A stochastic recursive optimal control problem under the G-expectation framework. (English) Zbl 1308.93225
Summary: In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by \(G\)-Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton-Jacobi-Bellman (HJB) equation in the framework of \(G\)-expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.
MSC:
| 93E20 | Optimal stochastic control |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 35K15 | Initial value problems for second-order parabolic equations |
Keywords:
\(G\)-expectation; backward stochastic differential equations; stochastic optimal control; dynamic programming principle; viscosity solutionReferences:
| [1] | Buckdahn, R., Li, J.: Stochastic differential games and viscosity solutions for Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim. 47, 444-475 (2008) · Zbl 1157.93040 · doi:10.1137/060671954 |
| [2] | Buckdahna, R., Labedb, B., Rainera, C., Tamer, L.: Existence of an optimal control for stochastic control systems with nonlinear cost functional. Stochastics: Int. J. Probab. Stoch. Process. 82, 241-256 (2010) · Zbl 1200.93137 |
| [3] | Denis, L., Martini, C.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16, 827-852 (2006) · Zbl 1142.91034 · doi:10.1214/105051606000000169 |
| [4] | Denis, L., Hu, M., Peng, S.: Function spaces and capacity related to a sublinear expectation: application to \[G\] G-Brownian motion paths. Potential Anal. 34, 139-161 (2011) · Zbl 1225.60057 · doi:10.1007/s11118-010-9185-x |
| [5] | Duffie, D., Epstein, L.: Stochastic differential utility. Econometrica 60, 353-394 (1992) · Zbl 0763.90005 · doi:10.2307/2951600 |
| [6] | El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997) · Zbl 0884.90035 · doi:10.1111/1467-9965.00022 |
| [7] | Epstein, L., Ji, S.: Ambiguous volatility, possibility and utility in continuous time. J. Math. Econ. (2013). http://hdl.handle.net/10.1093/rfs/hht018 · Zbl 1284.91148 |
| [8] | Epstein, L., Ji, S.: Ambiguous volatility and asset pricing in continuous time. Rev. Finan. Stud. 26, 1740-1786 (2013) · doi:10.1093/rfs/hht018 |
| [9] | Fleming, W.H., Soner, H.M.: Control Markov Processes and Viscosity Solutions. Springer, New York (1993) · Zbl 0773.60070 |
| [10] | Hu, M., Ji, S., Peng, S., Song, Y.: Backward stochastic differential equations driven by G-Brownian motion. Stoch. Process. Appl. 124, 759-784 (2014) · Zbl 1300.60074 · doi:10.1016/j.spa.2013.09.010 |
| [11] | Hu, M., Ji, S., Peng, S., Song, Y.: Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion. Stoch. Process. Appl. 124, 1170-1195 (2014) · Zbl 1300.60075 · doi:10.1016/j.spa.2013.10.009 |
| [12] | Hu, M., Peng, S.: On representation theorem of G-expectations and paths of \[G\] G-Brownian motion. Acta Math. Appl. Sin. Engl. Ser. 25, 539-546 (2009) · Zbl 1190.60043 · doi:10.1007/s10255-008-8831-1 |
| [13] | Matoussi, A., Piozin, L., Possamai, D.: Second-order BSDEs with general reflection and Dynkin games under uncertainty (2012). arXiv:1212.0476 · Zbl 1330.60074 |
| [14] | Matoussi, A., Possamai, D., Zhou, C.: Robust utility maximization in non-dominated models with 2BSDE, the uncertain volatility model. Math. Financ. (2012). arXiv:1201.0769v5 · Zbl 1335.91067 |
| [15] | Neufeld, A., Nutz, M.: Superreplication under volatility uncertainty for measurable claims (2012). arXiv:1208.6486v2 · Zbl 1282.91360 |
| [16] | Nutz, M.: Random G-expectations. Ann. Appl. Probab. 23(2013), 1755-1777 (2010) · Zbl 1273.93178 |
| [17] | Nutz, M.: A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17, 1-23 (2012) · Zbl 1244.93176 · doi:10.1214/EJP.v17-1892 |
