S. H. Babbs and K. B. Nowman, Kalman Filtering of Generalized Vasicek Term Structure Models, The Journal of Financial and Quantitative Analysis, vol.34, issue.1, pp.115-130, 1999.
DOI : 10.2307/2676248

H. F. Chen, Stochastic Approximation and its Applications, Nonconvex Optimization and its Applications, 2002.

H. F. Chen and Y. Zhu, Stochastic approximation procedures with randomly varying truncations, Sci. Sinica Ser. A, vol.29, issue.9, pp.914-926, 1986.

Q. Dai and K. J. Singleton, Specification Analysis of Affine Term Structure Models, The Journal of Finance, vol.11, issue.5, pp.1943-1978, 2000.
DOI : 10.1111/0022-1082.00278

M. Duflo, Random Iterative Models, Applications of Mathematics, vol.34, 1997.
DOI : 10.1007/978-3-662-12880-0

R. Gibson, F. S. Lhabitant, and D. Talay, Modeling the Term Structure of Interest Rates: A Review of the Literature, Foundations and Trends?? in Finance, vol.5, issue.1-2, 2010.
DOI : 10.1561/0500000032

URL : https://hal.archives-ouvertes.fr/hal-00602828

H. J. Kushner and G. G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, 2003.

J. Lelong, Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions, Statistics & Probability Letters, vol.78, issue.16, pp.2632-2636, 2008.
DOI : 10.1016/j.spl.2008.02.034

URL : https://hal.archives-ouvertes.fr/hal-00152255

J. Lelong, Asymptotic normality of randomly truncated stochastic algorithms , ESAIM Probab, Stat, vol.17, pp.105-119, 2013.

P. Malliavin, Integration and Probability, Graduate Texts in Mathematics, vol.157, 1995.
DOI : 10.1007/978-1-4612-4202-4

M. Musiela and M. Rutkowski, Martingale methods in financial modelling, Second, Stochastic Modelling and Applied Probability, 2005.

H. Robbins and S. Monro, A Stochastic Approximation Method, The Annals of Mathematical Statistics, vol.22, issue.3, pp.400-407, 1951.
DOI : 10.1214/aoms/1177729586

R. T. Rockafellar and R. J. Wets, Variational analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of, Mathematical Sciences], vol.317, 1998.

. Proof, Let us first prove the result for M = 0, that is X = exp (µ + ?G)