By Eckhard Platen

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Extra resources for Numerical Solution of Stochastic Differential Equations with Jumps in Finance

Example text

28) k=1 for t ≥ 0. Here we assume that the terms involved are almost surely ﬁnite, such pt pt ξ(τk −) ΔXτk and k=1 ΔXτk almost surely converge to that the sums k=1 a ﬁnite value for all t ≥ 0. Itˆ o Integral for Jump Measures In the case when jump sizes are continuously distributed and total jump intensities may not be ﬁnite, one may use the Itˆo integral with respect to a Poisson measure as introduced in Sect. 1. 32). 4 Itˆ o Integral 33 for all t ≥ 0. This means that if at a jump time τ the Poisson measure pϕ generates an event with mark v, then the change of the value of the corresponding Itˆ o integral is given by the value ξ(v, τ −), the integrand ξ for the mark v just before the jump time.

If X is a continuous martingale, then the maximal martingale inequality provides an estimate for the probability that a level a will be exceeded by the maximum of X. In particular the Doob inequality provides for p = 2 for the maximum of the square the estimate E ≤ 4 E |Xt |2 sup |Xs |2 s∈[0,t] for t ≥ 0. If X = {Xt , t ≥ 0} is a right continuous supermartingale, then it can be shown, see Doob (1953), that for any λ > 0 it holds λP sup Xt ≥ λ A0 ≤ E X0 A0 + E max(0, −X0 ) A0 . 20) t≥0 Standard Inequalities We list here some standard inequalities, see Ash (1972) and Ikeda & Watanabe (1989), that we will use later repeatedly.

Then the Itˆ o integral is again a semimartingale. If the integrator X is an (A, P )-local martingale with appropriate integrands, for example continuous or locally bounded integrands, then the Itˆ o integral is also an (A, P )-local martingale, see Protter (2005). In the case when X is of zero quadratic variation, the Itˆ o integral coincides with the random ordinary Riemann-Stieltjes integral. 25) 32 1 SDEs with Jumps does not vanish for all t ≥ 0. 26) for t ≥ 0. For example, if N is a Poisson process, then at the kth jump time τk we have ΔNτk = Nτk − Nτk−1 = 1 for k ∈ {1, 2, .