Ergodic BSDEs with Multiplicative and Degenerate Noise
Pages 2050 - 2077
Abstract
In this paper we study an Ergodic Markovian BSDE involving a forward process $X$ that solves an infinite dimensional forward stochastic evolution equation with multiplicative and possibly degenerate diffusion coefficient. A concavity assumption on the driver allows us to avoid the typical quantitative conditions relating the dissipativity of the forward equation and the Lipschitz constant of the driver. Although the degeneracy of the noise has to be of a suitable type, we can give a stochastic representation of a large class of Ergodic HJB equations; moreover, our general results can be applied to achieve the synthesis of the optimal feedback law in relevant examples of ergodic control problems for SPDEs.
References
[1]
J. P. Aubin, Applied Functional Analysis, 2nd ed., John Wiley & Sons, New York, Chichester, UK, Brisbane, AU, 1979, translated from the French by C. Labrousse, with exercises by B. Cornet and J.-M. Lasry.
[2]
S. N. Cohen and Y. Hu, Ergodic bsdes driven by markov chains, SIAM J. Control Optim., 51 (2013), pp. 4138--4168, https://doi.org/10.1137/120885875.
[3]
A. Cosso, M. Fuhrman, and H. Pham, Long time asymptotics for fully nonlinear Bellman equations: a backward SDE approach, Stochastic Process. Appl., 126 (2016), pp. 1932--1973, https://doi.org/10.1016/j.spa.2015.12.009.
[4]
A. Cosso, G. Guatteri, and G. Tessitore, Ergodic control of infinite-dimensional stochastic differential equations with degenerate noise, ESAIM Control Optim. Calc. Var., 25 (2019), 12, https://doi.org/10.1051/cocv/2018056.
[5]
G. da Prato and J. Zabczyk, Ergodicity for infinite-dimensional Systems, London Math. Soc. Lecture Note Ser. 229, Cambridge University Press, Cambridge, 1996.
[6]
R. W. R. Darling and E. Pardoux, Backwards SDE with random terminal time and applications to semilinear elliptic PDE, Ann. Probab., 25 (1997), pp. 1135--1159, https://doi.org/10.1214/aop/1024404508.
[7]
A. Debussche, Y. Hu, and G. Tessitore, Ergodic BSDEs under weak dissipative assumptions, Stochastic Process. Appl., 121 (2011), pp. 407--426, https://doi.org/10.1016/j.spa.2010.11.009.
[8]
M. Fuhrman, A class of stochastic optimal control problems in Hilbert spaces: BSDEs and optimal control laws, state constraints, conditioned processes, Stochastic Process. Appl., 108 (2003), pp. 263--298, https://doi.org/10.1016/j.spa.2003.09.002.
[9]
M. Fuhrman and G. Tessitore, The Bismut-Elworthy formula for backward SDEs and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces, Stoch. Stoch. Rep., 74 (2002), pp. 429--464, https://doi.org/10.1080/104S1120290024856.
[10]
M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 30 (2002), pp. 1397--1465, https://doi.org/10.1214/aop/1029867132.
[11]
M. Fuhrman and G. Tessitore, Generalized directional gradients, backward stochastic differential equations and mild solutions of semilinear parabolic equations, Appl. Math. Optim., 51 (2005), pp. 279--332, https://doi.org/10.1007/s00245-004-0814-x.
[12]
B. Goldys and B. Maslowsky, Ergodic control of semilinear stochastic equations and the Hamilton-Jacobi equation, J. Math. Anal. Appl., 234 (1999), pp. 592--631, https://doi.org/10.1006/jmaa.1999.6387.
[13]
M. Hu and F. Wang, Ergodic bsdes driven by g-brownian motion and applications, Stochastics and Dynamics, 18 (2018), 1850050, https://doi.org/10.1142/S0219493718500508.
[14]
Y. Hu and F. Lemonnier, Ergodic BSDE with an Unbounded and Multiplicative Underlying Diffusion and Application to Large Time Behavior of Viscosity Solution of HJB Equation, preprint, https://arxiv.org/abs/1801.01284, 2018.
[15]
Y. Hu, P. Y. Madec, and A. Richou, A probabilistic approach to large time behavior of mild solutions of HJB equations in infinite dimension, SIAM J. Control Optim., 53 (2015), pp. 378--398, https://doi.org/10.1137/140976091.
[16]
Y. Hu and G. Tessitore, BSDE on an infinite horizon and elliptic PDEs in infinite dimension, NoDEA Nonlinear Differential Equations Appl., 14 (2007), pp. 825--846, https://doi.org/10.1007/s00030-007-6029-5.
[17]
I. Kharroubi and H. Pham, Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE, Ann. Probab., 43 (2015), pp. 1823--1865, https://doi.org/10.1214/14-AOP920.
[18]
I. Lasiecka and R. Triggiani, Differential and algebraic Riccati equations with application to boundary/point control problems: continuous theory and approximation theory, in Lect. Notes Control Inf. Sci. 164, Springer-Verlag, Berlin, 1991, https://doi.org/10.1007/BFb0006880.
[19]
Y. Hu, M. Fuhrman, and G. Tessitore, Ergodic BSDEs and optimal ergodic control in Banach spaces, SIAM J. Control Optim., 48 (2009), pp. 1542--1566, https://doi.org/10.1137/07069849X.
[20]
Y. Hu, M. Fuhrman, and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, C. R. Math. Acad. Sci. Paris, 350 (2012), pp. 683--688, https://doi.org/10.1016/j.crma.2012.07.009.
[21]
P. Y. Madec, Ergodic bsdes and related pdes with neumann boundary conditions under weak dissipative assumptions, Stochastic Processes and their Applications, 125 (2015), pp. 1821--1860, https://doi.org/10.1016/j.spa.2014.11.015.
[22]
E. J. McShane and R. B. Warfield, Jr., On Filippov's implicit functions lemma, Proc. Amer. Math. Soc., 18 (1967), pp. 41--47.
[23]
E. Pardoux and A. Răşcanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, in Stochastic Modelling and Applied Probability 69, Springer, Cham, 2014, https://doi.org/10.1007/978-3-319-05714-9.
[24]
A. Richou, Ergodic bsdes and related pdes with neumann boundary conditions, Stochastic Process. Appl., 119 (2009), pp. 2945--2969, https://doi.org/10.1016/j.spa.2009.03.005.
[25]
M. Royer, BSDEs with a random terminal time driven by a monotone generator and their links with PDEs, Stoch. Stoch. Rep., 76 (2004), pp. 281--307, https://doi.org/10.1080/10451120410001696270.
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DOI:10.1137/sjcodc.58.4
Issue’s Table of Contents© 2020, Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
United States
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Published: 01 January 2020
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