Affine Transformations of Itô Diffusions and their Transition Densities


  • Sanae RUJIVAN Division of Mathematics, School of Science, Walailak University, Nakhon Si Thammarat 80161, Thailand


Itô diffusions, forward Kolmogorov equation, transition densities


For a given Itô diffusion, we derive the forward Kolmogorov equation (FKE) associated with the adjoint operator of the infinitesimal generator of an affine transformation of the given Itô diffusion. The fundamental solution obtained by solving the FKE is, in fact, the transition density of the transformed diffusion. Moreover, we prove that the transition density can be represented in terms of a product of two functions, a Jacobian term and a composition of the transition density of the given Itô diffusion and the inverse of the transformation. Finally, we present an application of our results in parameter estimation in commodity markets in which the commodity prices are assumed to follow an extended Black-Scholes model.


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L Arnold. Stochastic Differential Equations, Wiley, New York, 1974, p. 1-288.

NG Van Kampen. Stochastic Processes in Physics and Chemistry, North Holland, Amsterdam, 1981, p. 1-451.

W Dieterich, P Fulde and I Peschel. Theoretical models for super-ionic conductors. Adv. Physics. 1980; 29, 527-605.

A Gelb. Applied Optimal Estimation, MIT Press, Cambridge, MA, 1974, p. 51-84.

RI Jennrich and PB Bright. Fitting systems of linear differential equations using computer generated exact derivatives. Technometrics 1976; 18, 385-92.

RH Jones. Fitting multivariate models to unequally spaced data, In: E Parzen (ed.). Time Series Analysis of Irregularity Observed Data. Lecture Notes in Statistics 25 Springer-Verlag, New York, 1984, p. 155-88.

AR Bergstrom. Statistical Inference in Continuous Time Economic Models, North-Holland, Amsterdam, 1976, p. 267-327.

F Black and M Scholes. The pricing of options and corporate liabilities. J. Polit. Econ. 1973; 81, 637-54.

M Arato. Linear Stochastic Systems with Constant Coefficients: A Statistical Approach. In: Lecture Notes in Control and Information Sciences 45, Springer-Verlag, New York, 1982, p. 1-309.

R Adler, P Muller and BL Rozovskii. Stochastic Modeling in Physical Oceanography, Birkhauser, Boston - Berlin, 1996, p. 1-349.

JL Doob. Stochastic Processes, John Wiley, New York, 1953, p. 46-560.

V Genon-Catalot and J Jacod. Estimation of Diffusion Coefficient for Diffusion Processes: Random Sampling. Scand. J. Statist. 1994; 21, 193-221.

Y Aït-Sahalia. Maximum likelihood estimation of discretely-sampled diffusions: a closed form approximation approach. Econometrica 2002; 70, 223-62.

A Egorov, H Li and Y Xu. Maximum Likelihood of Time Inhomogeneous Diffusion. J. Econometrics 2003; 114, 107-39.

Y Aït-Sahalia. Closed-Form Likelihood Expansions for Multivariate Diffusions. Ann. Stat. 2008; 36, 906-37.

P Billingsley. Statistical Inference for Markov Processes, Chicago University Press, Chicago, 1961, p. 1-67.

MJ Crowder. Maximum Likelihood Estimation for Dependent Observations. Journal of Royal Statistical Society, Series B 1976; 38, 45-53.

IV Basawa and BLS Prakasa Rao. Statistical Inference for Stochastic Processes, Academic Press, New York, 1980, p. 67-143.

D Dacunha-Castelle and D Florens-Zmirou. Estimation of the coefficients of a diffusion from discrete observations. Stochastics 1986; 19, 263-84.

A Friedman. Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964, p. 1-268.

A Friedman. Stochastic Differential Equations, Vol. I, Academic Press, New York, 1975, p. 128-50.

D Heath and M Schweizer. Martingales versus PDEs in Finance: An Equivalence Result with Examples. J. Appl. Probab. 2000; 37, 947-57.

G Casella and LB Roger. Statistical Inference, Duxbury, California, 2002, p. 50-1.

H Geman. Commodities and Commodity Derivatives, John Wiley & Sons Ltd., West Sussex, 2005, p. 1-22.




How to Cite

RUJIVAN, S. (2011). Affine Transformations of Itô Diffusions and their Transition Densities. Walailak Journal of Science and Technology (WJST), 8(1), 71–79. Retrieved from



Research Article