## The Annals of Probability

### Stochastic control for a class of nonlinear kernels and applications

#### Abstract

We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimization, over a set of possibly nondominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are nonlinear generalizations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting, which we then use to provide a semimartingale characterization of the value function. We next explore several applications of our results. We first obtain a wellposedness result for second order BSDEs (as introduced in Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149–190]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a nonlinear optional decomposition in a robust setting, extending recent results of Nutz [Stochastic Process. Appl. 125 (2015) 4543–4555], which we then use to obtain a super-hedging duality in uncertain, incomplete and nonlinear financial markets. Finally, we relate, under additional regularity assumptions, the value function to a viscosity solution of an appropriate path–dependent partial differential equation (PPDE).

#### Article information

Source
Ann. Probab., Volume 46, Number 1 (2018), 551-603.

Dates
Revised: January 2017
First available in Project Euclid: 5 February 2018

https://projecteuclid.org/euclid.aop/1517821229

Digital Object Identifier
doi:10.1214/17-AOP1191

Mathematical Reviews number (MathSciNet)
MR3758737

Zentralblatt MATH identifier
06865129

#### Citation

Possamaï, Dylan; Tan, Xiaolu; Zhou, Chao. Stochastic control for a class of nonlinear kernels and applications. Ann. Probab. 46 (2018), no. 1, 551--603. doi:10.1214/17-AOP1191. https://projecteuclid.org/euclid.aop/1517821229

