. In a number of numerical simulations and theoretical calculations for specific choices of Lévy processes, vari... |

56 |
Optimal Stopping and Free-Boundary Problems
- Peskir, Shiryaev
- 2006
(Show Context)
Citation Context ...also observed for the case of spectrally negative processes that there was no smooth pasting if and only if the process is of bounded variation. In related work Peskir and Shiryaev [32, 33] (see also =-=[31]-=-) and Gapeev [20] studied a number of optimal stopping problems for special classes of Markov process of bounded variation with jumps such that the inter-arrival times of the jumps are independent and... |

42 | Russian and American put options under exponential phase-type Lévy models. Stochastic Process
- Asmussen, Avram, et al.
- 2004
(Show Context)
Citation Context ...age problem of a Lévy process. See Gerber and Shiu [21], Chan [15, 16], Boyarchenko and Levendorskiǐ [10, 11, 12, 13, 14], Mordecki [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius =-=[4]-=-, Chesney and Jeanblanc [17], Hirsa and Madan [23], Matache, Nitsche and Schwab [26], Almendral and Oosterlee [2] and Almendral [1]. Notably, Mordecki [28] handles the case when X is a general Lévy pr... |

37 |
Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval
- Bertoin
- 1997
(Show Context)
Citation Context ...ansforms in (2) and (3), we have that ∫ e −βx P(Xeα ∈ dx) = Φ(α) Φ(α) + β and ∫ [0,∞) [0,∞) e −βx P(−X eα ∈ dx) = α β − Φ(α) Φ(α) ψ(β) − α . Therefore, we see the known distributional identities (cf. =-=[7, 9]-=-), (6) and (7) P(−X eα ∈ dx) = P(Xeα ∈ dx) = Φ(α)e −Φ(α)x dx α Φ(α) dW (α) (x) − αW (α) (x)dx, where W (α) :[0, ∞) → [0, ∞) is the scale function (cf. [9]) and is characterized on (0, ∞) by ∫ ∞ e 0 −λ... |

35 | Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options
- Avram, Kyprianou, et al.
- 2004
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Citation Context ...ncides with unbounded variation. Note that similar conclusions for spectrally negative Lévy processes were also drawn for another optimal stopping problem related to the pricing of Russian options in =-=[6]-=-. Finally, within the class of compound Poisson models, Mordecki [27, 28] derives explicit formulae for the case of drifting Brownian motion, plus (mixed) exponential jumps which always has smooth pas... |

32 | Non-Gaussian Merton-Black-Scholes theory - Boyarchenko, Levendorksĭi |

27 |
S.Z.: Perpetual American Options under Lévy Processes
- Boyarchenko, Levendorskiǐ
(Show Context)
Citation Context ...e −rτ ∗ +Xτ∗ ], thus, linking the American perpetual put optimal stopping problem to the first passage problem of a Lévy process. See Gerber and Shiu [21], Chan [15, 16], Boyarchenko and Levendorskiǐ =-=[10, 11, 12, 13, 14]-=-, Mordecki [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan [23], Matache, Nitsche and Schwab [26], Almendral and Oosterlee [2] and ... |

24 |
Pricing American Options Under Variance
- Hirsa, Madan
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Citation Context ...u [21], Chan [15, 16], Boyarchenko and Levendorskiǐ [10, 11, 12, 13, 14], Mordecki [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan =-=[23]-=-, Matache, Nitsche and Schwab [26], Almendral and Oosterlee [2] and Almendral [1]. Notably, Mordecki [28] handles the case when X is a general Lévy process. In addition, these authors have observed th... |

23 |
Martingale Approach to Pricing Perpetual American Options
- Gerber, Shiu
- 1994
(Show Context)
Citation Context ... = inf{t ≥ 0:Xt < x ∗ } −rτ ∗ v(x) = KEx[e ] − Ex[e −rτ ∗ +Xτ∗ ], thus, linking the American perpetual put optimal stopping problem to the first passage problem of a Lévy process. See Gerber and Shiu =-=[21]-=-, Chan [15, 16], Boyarchenko and Levendorskiǐ [10, 11, 12, 13, 14], Mordecki [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan [23], ... |

