Geodesic
Progressive projection and log-optimal investment in the frictionless market
Teoriâ slučajnyh processov, Tome 25 (2020) no. 1, pp. 37-77 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, we introduce notion of progressive projection, closely related to the extended predictable projection. This notion is flexible enough to help us treat the problem of log-optimal investment without transaction costs almost exhaustively in case when the rate of return is not observed. We prove some results saying that the semimartingale property of a continuous process is preserved when changing the filtration to the one generated by the process under very general conditions. We also had to introduce a very useful and flexible notion of so called enriched filtration.
Keywords: Log-optimal investment, progressive projection, filtering. 
@article{THSP_2020_25_1_a2,
     author = {P. Dost\'al and T. Mach},
     title = {Progressive projection and log-optimal investment in the frictionless market},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {37--77},
     year = {2020},
     volume = {25},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2020_25_1_a2/}
}
TY  - JOUR
AU  - P. Dostál
AU  - T. Mach
TI  - Progressive projection and log-optimal investment in the frictionless market
JO  - Teoriâ slučajnyh processov
PY  - 2020
SP  - 37
EP  - 77
VL  - 25
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/THSP_2020_25_1_a2/
LA  - en
ID  - THSP_2020_25_1_a2
ER  - 
%0 Journal Article
%A P. Dostál
%A T. Mach
%T Progressive projection and log-optimal investment in the frictionless market
%J Teoriâ slučajnyh processov
%D 2020
%P 37-77
%V 25
%N 1
%U http://geodesic.mathdoc.fr/item/THSP_2020_25_1_a2/
%G en
%F THSP_2020_25_1_a2
P. Dostál; T. Mach. Progressive projection and log-optimal investment in the frictionless market. Teoriâ slučajnyh processov, Tome 25 (2020) no. 1, pp. 37-77. http://geodesic.mathdoc.fr/item/THSP_2020_25_1_a2/

[1] P. H. Algoet, T. M. Cover, “Asymptotic optimality and asymptotic equipartition properties of log-optimum investment”, Ann. Probab., 16:2 (1988), 876–898 | DOI | MR

[2] A. Bain, D. Crisan, Fundamentals of stochastic filtering, Stochastic Modelling and Applied Probability, 60, Springer, New York, 2009 | DOI | MR

[3] R. Bell, T. M. Cover, “Game-theoretic optimal portfolios”, Manage. Sci., 34:6 (1998), 724–733 | DOI | MR

[4] L. Breiman, “Optimal gambling system for flavorable games”, Proc. Fourth Berkeley Symp. on Math. Statist. and Prob., 1, eds. J. Neyman, ed., Univ. of Calif. Press, Berkley, 1961, 65–78 | MR

[5] S. Browne, W. Whitt, “Portfolio choice and the Bayesian Kelly criterion”, Adv. Appl. Probab., 28:4 (1996), 1145–1176 | DOI | MR

[6] D. Crisan, M. Kouritzin, J. Xiong, “Nonlinear filtering with signal dependent observation noise”, Electron. J. Probab., 14:63 (2009), 1863–1883 | DOI | MR

[7] D. Crisan, S. Ortiz-Latorre, “A high order time discretization of the solution of the non-linear filtering problem”, Stoch. PDE: Anal. Comp., 2019 | DOI | MR

[8] P. Dostál, “A note on weak convergence on martingale measures”, Acta Univ. Carol., Math. Phys., 43:1 (2002), 3–12 | MR

[9] P. Dostál, J. Klůjová, “Log-Optimal Investment in the Long Run with Proportional Transaction Costs When Using Shadow Price”, Kybernetika, 51:4 (2015), 588–628 | MR

[10] J. Dupačová, J. Hurt, J. Štěpán, Stochastic Modelling in Economics and Finance, Kluwer Academic Publishers, New York, 2003 | MR

[11] M. Fujisaki, G. Kallianpur, H. Kunita, “Stochastic differential equations for the non linear filtering problem”, Osaka J. Math., 9:1 (1972), 19–40 | MR

[12] J. Jacod, A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, Heidelberg, 2003 | MR

[13] O. Kallenberg, Foundations of Modern Probability, Springer-Verlag, New York, 1997 | MR

[14] G. Kallianpur, Stochastic Filtering Theory, Springer-Verlag, New York, 1980 | MR

[15] G. Kallianpur, C. Striebel, “Estimation of Stochastic Systems: Arbitrary System Process with Additive White Noise Observation Errors”, The Annals of Mathematical Statistics, 39:3 (1968), 785–801 | DOI | MR

[16] R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems”, Journal of Basic Engineering, 82:1 (1960), 35–45 | DOI | MR

[17] R. E. Kalman, R. S. Bucy, “New Results in Linear Filtering and Prediction Theory”, Journal of Basic Engineering, 83:1 (1961), 95–108 | DOI | MR

[18] I. Karatzas, K. Kardaras, “The numéraire portfolio in semimartingale financial models”, Finance and Stochastics, 11:4 (2007), 447–493 | DOI | MR

[19] I. Karatzas, S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin, New York, Heidelberg, 1988 | MR

[20] J. L. Kelly, “A new interpretation of information rate”, Bell Sys. Tech. J., 35:4 (1956), 917–926 | DOI | MR

[21] H. Kushner, “Dynamical equations for optimal nonlinear filtering”, J. Differ. Equations, 3 (1967), 179–190 | DOI | MR

[22] O. Mostovyi, P. Siorpaes, “Differentiation of measures on a non-separable space, and the Radon-Nikodym theorem”, 2019, arXiv: 1909.03505 [math.CA]

[23] R. Liptser, A. N. Shiryaev, Statistics of Random Processes. I General Theory, Springer-Verlag, New York, 1977 | MR

[24] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 1999 | MR

[25] L. C. G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, Second edition, Cambridge University Press, Cambridge, 2000 | MR