#### References

• [1] Aïd, R., Possamaï, D. and Touzi, N. (2016). A principal–agent model for pricing electricity volatility demand. Preprint.
• [2] Avellaneda, M., Levy, A. and Parás, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2 73–88.
• [3] Barrieu, P. and El Karoui, N. (2013). Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs. Ann. Probab. 41 1831–1863.
• [4] Bayraktar, E. and Sîrbu, M. (2012). Stochastic Perron’s method and verification without smoothness using viscosity comparison: The linear case. Proc. Amer. Math. Soc. 140 3645–3654.
• [5] Bayraktar, E. and Sîrbu, M. (2013). Stochastic Perron’s method for Hamilton–Jacobi–Bellman equations. SIAM J. Control Optim. 51 4274–4294.
• [6] Bertsekas, D. P. and Shreve, S. E. (1978). Stochastic Optimal Control: The Discrete Time Case, Vol. 23. Academic Press, New York.
• [7] Biagini, S., Bouchard, B., Kardaras, C. and Nutz, M. (2017). Robust fundamental theorem for continuous processes. Math. Finance. 27 963–987.
• [8] Bouchard, B., Moreau, L. and Nutz, M. (2014). Stochastic target games with controlled loss. Ann. Appl. Probab. 24 899–934.
• [9] Bouchard, B. and Nutz, M. (2012). Weak dynamic programming for generalized state constraints. SIAM J. Control Optim. 50 3344–3373.
• [10] Bouchard, B. and Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25 823–859.
• [11] Bouchard, B. and Nutz, M. (2016). Stochastic target games and dynamic programming via regularized viscosity solutions. Math. Oper. Res. 41 109–124.
• [12] Bouchard, B., Possamaï, D. and Tan, X. (2016). A general Doob–Meyer–Mertens decomposition for $g$-supermartingale systems. Electron. J. Probab. 21 1–21.
• [13] Bouchard, B., Possamaï, D., Tan, X. and Zhou, C. (2015). A unified approach to a priori estimates for supersolutions of BSDEs in general filtrations. Ann. Inst. Henri Poincaré B, Probab. Stat. To appear.
• [14] Bouchard, B. and Touzi, N. (2011). Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49 948–962.
• [15] Cheridito, P., Soner, H. M., Touzi, N. and Victoir, N. (2007). Second–order backward stochastic differential equations and fully nonlinear parabolic pdes. Comm. Pure Appl. Math. 60 1081–1110.
• [16] Claisse, J., Talay, D. and Tan, X. (2016). A pseudo–Markov property for controlled diffusion processes. SIAM J. Control Optim. 54 1017–1029.
• [17] Cohen, S. N. (2012). Representing filtration consistent nonlinear expectations as $g$-expectations in general probability spaces. Stochastic Process. Appl. 122 1601–1626.
• [18] Coquet, F., Hu, Y., Mémin, J. and Peng, S. (2002). Filtration–consistent nonlinear expectations and related $g$-expectations. Probab. Theory Related Fields 123 1–27.
• [19] Cvitanić, J., Karatzas, I. and Soner, H. M. (1998). Backward stochastic differential equations with constraints on the gains-process. Ann. Probab. 24 1522–1551.
• [20] Cvitanić, J., Possamaï, D. and Touzi, N. (2014). Moral hazard in dynamic risk management. Manage. Sci. To appear.
• [21] Cvitanić, J., Possamaï, D. and Touzi, N. (2015). Dynamic programming approach to principal–agent problems. Preprint. Available at arXiv:1510.07111.
• [22] Cvitanić, J. and Zhang, J. (2012). Contract Theory in Continuous–Time Models. Springer, Berlin.
• [23] Darling, R. W. R. and Pardoux, É. (1997). Backwards SDE with random terminal time and applications to semilinear elliptic PDE. Ann. Probab. 25 1135–1159.
• [24] Dellacherie, C. (1985). Quelques résultats sur les maisons de jeu analytiques. Séminaire de Probabilités de Strasbourg XIX 222–229.
• [25] Dellacherie, C. and Meyer, P.-A. (1979). Probabilities and Potential. Mathematics Studies 29. North-Holland, Amsterdam.
• [26] Denis, L. and Martini, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 827–852.
• [27] Dolinsky, Y. and Soner, H. M. (2014). Martingale optimal transport and robust hedging in continuous time. Probab. Theory Related Fields 160 391–427.
• [28] Doob, J. L. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Grundlehren der Mathematischen Wissenschaften 262. Springer, Berlin.
• [29] Dumitrescu, R., Quenez, M.-C. and Sulem, A. (2016). A weak dynamic programming principle for combined optimal stopping/stochastic control with ${\mathcal{E}}^{f}$-expectations. SIAM J. Control Optim. 54 2090–2115.
• [30] Ekren, I., Keller, C., Touzi, N. and Zhang, J. (2014). On viscosity solutions of path dependent PDEs. Ann. Probab. 42 204–236.
• [31] Ekren, I., Touzi, N. and Zhang, J. (2016). Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. Ann. Probab. 44 1212–1253.
• [32] Ekren, I., Touzi, N. and Zhang, J. (2016). Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. Ann. Probab. 44 2507–2553.
• [33] El Karoui, N. (1981). Les aspects probabilistes du contrôle stochastique. In École D’été de Probabilités de Saint–Flour IX—1979. Lecture Notes in Math. 876 73–238. Springer, Berlin.
• [34] El Karoui, N. and Huang, S.-J. (1997). A general result of existence and uniqueness of backward stochastic differential equations. In Backward Stochastic Differential Equations (N. El Karoui and L. Mazliak, eds.). Chapman & Hall/CRC Research Notes in Mathematics Series 364 27–36. Longman, Harlow.
• [35] El Karoui, N., Huu Nguyen, D. and Jeanblanc-Picqué, M. (1987). Compactification methods in the control of degenerate diffusions: Existence of an optimal control. Stochastics 20 169–219.
• [36] El Karoui, N., Kapoudjian, C., Pardoux, É., Peng, S. and Quenez, M.-C. (1997). Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 702–737.
• [37] El Karoui, N., Peng, S. and Quenez, M.-C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
• [38] El Karoui, N. and Quenez, M.-C. (1991). Programmation dynamique et évaluation des actifs contingents en marché incomplet. C. R. Acad. Sci., Sér. 1 Math. 313 851–854.
• [39] El Karoui, N. and Quenez, M.-C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 29–66.