22 | Optimal stopping and perpetual options for Lévy processes
- Mordecki
- 2002
(Show Context)
Citation Context ...he American perpetual put optimal stopping problem to the first passage problem of a Lévy process. See Gerber and Shiu [21], Chan [15, 16], Boyarchenko and Levendorskiǐ [10, 11, 12, 13, 14], Mordecki =-=[27, 28]-=-, Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan [23], Matache, Nitsche and Schwab [26], Almendral and Oosterlee [2] and Almendral [1]. Nota... |

21 |
Fluctuation identities for Lévy processes and splitting at the maximum
- Greenwood, Pitman
- 1980
(Show Context)
Citation Context ...oots. An important tool in the study of the fluctuations of Lévy processes is the Wiener–Hopf factorization which we now review for convenience. For a more detailed account, the reader is referred to =-=[22]-=- or [7]. Assume that r > 0. We have that Xer and Xer − Xer are independent where Xt = sup 0≤s≤t Xt and er is an independent exponentially distributed random variable with parameter r. As a consequence... |

18 | Wavelet Galerkin pricing of American options on Lévy driven assets, working paper
- Matache, Nitsche, et al.
- 2003
(Show Context)
Citation Context ... and Levendorskiǐ [10, 11, 12, 13, 14], Mordecki [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan [23], Matache, Nitsche and Schwab =-=[26]-=-, Almendral and Oosterlee [2] and Almendral [1]. Notably, Mordecki [28] handles the case when X is a general Lévy process. In addition, these authors have observed that the function v is continuous, e... |

17 | Solving the Poisson Disorder Problem
- Peskir, Shiryaev
- 2002
(Show Context)
Citation Context ...14]. Chan [15, 16] also observed for the case of spectrally negative processes that there was no smooth pasting if and only if the process is of bounded variation. In related work Peskir and Shiryaev =-=[32, 33]-=- (see also [31]) and Gapeev [20] studied a number of optimal stopping problems for special classes of Markov process of bounded variation with jumps such that the inter-arrival times of the jumps are ... |

14 | On American options under the variance gamma process, Working paper
- Almendral, Oosterlee
- 2006
(Show Context)
Citation Context ... 13, 14], Mordecki [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan [23], Matache, Nitsche and Schwab [26], Almendral and Oosterlee =-=[2]-=- and Almendral [1]. Notably, Mordecki [28] handles the case when X is a general Lévy process. In addition, these authors have observed that the function v is continuous, equals K − e x for x ≤ x ∗ and... |

12 |
Optimal stopping for partial sums
- Darling, Ligett, et al.
- 1972
(Show Context)
Citation Context ...ayed in the solution to the optimal stopping problem (1). Essentially, this identity is not new, as it appears implicitly in a number of texts dating back to at least the seventies. See, for example, =-=[19]-=-, page 1368, where one sees the same identity for random walks embedded in the proof of another result. To a fixed level x ∈ R we associate the first strict passage time τ + x (resp. τ − x ) above (re... |

11 | Sequential testing problems for Poisson processes
- PESKIR, SHIRYAEV
- 1998
(Show Context)
Citation Context ...14]. Chan [15, 16] also observed for the case of spectrally negative processes that there was no smooth pasting if and only if the process is of bounded variation. In related work Peskir and Shiryaev =-=[32, 33]-=- (see also [31]) and Gapeev [20] studied a number of optimal stopping problems for special classes of Markov process of bounded variation with jumps such that the inter-arrival times of the jumps are ... |

10 |
Regularity of the Half-Line for Lévy Processes
- Bertoin
- 1997
(Show Context)
Citation Context ...iation, d = 0 and ∫ 0− −1 (iii) X has unbounded variation. |x|Π(dx) ∫ = ∞. |x| 0 Π(y, ∞)dy Case (ii) was recently added to the class of processes exhibiting regularity of 0 for the lower half line in =-=[8]-=- and, for the other cases, we refer to the discussion at the beginning of [7], Section VI.3.12 L. ALILI AND A. E. KYPRIANOU In [12] a sufficient condition for smooth pasting and a sufficient conditio... |