| [18] | Nutz, M., Zhang, J.: Optimal stopping under adverse nonlinear expectation and related games (2012). arXiv:1212.2140v1 · Zbl 1322.60047 |
| [19] | Pardoux, E., Peng, S.: Adapted solutions of backward stochastic equations. Syst. Control Lett. 14, 55-61 (1990) · Zbl 0692.93064 · doi:10.1016/0167-6911(90)90082-6 |
| [20] | Peng, S.: Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math. Appl. Sin. 20, 1-24 (2004) · Zbl 1061.60063 · doi:10.1007/s10255-004-0161-3 |
| [21] | Peng, S.: Nonlinear expectations and nonlinear Markov chains. Chin. Ann. Math. 26B, 159-184 (2005) · Zbl 1077.60045 · doi:10.1142/S0252959905000154 |
| [22] | Peng, \[S.: G\] G-Expectation, \[G\] G-Brownian Motion and Related Stochastic Calculus of Itô Type, Stochastic Analysis and Applications, Abel Symp., 2, pp. 541-567. Springer, Berlin (2007) · Zbl 1131.60057 |
| [23] | Peng, \[S.: G\] G-Brownian motion and dynamic risk measure under volatility uncertainty (2007). arXiv:0711.2834v1 · Zbl 0884.90035 |
| [24] | Peng, S.: Multi-dimensional \[G\] G-Brownian motion and related stochastic calculus under \[G\] G-expectation. Stoch. Process. Appl. 118, 2223-2253 (2008) · Zbl 1158.60023 · doi:10.1016/j.spa.2007.10.015 |
| [25] | Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty (2010). arXiv:1002.4546v1 |
| [26] | Peng, S.: Backward stochastic differential equation, nonlinear expectation and their applications. In: Proceedings of the International Congress of Mathematicians, Hyderabad (2010) |
| [27] | Peng, S., Song, Y., Zhang, J.: A complete representation theorem for G-martingales (2012). arXiv:1201.2629v1 · Zbl 1337.60130 |
| [28] | Peng, S.: A generalized dynamic programming principle and Hamilton-Jacobi-Bellmen equation. Stoch. Stoch. Rep. 38, 119-134 (1992) · Zbl 0756.49015 · doi:10.1080/17442509208833749 |
| [29] | Peng, S.: Backward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJB equations. In: Yan, J., Peng, S., Fang, S., Wu, L. (eds.) Topics on Stochastic Analysis, Science Press, Beijing, pp. 85-138 (1997) (in Chinese) · Zbl 1252.60056 |
| [30] | Pham, T., Zhang, J.: Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation (2012). arXiv:1209.6605v1 · Zbl 1308.91029 |
| [31] | Wu, Z., Yu, Z.: Dynamic programming principle for one kind of stochastic recursive optimal control problem and Hamilton-Jacobi-Bellman equation. SIAM J. Control Optim. 47, 2616-2641 (2008) · Zbl 1171.49022 · doi:10.1137/060671917 |
| [32] | Soner, H.M., Touzi, N., Zhang, J.: Martingale representation theorem under G-expectation. Stoch. Process. Appl. 121, 265-287 (2011) · Zbl 1228.60070 · doi:10.1016/j.spa.2010.10.006 |
| [33] | Soner, H.M., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16, 1844-1879 (2011) · Zbl 1245.60062 · doi:10.1214/EJP.v16-950 |
| [34] | Soner, H.M., Touzi, N., Zhang, J.: Wellposedness of second order backward SDEs. Probab. Theory Relat. Fields 153, 149-190 (2012) · Zbl 1252.60056 · doi:10.1007/s00440-011-0342-y |
| [35] | Song, Y.: Some properties on G-evaluation and its applications to G-martingale decomposition. Sci. China Math. 54, 287-300 (2011) · Zbl 1225.60058 · doi:10.1007/s11425-010-4162-9 |
| [36] | Song, Y.: Uniqueness of the representation for \[G\] G-martingales with finite variation. Electron. J. Probab. 17, 1-15 (2012) · Zbl 1244.60046 · doi:10.1214/EJP.v17-1890 |
| [37] | Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999) · Zbl 0943.93002 |
| [38] | Zhang, D.F.: Stochastic optimal control problems under G-expectation. Optim. Control Appl. Meth. 34, 96-110 (2013) · Zbl 1273.93180 · doi:10.1002/oca.2012 |
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