Estimation of the Characteristics of the Jumps of a General Poisson-Diffusion Model

Abstract

Estimation of the characteristics of the jumps of a general Poisson-diffusion model. Scand. Actuarial J. 2004; 1: 42Á/52We consider a filtered probability space with a standard Brownian motion W, a simple Poisson process N with constant intensity l /0, and we consider the process Y such that Y 0 /R and dY t 0a t dt's t dW t 'g t dN t ; t0;(1)where a, s are predictable bounded stochastic processes, and g is a predictable process which is bounded away from zero. A discrete record of n'/1 observations {Y 0 , Y t 1 , . . ., Y t n(1 , Y t n } is available, with t i 0/ih. Using such observations, we construct estimators of N t i , i0/1, . . ., n , l and g t j , where t j are the instants of jump within [0, nh]. They are consistent and asymptotically controlled when the number of observations increases and the step h tends to zero. Key words: Stock price model, quadratic variation process, paths of a Brownian stochastic integral, simple Poisson process, estimators.

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