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Finite approximation schemes for Lévy processes, and their application to optimal stopping problems

Finite approximation schemes for Lévy processes, and their application to optimal stopping problems,10.1016/j.spa.2007.01.012,Stochastic Processes and

Finite approximation schemes for Lévy processes, and their application to optimal stopping problems   (Citations: 9)
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This paper proposes two related approximation schemes, based on a discrete grid on a finite time interval [0,T], and having a finite number of states, for a pure jump Lévy process Lt. The sequences of discrete processes converge to the original process, as the time interval becomes finer and the number of states grows larger, in various modes of weak and strong convergence, according to the way they are constructed. An important feature is that the filtrations generated at each stage by the approximations are sub-filtrations of the filtration generated by the continuous time Lévy process. This property is useful for applications of these results, especially to optimal stopping problems, as we illustrate with an application to American option pricing. The rates of convergence of the discrete approximations to the underlying continuous time process are assessed in terms of a “complexity” measure for the option pricing algorithm.By adding in a construction for a discrete approximation to Brownian motion, we also extend the approximation results to a general Lévy process.
Journal: Stochastic Processes and Their Applications - STOCH PROC APPL , vol. 117, no. 10, pp. 1422-1447, 2007
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    • ...Recently, Szimayer and Maller [22] suggested a new approximation scheme for pure jump L´ processes, the first jump approximation...
    • ...Szimayer and Maller [22] considered this approximation scheme for univariate L´ evy processes over a finite time horizon and used it to construct an approximation scheme for American option pricing...
    • ...We start by giving a d-dimensional and infinite time extension of the first jump approximation for a L´ process presented in [22] and an interesting refinement of it for a L´ evy process with zero mean...
    • ...Proof. (a) An inspection of the proof of [22, Theorem 3.1] shows that it immediately generalises to our multivariate set-up with no upper bound on the jump sizes and no binning of the jump sizes...
    • ...Using the time change l which is the obvious extension to [0, ¥) of the time change employed in the proof of [22, Theorem 3.2] and arguments analogous to theirs give dRd( ˜ L(n), ¯ L(n))• dl,Rd( ˜ L(n), ¯ L(n))•...
    • ...Note also that [22] provides convergence rates for the first jump approximation of a L´ evy process directly in terms of the Skorokhod metric by considering explicit expressions for the error...

    Robert Stelzer. First jump approximation of a Lévy-driven SDE and an application to mu...

    • ...Using the results in Subsections 2 and 3 it should be possible to exploit the first jump approximation by Szimayer and Maller [45] as a possible link between discretely and continuously observed L ´ evy processes (cf...

    B. Buchmann. Weighted empirical processes in the nonparametric inference for Lévy p...

    • ...The term '&2 results from the bracket process of B. For a related convergence result, see Theorem 2.2 of Szimayer and Maller [25]...
    • ...Our pathwise construction relies on a ‘first-jump’ approximation to a L´evy process developed by Szimayer and Maller [25], which we present here in a general notation...
    • ...Proof. (i) The claimed results follow immediately from Theorem 2.1 of [25]...
    • ...Our equation (5.3) then follows from equation (2.11) of [25]...

    Ross A. Malleret al. GARCH modelling in continuous time for irregularly spaced time series ...

    • ...Using a first-jump approximation of a Lévy process originally developed in Szimayer and Maller (2007) for an option pricing application, Maller et al. (2008) show that the COGARCH can be obtained as a limit of discrete time GARCH processes defined on the same probability space...

    Ross A. Malleret al. Ornstein–Uhlenbeck Processes and Extensions

    • ...The discrete time GARCH processes are constructed using a “first-jump" approximation for Lévy processes as developed by Szimayer and Maller (2007), which divides a compact interval into an increasing number of subintervals and for each subinterval takes the first jump exceeding a certain threshold...

    Alexander M. Lindner. Continuous Time Approximations to GARCH and Stochastic Volatility Mode...

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