Abstract
First, sufficient conditions are given for a triangular array of random vectors such that the sequence of related random step functions converges towards a (not necessarily time homogeneous) diffusion process. These conditions are weaker and easier to check than the existing ones in the literature, and they are derived from a very general semimartingale convergence theorem due to Jacod and Shiryaev, which is hard to use directly.
Next, sufficient conditions are given for the convergence of stochastic integrals of random step functions, where the integrands are functionals of the integrators. This result covers situations which cannot be handled by existing ones.
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References
P. Billingsley, Convergence of Probability Measures, Wiley (New York — London — Sydney, 1968). MR0233396
S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, Wiley (New York, 1986). MR0838085
I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, W. B. Saunders Co. (Philadelphia, Pa.-London-Toronto, Ont., 1969). MR0247660
M. Ispány and G. Pap, Weak convergence of step processes and an application for critical multitype branching processes with immigration (2007). http://arxiv.org/abs/math.PR/0701803
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, 2nd ed., Springer-Verlag (Berlin, 2003). MR1943877
A. Jakubowski, J. Mémin and G. Pagès, Convergence en loi des suites d’intégrales stochastiques sur l’espace D 1 de Skorokhod, Probab. Theory Related Fields, 81 (1989), 111–137. MR0981569
A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes, Adv. in Appl. Probab., 18 (1986), 20–65. MR0827331
T. G. Kurtz and Ph. E. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab., 19 (1991), 1035–1070. MR1112406
T. G. Kurtz and Ph. E. Protter, Weak convergence of stochastic integrals and differential equations, in: Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995), Lecture Notes in Math., 1627, Springer (Berlin, 1996), pp. 1–41. MR1431298
T. G. Kurtz and Ph. E. Protter, Weak convergence of stochastic integrals and differential equations, Working paper, 2004, http://www.orie.cornell.edu/-protter/WebPapers/KPwkConvStochIntI.pdf
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Ispány, M., Pap, G. A note on weak convergence of random step processes. Acta Math Hung 126, 381–395 (2010). https://doi.org/10.1007/s10474-009-9099-5
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DOI: https://doi.org/10.1007/s10474-009-9099-5