McCauley, Joseph L. (2007): Ito Processes with Finitely Many States of Memory.

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Abstract
We show that Ito processes imply the FokkerPlanck (K2) and Kolmogorov backward time (K1) partial differential eqns. (pde) for transition densities, which in turn imply the ChapmanKolmogorov equation without approximations. This result is not restricted to Markov processes. We define ‘finite memory’ and show that Ito processes admit finitely many states of memory. We then provide an example of a Gaussian transition density depending on two past states that satisfies both K1, K2, and the ChapmanKolmogorov eqn. Finally, we show that transition densities of BlackScholes type pdes with finite memory are martingales and also satisfy the ChapmanKolmogorov equation. This leads to the shortest possible proof that the transition density of the BlackScholes pde provides the socalled ‘martingale measure’ of option pricing.
Item Type:  MPRA Paper 

Original Title:  Ito Processes with Finitely Many States of Memory 
Language:  English 
Keywords:  Ito process, martingale, stochastic differential eqn., Langevin eqn., memory, nonMarkov process, FokkerPlanck eqn., Kolmogorov’s backward time eqn., ChapmanKolmogorov eqn., BlackScholes eqn 
Subjects:  G  Financial Economics > G1  General Financial Markets C  Mathematical and Quantitative Methods > C2  Single Equation Models; Single Variables > C20  General 
Item ID:  5811 
Depositing User:  Joseph L. McCauley 
Date Deposited:  18. Nov 2007 16:50 
Last Modified:  13. Feb 2013 04:05 
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URI:  http://mpra.ub.unimuenchen.de/id/eprint/5811 