Stochastic differential games in a non-markovian setting

SIAM Journal on Control and Optimization

Volumn 43, Issue 5, 2005, Pages 1737-1756

  Bayraktar, Erhan ^a   Poor, H Vincent ^b  

^a UNIVERSITY OF MICHIGAN   (United States)

^b Princeton University   (United States)

Author keywords

Fractional Brownian motion; Fractional noise theory; Non Markovian games; Stochastic differential games

Indexed keywords

FRACTIONAL BROWNIAN MOTION; FRACTIONAL NOISE THEORY; NON-MARKOVIAN GAMES; STOCHASTIC DIFFERENTIAL GAMES;

BROWNIAN MOVEMENT; COMPUTER SIMULATION; DIFFERENTIATION (CALCULUS); DIFFUSION; GAME THEORY; MARKOV PROCESSES; MATHEMATICAL MODELS; PARETO PRINCIPLE;

STOCHASTIC CONTROL SYSTEMS;

EID: 18644371201     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012902417632     Document Type: Article

References (34)

1. K. AASE, B. ØKSENDAL, AND J. UBØE, White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance, Finance Stoch., 4 (2000), pp. 465-496.
2. R. J. BARTON AND H. V. POOR, Signal detection in fractional Gaussian noise, IEEE Trans. Inform. Theory, 34 (1988), pp. 943-959.
3. E. BAYRAKTAR, U. HORST, AND R. SIRCAR, A Limit Theorem for Financial Markets with Inert Investors, Princeton University, Princeton, NJ, 2003, preprint.
4. E. BAYRAKTAR AND H. V. POOR, Stochastic differential games in a non-Markovian setting (longer version), 2004, available at http://www.math.lsa. umich.edu/~erhan/.
5. D. C. BRODY, J. SYROKA, AND M. ZERVOS, Dynamical pricing of weather derivatives, Quant. Finance, 2 (2002), pp. 189-198.
6. S. BROWNE, Stochastic differential portfolio games, J. Appl. Probab., 37 (2000), pp. 126-147.
7. T. E. DUNCAN, Y. Hu, AND B. PASIK-DUNCAN, Stochastic calculus for fractional Brownian motion I. Theory, SIAM J. Control Optim., 38 (2000), pp. 582-612.
8. R. J. ELLIOTT AND J. VAN DER HOEK, A general fractional white noise theory and applications to finance, Math. Finance, 13 (2003), pp. 301-330.
9. A. FRIEDMAN, Stochastic Differential Equations and Applications, Academic Press, New York, 1976.
10. S. D. GAIDOV, Pareto-optimality in stochastic differential games, Problems Control Inform. Theory, 15 (1986), pp. 439-450.
11. S. D. GAIDOV, Nash equilibrium in stochastic differential games, Computers Math. Appl., 12A (1986), pp. 761-768.
12. S. D. GAIDOV, Z-equilibria in many-player stochastic differential games, Arch. Math. (Brno), 29 (1993), pp. 123-133.
13. G. GRIPENBERG AND I. NORROS, On the prediction of fractional Brownian motion, J. Appl. Probab., 33 (1996), pp. 400-410.
14. H. HOLDEN, B. ØKSENDAL, J. UBØE, AND T. ZHANG, Stochastic Partial Differential Equations, Probab. Appl., Birkhäuser Boston, Boston, 1996.
15. Y. HU, Prediction and translation of fractional Brownian motions, in Stochastics in Finite and Infinite Dimensions, Trends Math., T. Hida et al., eds., Birkhäuser Boston, Boston, 2001, pp. 153-171.
16. Y. Hu AND B. ØKSENDAL, Fractional white noise calculus and applications to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6 (2003), pp. 1-32.
17. Y. Hu, B. ØKSENDAL, AND A. SULEM, Optimal consumption and portfolio in a Black-Scholes market driven by fractional Brownian motion, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6 (2003), pp. 519-536.
18. S. T. HUANG AND S. CAMBANIS, Stochastic and multiple Wiener integrals for Gaussian processes, Ann. Probab., 6 (1978), pp. 585-614.
19. G. KALLIANPUR, The role of reproducing kernel Hubert spaces in the study of Gaussian processes, in Advances in Probability and Related Topics, Vol. 2, Dekker, NY, 1970, pp. 49-83.
20. A. LE BRETON, Filtering and parameter estimation in a simple linear system driven by fractional Brownian motion, Statist. Probab. Lett., 39 (1998), pp. 263-274.
21. M. LOÈVE, Probability Theory, Van Nostrand, New York, 1955.
22. C. MARINELLI, The Stochastic Goodwill Problem, preprint, http://www.arxiv.org/abs/math.OC/0310316 (2003).
23. L. NILAKANTAN, Continuous Time Stochastic Games, Ph.D. dissertation, School of Statistics, University of Minnesota, Minneapolis, MN, 1993.
24. I. NORROS, On the use of fractional Brownian motion in the theory of connectionless networks, IEEE J. on Selected Areas in Communications, 13 (1995), pp. 953-962.
25. C. J. NUZMAN AND H. V. POOR, Linear estimation of self-similar processes via Lamberti's transformation, J. Appl. Probab., 37 (2000), pp. 429-452.
26. C. J. NUZMAN AND H. V. POOR, Reproducing kernel Hilbert space methods for wide-sense self-similar processes, Ann. Appl. Probab., 11 (2001), pp. 1199-1219.
27. B. ØKSENDAL AND K. REIKVAM, Stochastic differential games with controls - discussion of a specific example, in Proceedings of the First Symposium on Mathematical Finance, Gaborone, Botswana, 1997, E. Lungu, ed., pp. 74-82; also available at http://www.math.uio.no/ eprint/pure_math/1998/ pure_1998.html.
28. V. PIPIRAS AND M. S. TAQQU, Integration questions related to fractional Brownian motion, Probab. Theory Related Fields, 118 (2000), pp. 251-291.
29. J. POTHOFF AND M. TIMPEL, On a dual pair of smooth and generalized random variables, Potential Anal., 4 (1995), pp. 637-654.
30. S. G. SAMKO, A. A. KILBAS, AND O. I. MARICHEV, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, Switzerland, 1993.
31. G. SAMORODNITSKY AND M. S. TAQQU, Stable Non-Gaussian Random Processes, Chapman and Hall, London, 1994.
32. P. VARAIYA, N-player stochastic differential games, SIAM J. Control Optim., 14 (1976), pp. 538-545.
33. M. ZÄHLE, Integration with respect to fractal functions and stochastic calculus, Probab. Theory Related Fields, 21 (1998), pp. 333-374.
34. T. S. ZHANG, Characterizations of the white noise test functionals and Hida distributions, Stochastics, Stochastic Rep. 41 (1992), pp. 71-87.