Stochastic approximation with averaging innovation applied to Finance

Monte Carlo Methods and Applications

Volumn 18, Issue 1, 2012, Pages 1-51

  Laruelle, Sophie ^a   Pagès, Gilles ^a  

^a University of Paris VII LPSM partially supported by ERC Starting Grant 680275 MALIG   (France)

Author keywords

mixing process; Calibration; Conditional Value at Risk; Ergodic control; G lKoksma theorem; Optimal asset allocation; Quasi Monte Carlo; Sequence with low discrepancy; Stationary process; Stochastic approximation; Two armed bandit algorithm; Value at Risk

Indexed keywords

EID: 84860535634     PISSN: 09299629     EISSN: 15693961     Source Type: Journal    
DOI: 10.1515/mcma-2011-0018     Document Type: Article

References (38)

1. C. Andrieu, É. Moulines and P. Priouret, Stability of stochastic approximation under verifiable conditions, SIAM J. Control Optim. 44 (2005), 283-312.
2. O. Bardou, N. Frikha and G. Pagès, Computing VaR and CVaR using stochastic approximation and adaptive unconstrained importance sampling, Monte Carlo Methods Appl. 15 (2009), 173-210.
3. M. Ben Alaya, Sur la méthode du shift en simulation, in: Probabilités Numériques, Collection Didactique 10, INRIA, Rocquencourt (1992), 61-66.
4. M. Ben Alaya, On the simulation of expectations of random variables depending on a stopping time, Stoch. Anal. Appl. 11 (1993), 133-153.
5. M. Ben Alaya and G. Pagès, Rate of convergence for computing expectations of stopping functionals of an-mixing process, Adv. Appl. Probab. 30 (1998), 425-448. (Pubitemid 128482491)
6. A. Benveniste, M. Métivier and P. Priouret, Adaptive Algorithms and Stochastic Approximations, Translated from the French by Stephen S. Wilson, Applications of Mathematics (New York) 22, Springer-Verlag, Berlin, 1990.
7. N. Bouleau and D. Lépingle, Numerical Methods for Stochastic Processes, First edition, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1994.
8. M.-F. Bru, Wishart processes, J. Theoret. Probab. 4 (1991), 725-751.
9. J. Dedecker, P. Doukhan, G. Lang, J. R. R. León, S. Louhichi and C. Prieur, Weak Dependence: With Examples and Applications, Lecture Notes in Statistics 190, Springer-Verlag, New York, 2007.
10. J. Dedecker, F. Merlevède and D. Volný, On the weak invariance principle for nonadapted sequences under projective criteria, J. Theoret. Probab. 20 (2007), 971-1004.
11. R. Douc, G. Fort, É. Moulines and P. Priouret, Forgetting the initial distribution for hidden Markov models, Stochastic Process. Appl. 119 (2009), 1235-1256.
12. P. Doukhan, Mixing, Lecture Notes in Statistics 85, Springer-Verlag, New York, 1994.
13. M. Duflo, Algorithmes Stochastiques, Mathématiques & Applications (Berlin) 23, Springer-Verlag, Berlin, 1996.
14. H. Föllmer and A. Schied, Stochastic Finance, de Gruyter Studies in Mathematics 27, Walter de Gruyter & Co., Berlin, 2002
15. N. Frikha, Contribution à la modélisation et à la gestion dynamique du risque des marchés de l'énergie, Ph.D. thesis, UPMC, 2010.
16. I. S. Gál and J. F. Koksma, Sur l'ordre de grandeur des fonctions sommables, C. R. Acad. Sci. Paris 227 (1948), 1321-1323.
17. C. Gouriéroux, J. Jasiak and R. Sufana, The Wishart autoregressive process of multivariate stochastic volatility, J. Econometrics 150 (2009), 167-181.
18. H. J. Kushner and D. S. Clark, Stochastic Approximation Methods for Constrained and Unconstrained Systems, Applied Mathematical Sciences 26, Springer-Verlag, New York, 1978.
19. H. J. Kushner and G. G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, Second edition, Applications of Mathematics (New York) 35, Springer-Verlag, New York, 2003.
20. D. Lamberton and G. Pagès, Recursive computation of the invariant distribution of a diffusion, Bernoulli 8 (2002), 367-405. (Pubitemid 44242679)
21. D. Lamberton and G. Pagès, A penalized bandit algorithm, Electron. J. Probab. 13 (2008), no. 13, 341-373.
22. D. Lamberton and G. Pagès, How fast is the bandit?, Stoch. Anal. Appl. 26 (2008), 603-623.
23. D. Lamberton, G. Pagès and P. Tarrès, When can the two-armed bandit algorithm be trusted?, Ann. Appl. Probab. 14 (2004), 1424-1454. (Pubitemid 41449224)
24. B. Lapeyre, G. Pagès and K. Sab, Sequences with low discrepancy-generalisation and application to Robbins-Monro, Statistics 21 (1990), 251-272.
25. S. Laruelle, Analyse d'algorithmes stochastiques appliqués à la Finance, Ph.D. thesis, UPMC, 2011.
26. S. Laruelle, C.-A. Lehalle and G. Pagès, Optimal split of orders across liquidity pools: A stochastic algorithm approach, SIAM J. Financial Math. 2 (2011), 1042-1076.
27. V. Lemaire, Estimation récursive de la mesure invariante d'un processus de diffusion, Ph.D. thesis, Université de Marne-La- Vallée (Paris), 2005.
28. V. Lemaire, An adaptive scheme for the approximation of dissipative systems, Stochastic Process. Appl. 117 (2007), 1491-1518. (Pubitemid 47334858)
29. V. Lemaire and G. Pagès, Unconstrained recursive importance sampling, Ann. Appl. Probab. 20 (2010), 1029-1067.
30. A. Løkka and M. Zervos, A model for the long-term optimal capacity level of an investment project, Int. J. Theor. Appl. Finance 14 (2011), 187-196.
31. K. S. Narendra and I. J. Shapiro, Use of stochastic automata for parameter self-optimization with multimodal performance criteria, IEEE Trans. Syst. Sci. Cybern. 5 (1969), 352-360.
32. H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Regional Conference Series in Applied Mathematics 63, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
33. M. F. Norman, On the linear model with two absorbing barriers, J. Math. Psych. 5 (1968), 225-241.
34. M. Peligrad and S. Utev, Central limit theorem for stationary linear processes, Ann. Probab. 34 (2006), 1608-1622. (Pubitemid 44598882)
35. P. D. Proínov, Discrepancy and integration of continuous functions, J. Approx. Theory 52 (1988), 121-131.
36. H. Robbins and S. Monro, A stochastic approximation method, Ann. Math. Stat. 22 (1951), 400-407.
37. R. T. Rockafellar and S. Uryasev, Conditional value-at-risk: Optimization approach, in: Stochastic Optimization: Algorithms and Applications (Gainesville, FL, 2000), Applied Optimization 54, Kluwer Academic Publishers, Dordrecht (2001), 411-435.
38. P. Tarrès and P. Vandekerkhove, On ergodic two-armed bandits, Ann. Appl. Probab., to appear.