8 | On the valuation of constant barrier options under spectrally one-sided exponential Lévy models and Carr’s approximation for American puts - Avram, Chan, et al. |

8 | The distribution of the maximum of a Lévy process with positive jumps of phase-type
- Mordecki
- 2002
(Show Context)
Citation Context ...se Xeα = Xeα, identity (4) simply reads E[e −ατx−βX τ + ∫ (8) x ] = (α + φ(β)) for all α,β,x ≥ 0. e [x,∞) −βz U (α) (dz) 3.4. A general phase-type process. Here we borrow an example which appeared in =-=[29]-=- and then with a different proof in [4]. The Lévy process X is taken to be the independent sum of a spectrally positive process with a compound Poisson process having negative phase-type jumps. Recall... |

7 |
Some applications of Lévy processes in insurance and finance
- Chan
- 2004
(Show Context)
Citation Context ... so that [ −rτ v(x) = KEx e ∗] [ −rτ − Ex e ∗ +Xτ∗ ] , thus, linking the American perpetual put optimal stopping problem to the first passage problem of a Lévy process. See Gerber and Shiu [21], Chan =-=[15, 16]-=-, Boyarchenko and Levendorskiǐ [10–14], Mordecki [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan [23], Matache, Nitsche and Schwab ... |

6 | Numerical valuation of American options under the CGMY process. In Exotic option pricing and advanced Lévy models
- Almendral
- 2005
(Show Context)
Citation Context ... [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan [23], Matache, Nitsche and Schwab [26], Almendral and Oosterlee [2] and Almendral =-=[1]-=-. Notably, Mordecki [28] handles the case when X is a general Lévy process. In addition, these authors have observed that the function v is continuous, equals K − e x for x ≤ x ∗ and is bounded below ... |

6 | Perpetual options and Canadization through fluctuation theory
- Kyprianou, Pistorius
- 2003
(Show Context)
Citation Context ...y) + is convex, it follows that the optimal value function v is convex in the variable y = e x and, hence, v is continuous and has left and right derivatives in x. See, for example, the discussion in =-=[25]-=-. In Section 6 we show how the identity given in Corollary 2 can be used to give a proof of Theorem 3 which avoids the necessity of a random walk approximation. Embedded in the proof is a clearer indi... |

6 |
Electronic foreign-exchange markets and passage events of independent subordinators
- Winkel
(Show Context)
Citation Context ...ascending subordinator with Laplace exponent φ(q) := −log E(e −qX1 ), q ≥ 0, then we see that ∫ ∞ e 0 −qx E[e −ατ+ x −β(X τ + x −x) ]dx = φ(q) − φ(β) (q − β)(α + φ(q)) , which was also established in =-=[37]-=-. Otherwise, if X is not a descending subordinator, with the help of (2), we come to rest at the Pecherskii–Rogozin identity ∫ ∞ e 0 −qx E[e −ατ+ x −β(X τ + x ( −x) 1 ]dx = q − β . 1 − Ψ+ α(−q) Ψ + α(... |

5 |
Pricing American currency options in an exponential Lévy
- Chesney, Jeanblanc
- 2004
(Show Context)
Citation Context ...ss. See Gerber and Shiu [21], Chan [15, 16], Boyarchenko and Levendorskiǐ [10, 11, 12, 13, 14], Mordecki [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc =-=[17]-=-, Hirsa and Madan [23], Matache, Nitsche and Schwab [26], Almendral and Oosterlee [2] and Almendral [1]. Notably, Mordecki [28] handles the case when X is a general Lévy process. In addition, these au... |

5 |
Optimal stopping for a diffusion with jumps
- Mordecki
- 1997
(Show Context)
Citation Context ...he American perpetual put optimal stopping problem to the first passage problem of a Lévy process. See Gerber and Shiu [21], Chan [15, 16], Boyarchenko and Levendorskiǐ [10, 11, 12, 13, 14], Mordecki =-=[27, 28]-=-, Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan [23], Matache, Nitsche and Schwab [26], Almendral and Oosterlee [2] and Almendral [1]. Nota... |

