scholarly journals Asymptotic behaviour of first passage time distributions for Lévy processes

2012 ◽  
Vol 157 (1-2) ◽  
pp. 1-45 ◽  
Author(s):  
R. A. Doney ◽  
V. Rivero
Keyword(s):  
Asymptotic Behaviour ◽  
Lévy Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
First Passage

scholarly journals Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications

2015 ◽  
Vol 52 (04) ◽  
pp. 1076-1096
Author(s):  
Aleksandar Mijatović ◽  
Martijn R. Pistorius ◽  
Johannes Stolte
Keyword(s):  
Monte Carlo ◽  
Lévy Processes ◽  
Variance Reduction ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
Weak Approximation ◽  
Occupation Times ◽  
First Passage ◽  
Exponential Lévy Models

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Lévy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Lévy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Lévy processes, which includes Brownian motion with drift, Kou's double-exponential model, and hyperexponential jump-diffusion models. We present a highly accurate numerical realisation and derive error estimates. By way of illustration the method is applied to the valuation of range accruals and barrier options under exponential Lévy models and Bates-type stochastic volatility models with exponential jumps. Compared with standard Monte Carlo methods, we find that the method is significantly more efficient.

scholarly journals Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications

2015 ◽  
Vol 52 (4) ◽  
pp. 1076-1096 ◽  
Author(s):  
Aleksandar Mijatović ◽  
Martijn R. Pistorius ◽  
Johannes Stolte
Keyword(s):  
Monte Carlo ◽  
Lévy Processes ◽  
Variance Reduction ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
Weak Approximation ◽  
Occupation Times ◽  
First Passage ◽  
Exponential Lévy Models

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Lévy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Lévy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Lévy processes, which includes Brownian motion with drift, Kou's double-exponential model, and hyperexponential jump-diffusion models. We present a highly accurate numerical realisation and derive error estimates. By way of illustration the method is applied to the valuation of range accruals and barrier options under exponential Lévy models and Bates-type stochastic volatility models with exponential jumps. Compared with standard Monte Carlo methods, we find that the method is significantly more efficient.

scholarly journals First passage time law for some Lévy processes with compound Poisson: Existence of a density

Bernoulli ◽  
2011 ◽  
Vol 17 (4) ◽  
pp. 1127-1135 ◽  
Author(s):  
Laure Coutin ◽  
Diana Dorobantu
Keyword(s):  
Lévy Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
First Passage ◽  
Compound Poisson

Exponential trends of Ornstein–Uhlenbeck first-passage-time densities

1985 ◽  
Vol 22 (02) ◽  
pp. 360-369 ◽  
Author(s):  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
L. Sacerdote
Keyword(s):  
Asymptotic Behaviour ◽  
Excellent Agreement ◽  
First Passage Time ◽  
Passage Time ◽  
Computational Results ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Exponential Behaviour

The asymptotic behaviour of the first-passage-time p.d.f. through a constant boundary for an Ornstein–Uhlenbeck process is investigated for large boundaries. It is shown that an exponential p.d.f. arises, whose mean is the average first-passage time from 0 to the boundary. The proof relies on a new recursive expression of the moments of the first-passage-time p.d.f. The excellent agreement of theoretical and computational results is pointed out. It is also remarked that in many cases the exponential behaviour actually occurs even for small values of time and boundary.

On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries

1990 ◽  
Vol 22 (4) ◽  
pp. 883-914 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Asymptotic Behaviour ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Sufficient Conditions ◽  
Passage Time ◽  
Asymptotic Results ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Periodic Boundaries

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.'s through certain time-varying boundaries, including periodic boundaries, is determined for a class of one-dimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Ornstein–Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.

Sign in / Sign up

Export Citation Format

Share Document