McCauley, Joseph L. (2007): FokkerPlanck and ChapmanKolmogorov equations for Ito processes with finite memory.

PDF
MPRA_paper_2128.pdf Download (897kB)  Preview 
Abstract
The usual derivation of the FokkerPlanck partial differential eqn. (pde) assumes the ChapmanKolmogorov equation for a Markov process [1,2]. Starting instead with an Ito stochastic differential equation (sde), we argue that finitely many states of memory are allowed in Kolmogorov’s two pdes, K1 (the backward time pde) and K2 (the FokkerPlanck pde), and show that a ChapmanKolmogorov eqn. follows as well. We adapt Friedman’s derivation [3] to emphasize that finite memory is not excluded. We then give an example of a Gaussian transition density with 1state memory satisfying both K1, K2, and the ChapmanKolmogorov eqns. We begin the paper by explaining the meaning of backward time diffusion, and end by using our interpretation to produce a very short proof that the Green function for the BlackScholes pde describes a Martingale in the risk neutral discounted stock price.
Item Type:  MPRA Paper 

Institution:  University of Houston 
Original Title:  FokkerPlanck and ChapmanKolmogorov equations for Ito processes with finite memory 
Language:  English 
Keywords:  Stochastic process; martingale; Ito process; stochastic differential eqn.; memory; nonMarkov process; 2 backward time diffusion; FokkerPlanck; Kolmogorov’s partial differential eqns.; ChapmanKolmogorov eqn.; Black Scholes eqn 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C69  Other G  Financial Economics > G0  General 
Item ID:  2128 
Depositing User:  Joseph L. McCauley 
Date Deposited:  09. Mar 2007 
Last Modified:  11. Feb 2013 19:27 
References:  1. B. V. Gnedenko, The Theory of Probability, tr. by B.D. Seckler (Chelsea, N.Y., 1967). 2. R.L. Stratonovich. Topics in the Theory of Random Noise, tr. By R. A. Silverman (Gordon & Breach: N.Y 1963). 3. A. Friedman, Stochastic Differential Equations and Applications (Academic, N.Y., 1975). 4. L. Arnold, Stochastic Differential Equations (Krieger, Malabar, 1992). 18 5. J.M. Steele, Stochastic Calculus and Financial Applications (SpringerVerlag, N.Y., 2000). 6. K.E. Bassler, G.H. Gunaratne, & J. L. McCauley, Hurst Exponents, Markov Processes, and Nonlinear Diffusion Equations, Physica A 369: 343 (2006). 7. J. L. McCauley , G.H. Gunaratne, & K.E. Bassler, Martingales, Detrending Data, and the Efficient Market Hypothesis, submitted (2007). 8. P. Hänggi and H. Thomas, Time Evolution, Correlations, and Linear Response of NonMarkov Processes, Zeitschr. Für Physik B26: 85 (1977). 9. J. L. McCauley, Markov vs. nonMarkovian processes: A comment on the paper ‘Stochastic feedback, nonlinear families of Markov processes, and nonlinear FokkerPlanck equations’ by T.D. Frank, submitted (2007). 10. J. L. McCauley , G.H. Gunaratne, & K.E. Bassler, Hurst Exponents, Markov Processes, and Fractional Brownian Motion, Physica A (2007). 11. P. Hänggi, H. Thomas, H. Grabert, and P. Talkner, Note on time Evolution of NonMarkov Processes, J. Stat. Phys. 18: 155 (1978). 12. T.D. Frank, Stochastic feedback, nonlinear families of Markov processes, and nonlinear FokkerPlanck equations, Physica A331: 391 (2004). 13. J.L. McCauley, Dynamics of Markets: Econophysics and Finance (Cambridge, Cambridge, 2004). 19 14. J. L. McCauley, G.H. Gunaratne, & K.E. Bassler, Martingale Option Pricing, Physica A (2007). 15. M.C. Wang & G.E. Uhlenbeck in Selected Papers on Noise and Stochastic Processes, ed. N. Wax, Dover: N.Y., 1954. 16. D. Duffie, An Extension of the BlackScholes Model of Security Valuation, J. Econ. Theory 46,194, 1988. 17. W. Feller, The Annals of Math. Statistics 30, No. 4, 1252, 1959. 18. J. L. Snell, A Conversation with Joe Doob, http://www.dartmouth.edu/~chance/Doob/conversation. html; Statistical Science 12, No. 4, 301, 1997. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/2128 