G-Gaussian processes under sublinear expectations and $ q $-Brownian motion in quantum mechanics

Shige Peng

doi:10.3934/naco.2022034

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2023, Volume 13, Issue 3&4: 583-603. Doi: 10.3934/naco.2022034

This issue Previous Article A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection Next Article Approximation of

$ N $

-player stochastic games with singular controls by mean field games

G-Gaussian processes under sublinear expectations and $ q $-Brownian motion in quantum mechanics

Shige Peng

School of Mathematics, Shandong University, 250100 China

Received: February 2022

Revised: October 2022

Early access: November 2022

Published: September 2023

This paper is partially supported by [NSFC1114005132101].

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Abstract

An important observation of this paper is that a non-trivial $ G $-Brownian motion is not a Gaussian process, e.g., finite dimensional distributions of $ G $-Brownian motion is not G-normal, or G-Gaussian. We then have to start from the very beginning, to establish the foundation of $ G $-Gaussian processes which is more suitable for space-parameter systems. It is known that the notion of classical Brownian motion is not suitable to model the random propagation of a quantum particle. In this paper we have rigorously defined a new stochastic process called $ q $-Brownian who's propagator exactly coincides with the one proposed by Feynman based on the solution of Schrödinger equation. The notion of expectation plays a fundamental base of the above results. This paper was originally published on [38].

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Mathematics Subject Classification: Primary: 60A, 60H10, 60H05, 60H30, 60H05, 60H30, 60E05.

