Numerical methods for Lévy processes

Hilber, Norbert; Reich, Nils; Schwab, Christoph; Winter, Christoph

doi:10.1007/s00780-009-0100-5

Cited by 33 publications

(20 citation statements)

References 114 publications

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“…We conclude this introductory part by illustrating this approach in case the underlying process X is a Lévy process (for a more general survey we refer to [32,33]). …”

Section: Pricing Equationsmentioning

confidence: 99%

Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

Reich¹

2009

ESAIM: M2AN

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Abstract. For a class of anisotropic integrodifferential operators B arising as semigroup generators ofMarkov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations Bu = f on [0,1] n with possibly large n. Under certain conditions on B, the scheme is of essentially optimal and dimension independent complexity O(h −1 | log h| 2(n−1) ) without corrupting the convergence or smoothness requirements of the original sparse tensor finite element scheme. If the conditions on B are not satisfied, the complexity can be bounded by O(h −(1+ε) ), where ε 1 tends to zero with increasing number of the wavelets' vanishing moments. Here h denotes the width of the corresponding finite element mesh. The operators under consideration are assumed to be of non-negative (anisotropic) order and admit a non-standard kernel κ(·, ·) that can be singular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented.Mathematics Subject Classification. 47A20, 65F50, 65N12, 65Y20, 68Q25, 45K05, 65N30.

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“…We conclude this introductory part by illustrating this approach in case the underlying process X is a Lévy process (for a more general survey we refer to [32,33]). …”

Section: Pricing Equationsmentioning

confidence: 99%

Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

Reich¹

2009

ESAIM: M2AN

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“…Likewise, γ τ is a valid Lévy measure. Consequently, by Corollary 1 and the symmetry of λ, (19) is satisfied with q = 10.…”

Section: Now Define Lévy Measuresmentioning

confidence: 87%

“…Normal approximation of i.d. distributions, which was studied in [26] and later developed in [1,11] in the framework of small jump approximation, has received much attention in the literature [2,13,16,19,22,23,33].…”

Section: Introductionmentioning

confidence: 99%

Nonnormal Small Jump Approximation of Infinitely Divisible Distributions

Chi¹

2014

Advances in Applied Probability

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We consider a type of nonnormal approximation of infinitely divisible distributions that incorporates compound Poisson, Gamma, and normal distributions. The approximation relies on achieving higher orders of cumulant matching, to obtain higher rates of approximation error decay. The parameters of the approximation are easy to fix. The computational complexity of random sampling of the approximating distribution in many cases is of the same order as normal approximation. Error bounds in terms of total variance distance are derived. Both the univariate and the multivariate cases of the approximation are considered. is 0.4785 ([28], p. 71). Also, in some symmetric cases, since the third cumulants of X r and the Gaussian random variable are 0, |κ| 3,Xr in the bound can be more or less replaced with |κ| 4,Xr , while the power of κ 2,Xr is raised to 2 [1]. From the pattern of the bound, one could guess that, if X r and some Y r have the same cumulants of order 1, . . . , q − 1 with q ≥ 5, then X r could be approximated by Y r with the error being bounded by C(r)(|κ| q,Xr + |κ| q,Yr )/κ q/2 2,Xr , where C(r) is a near-constant, at least when r is small, or most ideally, a universal constant which may depend on the dimension of X but is not very large. Elementary calculations indicates that in many cases, the above bound vanishes at a genuinely higher rate than the bound for normal approximation as r → 0+. Since the qth cumulant of a Gaussian random variable is 0, the above bound, if true, is consistent with the bound for normal approximation.Even by some rough analysis on characteristic functions, there is good reason to expect that the bound is true for both univariate and multivariate cases. However, before attempting to work out the detail, perhaps one should first ask if such an approximation can possibly be implemented easily. The meaning of the question is twofold. First, the distribution of the approximating random variable should be easy to identify; preferably, it is i.d. Second, the approximating random variable should be easy to sample; preferably, the computational complexity of the sampling is of the same order as the normal approximation. If the answer to the question is positive, then the next question is how large q can be. It can be anticipated that the larger q is, the faster the error of approximation vanishes as r → 0+. After the answers to the two questions are in place, a wide range of available techniques can potentially be modified to establish the error bound (e.g. [3,5,8,28]).Clearly, cumulant matching is equivalent to moment matching. Actually, our proof of the above type of bound eventually will be based on moment matching. However, thanks to the Lévy-Khintchine representation, it is more natural and convenient to consider cumulants than moments. We shall show that it is fairly easy to construct approximating i.d. random variables with matching cumulants up to at least the fourth order, in other words, we can get at least q = 5. In many important cases, we can get q = 6, and in the symmetric ...

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“…The form of the integral term in (5) may seem different from the integral term appearing in backward PIDEs [14,25]. The following lemma expresses χ T,y (z) in a more familiar form in terms of call payoffs: Lemma 1.…”

Section: Remark 2 the Discounted Asset Pricêmentioning

confidence: 99%