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On Poisson mixture of lognormal distributions. (English) Zbl 1465.60015

Generally, jump-diffusion processes used in finance are confined to the processes with Brownian motion, constant trend and jump component, described by compound Poisson processes (CPP). CPP is usually defined by a sum of standard normal distributions. In most applications one either needs moments or characteristic function of the process. The authors provide several useful properties for discretized processes and analyze parameters via the moments method. They prove that the distribution cannot be uniquely determined via its moments, as is the case for the lognormal distribution. Utilizing the expectation maximization method described in [G. Y. Ait-Sahalia, “Telling from discrete data whether the underlying continuous-time model is diffusion”, J. Finance 57, No. 5, 2075–2112 (2002; doi:10.1111/1540-6261.00489)] and [S. Nadarajah and C. S. Withers, Ann. Sci. Math. Qué. 31, No. 2, 187–192 (2007; Zbl 1158.60347)], they try to evaluate the jump part distribution of discretized process directly. Two forms of discretization [Zbl 1158.60347] are possible: One constrained by the period structure, i.e., not letting other than one exponential jump in an infinitesimal period and the second one which uses weights for the number of exponential jumps.
Theorem 1: The process \(S\) has all finite moments and all moments satisfy the following inequality \[ E((\Sigma X_{k})^n)<\exp(-n\lambda +\lambda ne^{n/2}). \] Theorem 2: The process \(S\) has all finite moments and all moments satisfy the following inequality \[ E((\Sigma X_{k})^n)>e^{-n\lambda}_0F_{n-1}(1,\dots,1,(e^{n/2}\lambda)^n), \] where \(_0F_{n-1}\) is a generalized hypergeometric function (see [A. M. Mathai and R. K. Saxena, Generalized hypergeometric functions with applications in statistics and physical sciences. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0272.33001)]). From this condition we have \[ \Sigma M_{n}^{-(1/(2n))}<\Sigma e^{(\lambda/2)}(_0F_{n-1}(;1,\dots,1,(e^{n/2}\lambda)^n))^{-(1/(2n))}. \] Using Krein’s condition, the authors prove the following results:
Theorem 3: The distribution of the random variable \(S\) cannot be uniquely determined by its moments, by Krein’s condition [G. D. Lin, J. Stat. Distrib. Appl. 4, Paper No. 5, 17 p. (2017; Zbl 1386.60058); H. L. Pedersen, J. Approx. Theory 95, No. 1, 90–100 (1998; Zbl 0924.44009); J. Stoyanov, Bernoulli 6, No. 5, 939–949 (2000; Zbl 0971.60017)].
By examining the structure of central moments of lognormal distribution [E. L. Crow (ed.) and K. Shimizu (ed.), Lognormal distributions. Theory and applications. New York etc.: Marcel Dekker, Inc. (1988; Zbl 0644.62014)] they show that why they cannot use the central limit theorem.

MSC:

60E05 Probability distributions: general theory
60J60 Diffusion processes
62E17 Approximations to statistical distributions (nonasymptotic)
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