Applying the Wiener-Hopf Monte Carlo simulation technique for Lévy processes to path functionals. (English) Zbl 1319.65006

Ferreiro-Castilla, Albert; van Schaik, Kees

J. Appl. Probab. 52, No. 1, 129-148 (2015).

The authors apply the Wiener-Hopf Monte Carlo (WHMC) simulation method for Lévy processes to a more general path functionals Lévy processes.
The WHMC simulation techniques for Lévy processes has been introduced by A. Kuznetsov et al. [Ann. Appl. Probab. 21, No. 6, 2171–2190 (2011; Zbl 1245.65005)] to simulate the joint distribution of the position \(X_t\) at time \(t\) of a general Lévy process and its running maximum \(\bar X_t\) up to time \(t\).
The WHMC algorithm is built using two main ideas. Firstly, on the observation that (see [P. Carr, “Randomization and the American put”, Rev. Financial Studies 11, 597–626 (1998)]) for sufficiently large \(n\), the distribution of \((X_t,\bar X_t)\) may be approximated by the distribution of \((X_{g(n,n/t)}, \bar X_{g(n,n/t)})\), where \(g(n,n/t)\) is a gamma random variable with shape parameter \(n\) and scale parameter \(n/t\). Then, using the Wiener-Hopf factorization, it has been shown by Kuznetsov et al. [loc. cit.] that for every \(\lambda>0\) the random vector \((X_{g(n,\lambda)}, \bar X_{g(n,\lambda)})\) has the same distribution as a random vector \((V(n,\lambda), J(n,\lambda))\), which components are computed explicitly from simple recursive formulas.
This paper extends the WHMC method to path functional Lévy processes of the form \(f(\tau_u, X_{\tau_u}-u, u-X_{\tau_u-},u-\bar X_{\tau_u-})\), where \(\tau_u\) is the hitting time of the level \(u\). More specifically, it is shown that for any Lévy process \(X\), the distribution of \((\tau_u, X_{\tau_u}-u, u-X_{\tau_u-},u-\bar X_{\tau_u-})\) may be approximated, for large enough \(n\), by the distribution of \[ \Big(\frac{t}{n} (\kappa_u^{(n)} \wedge n), V(\kappa_u^{(n)} \wedge n, n/t ) -u , u- V((\kappa_u^{(n)} -1)\wedge n, n/t) , u - J((\kappa_u^{(n)} -1)\wedge n, n/t) \Big), \] where all the components depend on the discrete version \(\tau_u^{(n)}\) of the hitting time \(\tau_u\) defined by \(\tau_u^{(n)} = \inf\{ k \in \{0, \dots,n \}, \;J(k,n/t) >u\}\). Furthermore, the quadratic mean convergence rate of \(\frac{t}{n} (\kappa_u^{(n)} \wedge n)\) towards \(\tau_u \wedge n\) is investigated and is shown to be of order \(2t^2/n\).
This allows to approximate quantities of the form \(f(\tau_u, X_{\tau_u}-u, u-X_{\tau_u-},u-\bar X_{\tau_u-})\) using Monte Carlo or multilevel Monte Carlo simulation techniques. Numerical simulations are performed for the \(\beta\)-family of Lévy processes and for the first passage time of the Brownian motion to illustrate the advantage of the multilevel Monte Carlo using the Wiener-Hopf method with respect to the ‘plain’ Monte Carlo method.

Reviewer: Abass Sagna (Evry)

MSC:

65C05	Monte Carlo methods
60G51	Processes with independent increments; Lévy processes
65C50	Other computational problems in probability (MSC2010)
91G60	Numerical methods (including Monte Carlo methods)
91B24	Microeconomic theory (price theory and economic markets)
91B30	Risk theory, insurance (MSC2010)

Keywords:

Wiener-Hopf decomposition; Monte Carlo simulation; multilevel Monte Carlo; Lévy process; exotic option pricing; first passage time; overshoot; insurance risk process; algorithm; Brownian motion

Citations:

Zbl 1245.65005