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References

[1]	Ph. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228. doi: 10.1111/1467-9965.00068.
[2]	P. Barrieu and N. El Karoui, Pricing, hedging and optimally designing derivatives via minimization of risk measures, Preprint, in Contemporary Mathematics, 2004.
[3]	Z. Chen and L. Epstein, Ambiguity, risk and asset returns in continuous time, Econometrica, 70 (2002), 1403-1443. doi: 10.1111/1468-0262.00337.
[4]	G. Choquet, Theory of capacities, Annales de Institut Fourier, 5 (1953), 131-295.
[5]	P. J. Daniell, A general form of integral, Annals of Mathematics, 19 (1918), 279-294. doi: 10.2307/1967495.
[6]	M. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of The American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[7]	L. Denis, M. Hu and S. Peng, Function spaces and capacity related to a sublinear expectation: application to G-brownian motion pathes, Potential Analysis, 34 (2011), 139-161. doi: 10.1007/s11118-010-9185-x.
[8]	L. Denis and C. Martini, A theoretical framework for the pricing of contingent claims in the presence of model uncertainty, The Annals of Applied Probability, 16 (2006), 827-852. doi: 10.1214/105051606000000169.
[9]	F. Delbaen, E. Rosazza Gianin and S. Peng, Representation of the penalty term of dynamic concave utilities, Finance and Stochastics, 14 (2010), 449-472. doi: 10.1007/s00780-009-0119-7.
[10]	L. Denis and C. Martini, A theoretical framework for the pricing of contingent claims in the presence of model uncertainty, The Ann. of Appl. Probability, 16 (2006), 827-852. doi: 10.1214/105051606000000169.
[11]	L. Epstein and S. Ji, Ambiguous volatility, possibility and utility in continuous time, Journal of Mathematical Economics, 50 (2014), 269-282. doi: 10.1016/j.jmateco.2013.09.005.
[12]	N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equation in finance, Mathematical Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.
[13]	Föllmer and Schied, Statistic Finance, Walter de Gruyter, 2004.
[14]	F. Q. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion, Stochastic Processes and their Applications, 119 (2009), 3356-3382. doi: 10.1016/j.spa.2009.05.010.
[15]	M. Hu, Explicit solutions of the G-heat equation for a class of initial conditions, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 6588-6595. doi: 10.1016/j.na.2012.08.002.
[16]	M. Hu and S. Peng, On representation theorem of g-expectations and paths of G-brownian motion, Acta Mathematicae Applicatae Sinica, English Series, 25 (2009), 539-546. doi: 10.1007/s10255-008-8831-1.
[17]	M. Hu and S. Peng, G-Lévy Processes under sublinear expectations, Probability, Uncertainty and Quantitative Risk, 6 (2021), 1-22. doi: 10.3934/puqr.2021001.
[18]	P. J. Huber, Robust Statistics, John Wiley & Sons, 1981.
[19]	Magali Kervarec, Etude des Modèles non dominé en Mathématiques Financières, , Thèse, Université d'Evry Val d'Esonne, 2008.
[20]	P. Lévy, Processus Stochastic et Mouvement Brownian, Jacques Gabay, 2ème édition, Gautier-Villars, 1965.
[21]	T. Lyons, Uncertain volatility and the risk free synthesis of derivatives, Applied Mathematical Finance, 2 (1995), 117-133.
[22]	C. Miao and B. Zhang, Method of Harmonic Analysis in Partial Differential Equation (Chinese), Science Press, 2008.
[23]	M. Nisio, On a nonlinear semigroup attached to optimal stochastic control, Publ. RIMS, Kyoto Univ., 13 (1976), 513-537. doi: 10.2977/prims/1195190727.
[24]	M. Nisio, On stochastic optimal controls and envelope of Markovian semi–groups., Proc. of Int. Symp. Kyoto, (1976), 297–325.
[25]	E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems and Control Letters, 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.
[26]	S. Peng, Backward SDE and related g–expectation, in Backward Stochastic Differential Equations (ed. El Karoui Mazliak), Pitman Research Notes in Math. Series, No.364, (1997) 141-159.
[27]	S. Peng, BSDE and Stochastic Optimizations, Topics in Stochastic Analysis (eds. J. Yan, S. Peng, S. Fang and L. M. Wu), (Chinese Vers.), Ch.2, Science Press, Beijing, 1997
[28]	S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type, Prob. Theory Rel. Fields, 113 (1999), 473-499. doi: 10.1007/s004400050214.
[29]	S. Peng, Nonlinear expectation, nonlinear evaluations and risk measurs, in Stochastic Methods in Finance Lectures(eds. K. Back T. R. Bielecki, C. Hipp, S. Peng, W. Schachermayer), 143–217, LNM 1856, Springer-Verlag, 2004. doi: 10.1007/978-3-540-44644-6_4.
[30]	S. Peng, Filtration consistent nonlinear expectations and evaluations of contingent claims, Acta Mathematicae Applicatae Sinica, English Series 20 (2004), 1-24. doi: 10.1007/s10255-004-0161-3.
[31]	S. Peng, Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 26B (2005), 159-184. doi: 10.1142/S0252959905000154.
[32]	S. Peng, G–expectation, G–Brownian motion and related stochastic calculus of Itô's type, Stochastic Analysis and Applications, (2006), 541–567. doi: 10.1007/978-3-540-70847-6_25.
[33]	S. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Processes and Their Applications, 118 (2008), 2223-2253. doi: 10.1016/j.spa.2007.10.015.
[34]	S. Peng, Law of large numbers and central limit theorem under nonlinear expectations, Probability, Uncertainty and Quantitative Risk, 4 (2019), Article Number: 4. doi: 10.1186/s41546-019-0038-2.
[35]	S. Peng, G–Brownian motion and dynamic risk measure under volatility uncertainty, arXiv: 0711.2834v1.
[36]	S. Peng, A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1.
[37]	S. Peng, Nonlinear expectations and stochastic calculus under uncertainty —with robust central limit theorem and G-Brownian motion, arXiv: 1002.4546v1.
[38]	S. Peng, G-Gaussian processes under sublinear expectations and $q$-Brownian motion in quantum mechanics, arXiv: 1105.1055.
[39]	Y. Song, Properties of hitting times for G-martingale and their applications, Stochastic Processes and Their Applications, 121 (2011), 1770-1784. doi: 10.1016/j.spa.2011.04.007.
[40]	H. M. Soner, N. Touzi and J. Zhang, Martingale representation theorem for the G-expectation, Stochastic Processes and Their Applications, 121 (2011), 265-287. doi: 10.1016/j.spa.2010.10.006.
[41]	M. Soner, N. Touzi and J. Zhang, Quasi-sure stochastic analysis through aggregation, Electron. J. Probab., 16 (2011), 1844-1879. doi: 10.1214/EJP.v16-950.
[42]	M. Soner, N. Touzi and J. Zhang, Dual formulation of second order target problems, Ann. Appl. Probab., 23 (2013), 308-347. doi: 10.1214/12-AAP844.
[43]	P. Walley, Statistical Reasoning with Imprecise Probabilities, Chapman and Hall, London, 1991.
[44]	J. Xu and B. Zhang, Martingale characterization of G-Brownian motion, Stochastic Processes and their Applications, 119 (2009), 232-248. doi: 10.1016/j.spa.2008.02.001.
[45]	J.-A. Yan, Lecture Note on Measure Theory, Science Press, Beijing (Chinese version), 1998.