5 | Electronic foreign exchange markets and level passage events of multivariate subordinators - Winkel - 2001 |

4 |
Lévy processes in finance distinguished by their coarse and fine path properties
- Kyprianou, Loeffen
- 2005
(Show Context)
Citation Context ...which is included in their class of Lévy processes, it is easy to confirm that the integral test is indeed automatically satisfied when X has bounded variation and d = 0; see also the calculations in =-=[24]-=-. For a general RLPE Lévy process, however, to see why the integral test in Proposition 7 is satisfied, one may reason intuitively as follows. The small jump structure of an RLPE process is essentiall... |

2 |
Problems of the sequential discrimination of hypotheses for a compound Poisson process with exponential jumps
- Gapeev
- 2002
(Show Context)
Citation Context ... the case of spectrally negative processes that there was no smooth pasting if and only if the process is of bounded variation. In related work Peskir and Shiryaev [32, 33] (see also [31]) and Gapeev =-=[20]-=- studied a number of optimal stopping problems for special classes of Markov process of bounded variation with jumps such that the inter-arrival times of the jumps are independent and exponentially di... |

1 |
Models of investment under uncertaintly when shocks are non-Gaussian. Working paper series EERC, 98/02, EERC/Eurasia Foundation Moscow
- Boyarchenko, Levendorskiǐ
- 1998
(Show Context)
Citation Context ...e −rτ ∗ +Xτ∗ ], thus, linking the American perpetual put optimal stopping problem to the first passage problem of a Lévy process. See Gerber and Shiu [21], Chan [15, 16], Boyarchenko and Levendorskiǐ =-=[10, 11, 12, 13, 14]-=-, Mordecki [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan [23], Matache, Nitsche and Schwab [26], Almendral and Oosterlee [2] and ... |

1 |
Pricing of a Perpetual American Put for Truncated Lévy Processes, manuscript
- Boyarchenko, Levendorskiǐ
- 1999
(Show Context)
Citation Context ...e −rτ ∗ +Xτ∗ ], thus, linking the American perpetual put optimal stopping problem to the first passage problem of a Lévy process. See Gerber and Shiu [21], Chan [15, 16], Boyarchenko and Levendorskiǐ =-=[10, 11, 12, 13, 14]-=-, Mordecki [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan [23], Matache, Nitsche and Schwab [26], Almendral and Oosterlee [2] and ... |

1 |
American Options: The EPV Pricing Model. Preprint posted on SSRN web directory; http://ssrn.com/abstract=547863
- Boyarchenko, Levendorskiǐ
- 2004
(Show Context)
Citation Context |

1 |
American options driven spectrally by one sided Lévy processes. Original unpublished manuscript. See also Exotic option pricing and advanced Lévy models
- Chan
- 2000
(Show Context)
Citation Context ...0:Xt < x ∗ } −rτ ∗ v(x) = KEx[e ] − Ex[e −rτ ∗ +Xτ∗ ], thus, linking the American perpetual put optimal stopping problem to the first passage problem of a Lévy process. See Gerber and Shiu [21], Chan =-=[15, 16]-=-, Boyarchenko and Levendorskiǐ [10, 11, 12, 13, 14], Mordecki [27, 28], Avram, Chan and Usabel [5], Asmussen, Avram and Pistorius [4], Chesney and Jeanblanc [17], Hirsa and Madan [23], Matache, Nitsch... |

1 |
On the joint distribution of rnadom variables associated with fluctuations of a process with independent increments
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- 1969
(Show Context)
Citation Context ...t a descending subordinator, with the help of (2), we come to rest at the Pecherskii–Rogozin identity ∫ ∞ e 0 −qx E[e −ατ+ x −β(X τ + x ( −x) 1 ]dx = q − β . 1 − Ψ+ α(−q) Ψ + α(−β) for any q > 0. See =-=[30]-=- and [34] for a comparison with existing proofs. )6 L. ALILI AND A. E. KYPRIANOU 3.2. Spectrally one-sided processes. Suppose that X is spectrally negative, but not a negative subordinator, with Lapl... |