Ito Processes with Finitely Many States of Memory
AbstractWe show that Ito processes imply the Fokker-Planck (K2) and Kolmogorov backward time (K1) partial differential eqns. (pde) for transition densities, which in turn imply the Chapman-Kolmogorov 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 Chapman-Kolmogorov eqn. Finally, we show that transition densities of Black-Scholes type pdes with finite memory are martingales and also satisfy the Chapman-Kolmogorov equation. This leads to the shortest possible proof that the transition density of the Black-Scholes pde provides the so-called ‘martingale measure’ of option pricing.
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Bibliographic InfoPaper provided by University Library of Munich, Germany in its series MPRA Paper with number 5811.
Date of creation: 16 Nov 2007
Date of revision:
Ito process; martingale; stochastic differential eqn.; Langevin eqn.; memory; nonMarkov process; Fokker-Planck eqn.; Kolmogorov’s backward time eqn.; Chapman-Kolmogorov eqn.; Black-Scholes eqn;
Find related papers by JEL classification:
- G1 - Financial Economics - - General Financial Markets
- C20 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - General
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- McCauley, Joseph L. & Gunaratne, Gemunu H. & Bassler, Kevin E., 2007. "Martingale option pricing," MPRA Paper 2151, University Library of Munich, Germany.
- J. L. McCauley & G. H. Gunaratne & K. E. Bassler, 2006. "Martingale Option Pricing," Papers physics/0606011, arXiv.org, revised Feb 2007.
- McCauley, J.L. & Gunaratne, G.H. & Bassler, K.E., 2007. "Martingale option pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 351-356.
- Duffie, Darrell, 1988. "An extension of the Black-Scholes model of security valuation," Journal of Economic Theory, Elsevier, vol. 46(1), pages 194-204, October.
- Bassler, Kevin E. & Gunaratne, Gemunu H. & McCauley, Joseph L., 2007. "Empirically Based Modeling in the Social Sciences and Spurious Stylized Facts," MPRA Paper 5813, University Library of Munich, Germany.
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