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Discrete-type approximations for non-Markovian optimal stopping problems: Part I

Part of: Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  11 December 2019

Dorival Leão ,
Alberto Ohashi  and
Francesco Russo
Dorival Leão*
Affiliation:
Estatcamp
Alberto Ohashi*
Affiliation:
Universidade de Brasília, Universidade Federal da Paraíba
Francesco Russo*
Affiliation:
ENSTA ParisTech
*
* Postal address: Estatcamp. Rua Maestro João Seppe, 900, 13561-180, São Carlos, Brazil.
*** Postal address: Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, Brazil.
***** Postal address: ENSTA ParisTech, Unité de Mathématiques appliquées, 828, Boulevard des Maréchaux, F-91120, Palaiseau, France.
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Abstract

We present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\varepsilon$ -optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent stochastic differential equations driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra et al.

Keywords

Optimal stoppingstochastic optimal controlfractional Brownian motion

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60G22: Fractional processes, including fractional Brownian motion 60H99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 4 , December 2019 , pp. 981 - 1005
Copyright
© Applied Probability Trust 2019 

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References

Agarwal, A., Juneja, S. and Sircar, R. (2016). American options under stochastic volatility: Control variates, maturity randomization and multiscale asymptotics. Quant. Finance 16, 17–30.CrossRefGoogle Scholar
Bally, V. and Pagès, G. (2003). A quantization algorithm for solving discrete time multidimensional optimal stopping problems, Bernoulli 9, 1003–1049.CrossRefGoogle Scholar
Bally, V. and Pagès, G. (2003). Error analysis of the quantization algorithm for obstacle problems. Stoch. Process. Appl. 106, 1–40.CrossRefGoogle Scholar
Bayer, C., Friz, P. and Gatheral, J. (2016). Pricing under rough volatility. Quant. Finance 16, 887–904.CrossRefGoogle Scholar
Belomestny, D. (2013). Solving optimal stopping problems via dual empirical dual optimization. Ann. Appl. Prob. 23, 1988–2019.CrossRefGoogle Scholar
Bertsekas, D. P. and Shreve, S. (1996). Stochastic Optimal Control: The Discrete-Time Case. Athena Scientific, Belmont Massachusetts.Google Scholar
Bezerra, S. C., Ohashi, A., Russo, F. and Souza, F. (2017). Discrete-type approximations for non-Markovian optimal stopping problems: Part II. Preprint, arXiv:1707.05250.Google Scholar
Bonetti, D., Leão, D., Ohashi, A. and Siqueira, V. (2015). A general multidimensional Monte Carlo approach for dynamic hedging under stochastic volatility, Internat. J. Stoch. Anal., Article ID 863165.Google Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion: Facts and Formulae. Birkhauser, Basel.CrossRefGoogle Scholar
Bouchard, B. and Chassagneux, J. F. (2008). Discrete time approximation for continuously and discretely reflected BSDEs. Stoch. Process. Appl. 118, 2269–2293.CrossRefGoogle Scholar
Bouchard, B. and Touzi, N. (2004). Discrete-time approximation and Monte Carlo simulation of backward stochastic differential equations. Stoch. Process. Appl. 111, 175–206.CrossRefGoogle Scholar
Bouchard, B. and Warin, X. (2012). Monte Carlo valuation of American options: Facts and new algorithms to improve existing methods. In Numerical Methods in Finance, eds Carmona, R., Del Moral, P., Hu, P., and Oudjane, N.. Springer, Berlin, pp. 215–255.CrossRefGoogle Scholar
Burq, Z. A. and Jones, O. D. (2008). Simulation of Brownian motion at first-passage times. Math. Comput. Simul. 77, 64–71.CrossRefGoogle Scholar
Carmona, R., Del Moral, P., Hu, P. and Oudjane, N. (eds) (2012). Numerical Methods in Finance. Springer, Berlin.CrossRefGoogle Scholar
Chronopoulou, A. and Viens, F. (2012). Stochastic volatility models with long memory in discrete and continuous time. Quant. Finance 12, 635–649.CrossRefGoogle Scholar
Dellacherie, C. and Meyer, P. A. (1978). Probabilities and Potential. North-Holland Publishing Co., Amsterdam.Google Scholar
Ekren, I. (2017). Viscosity solutions of obstacle problems for fully nonlinear path-dependent PDEs. Stoch. Process. Appl. 127, 3966–3996.CrossRefGoogle Scholar
Fischer, M. and Nappo, G. (2009). On the moments of the modulus of continuity of Itô processes. Stoch. Anal. Appl. 28, 103–122.CrossRefGoogle Scholar
Forde, M. and Zhang, H. (2017). Asymptotics for rough stochastic volatility models. SIAM J. Finan. Math 8, 114–145.CrossRefGoogle Scholar
Fuhrman, M., Pham, H. and Zeni, F. (2016). Representation of non-Markovian optimal stopping problems by constrained BSDEs with a single jump. Electron. Commun. Probab. 21, 3.CrossRefGoogle Scholar
Gatheral, J. (2014). Volatility is rough. Available at SSRN 2509457.CrossRefGoogle Scholar
He, S-w., Wang, J-g. and Yan, J-a. (1992). Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton, FL.Google Scholar
Hu, Y. (2005). Integral transformations and anticipative calculus for fractional Brownian motions. Mem. Amer. Math. Soc 175, 825.Google Scholar
Khoshnevisan, D. and Lewis, T. M. (1999). Stochastic calculus for Brownian motion on a Brownian fracture. Ann. Appl. Probab. 9, 629–667.Google Scholar
Lamberton, D. (2008). Optimal stopping and American options. Daiwa Lecture Series, Kyoto.Google Scholar
Leão, D. and Ohashi, A. (2013). Weak approximations for Wiener functionals. Ann. Appl. Probab. 23, 1660–1691.CrossRefGoogle Scholar
Leão, D. Ohashi, A. and Simas, A. B. (2018). A weak version of path-dependent functional Itô calculus. Ann. Probab. 46, 3399–3441.CrossRefGoogle Scholar
Leão, D., Ohashi, A. and Souza, F. (2018). Stochastic near-optimal controls for path-dependent systems. Preprint, arXiv:1707.04976v2.Google Scholar
Leão, D., Ohashi, A. and Souza, F. Mean variance hedging with rough stochastic volatility. In preparation.Google Scholar
Rambharat, B. R., and Brockwell, A. E. (2010). Sequential Monte Carlo pricing of American-style options under stochastic volatility models. Ann. Appl. Stat. 4, 222–265.CrossRefGoogle Scholar
Rogers, L. C. G. (2002). Monte Carlo valuation of American options. Math. Finance 12, 271–286.CrossRefGoogle Scholar
Schoenmakers, J., Zhang, J. and Huang, J. (2013). Optimal dual martingales, their analysis, and application to new algorithms for Bermudan products. Siam J. F. Math. 4, 86–116.CrossRefGoogle Scholar
Zanger, D. (2018). Convergence of a least-squares Monte Carlo algorithm for American option pricing with depedendent sample data. Math. Finance 28, 447–479.CrossRefGoogle Scholar