• [40] El Karoui, N. and Tan, X. (2013). Capacities, measurable selection and dynamic programming part I: Abstract framework. Preprint. Available at arXiv:1310.3363.
• [41] El Karoui, N. and Tan, X. (2013). Capacities, measurable selection and dynamic programming part II: Application in stochastic control problems. Preprint. Available at arXiv:1310.3364.
• [42] Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25. Springer, New York.
• [43] Föllmer, H. and Kabanov, Yu. M. (1997). Optional decomposition and Lagrange multipliers. Finance Stoch. 2 69–81.
• [44] Föllmer, H. and Kramkov, D. O. (1997). Optional decompositions under constraints. Probab. Theory Related Fields 109 1–25.
• [45] Fremlin, D. H. (2008). Measure Theory, Vol. 5: Set–Theoretic Measure Theory. Torres Fremlin.
• [46] Hamadène, S. and Lepeltier, J.-P. (1995). Backward equations, stochastic control and zero–sum stochastic differential games. Stochastics 54 221–231.
• [47] Hu, M., Ji, S., Peng, S. and Song, Y. (2014). Backward stochastic differential equations driven by ${G}$-Brownian motion. Stochastic Process. Appl. 124 759–784.
• [48] Hu, M., Ji, S., Peng, S. and Song, Y. (2014). Comparison theorem, Feynman–Kac formula and Girsanov transformation for BSDEs driven by ${G}$-Brownian motion. Stochastic Process. Appl. 124 1170–1195.
• [49] Hu, Y., Ma, J., Peng, S. and Yao, S. (2008). Representation theorems for quadratic $f$-consistent nonlinear expectations. Stochastic Process. Appl. 118 1518–1551.
• [50] Jacod, J. and Shiryaev, A. (2013). Limit Theorems for Stochastic Processes, Vol. 288. Springer, Berlin.
• [51] Karandikar, R. L. (1995). On pathwise stochastic integration. Stochastic Process. Appl. 57 11–18.
• [52] Kazi-Tani, N., Possamaï, D. and Zhou, C. (2015). Second–order BSDEs with jumps: Formulation and uniqueness. Ann. Appl. Probab. 25 2867–2908.
• [53] Kazi-Tani, N., Possamaï, D. and Zhou, C. (2015). Second order BSDEs with jumps: Existence and probabilistic representation for fully–nonlinear PIDEs. Electron. J. Probab. 20 1–31.
• [54] Kobylanski, M. (1997). Résultats d’existence et d’unicité pour des équations différentielles stochastiques rétrogrades avec des générateurs à croissance quadratique. C. R. Acad. Sci., Ser. 1 Math. 324 81–86.
• [55] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558–602.
• [56] Kramkov, D. O. (1996). Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Related Fields 105 459–479.
• [57] Larson, P. B. (2009). The filter dichotomy and medial limits. J. Math. Log. 9 159–165.
• [58] Lenglart, É. (1980). Tribus de Meyer et théorie des processus. Séminaire de Probabilités de Strasbourg XIV 500–546.
• [59] Lepeltier, J.-P. and San Martín, J. (1997). Backward stochastic differential equations with continuous coefficient. Statist. Probab. Lett. 32 425–430.
• [60] Lyons, T. J. (1995). Uncertain volatility and the risk–free synthesis of derivatives. Appl. Math. Finance 2 117–133.
• [61] Mastrolia, T. and Possamaï, D. (2015). Moral hazard under ambiguity. Preprint. Available at arXiv:1511.03616.
• [62] Matoussi, A., Piozin, L. and Possamaï, D. (2014). Second–order BSDEs with general reflection and game options under uncertainty. Stochastic Process. Appl. 124 2281–2321.
• [63] Matoussi, A., Possamaï, D. and Zhou, C. (2013). Second order reflected backward stochastic differential equations. Ann. Appl. Probab. 23 2420–2457.
• [64] Matoussi, A., Possamaï, D. and Zhou, C. (2015). Robust utility maximization in nondominated models with 2BSDE: The uncertain volatility model. Math. Finance 25 258–287.
• [65] Neufeld, A. and Nutz, M. (2013). Superreplication under volatility uncertainty for measurable claims. Electron. J. Probab. 18 1–14.
• [66] Neufeld, A. and Nutz, M. (2014). Measurability of semimartingale characteristics with respect to the probability law. Stochastic Process. Appl. 124 3819–3845.
• [67] Neveu, J. (1975). Discrete–Parameter Martingales, Vol. 10. North-Holland, Amsterdam.
• [68] Nutz, M. (2012). Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17 1–7.
• [69] Nutz, M. (2015). Robust superhedging with jumps and diffusion. Stochastic Process. Appl. 125 4543–4555.
• [70] Nutz, M. and Soner, H. M. (2012). Superhedging and dynamic risk measures under volatility uncertainty. SIAM J. Control Optim. 50 2065–2089.
• [71] Nutz, M. and van Handel, R. (2013). Constructing sublinear expectations on path space. Stochastic Process. Appl. 123 3100–3121.
• [72] Nutz, M. and Zhang, J. (2015). Optimal stopping under adverse nonlinear expectation and related games. Ann. Appl. Probab. 25 2503–2534.
• [73] Pardoux, É. (1999). BSDEs, weak convergence and homogenization of semilinear PDEs. In Nonlinear Analysis, Differential Equations and Control (F. H. Clarke, R. J. Stern and G. Sabidussi, eds.). NATO Science Series 528 503–549. Springer, Berlin.
• [74] Peng, S. (1997). Backward SDE and related $g$-expectation. In Backward Stochastic Differential Equations (N. El Karoui and L. Mazliak, eds.). Pitman Research Notes in Mathematics 364 141–160. Longman, Harlow.
• [75] Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Probab. Theory Related Fields 113 473–499.
• [76] Peng, S. (2013). Nonlinear expectation theory and stochastic calculus under Knightian uncertainty. In Real Options, Ambiguity, Risk and Insurance (A. Bensoussan, S. Peng and J. Sung, eds.). Studies in Probability, Optimization and Statistics 5. IOS Press, Amsterdam.
• [77] Possamaï, D. (2013). Second order backward stochastic differential equations under a monotonicity condition. Stochastic Process. Appl. 123 1521–1545.
• [78] Possamaï, D., Royer, G. and Touzi, N. (2013). On the robust superhedging of measurable claims. Electron. Commun. Probab. 18 1–13.
• [79] Possamaï, D. and Zhou, C. (2013). Second order backward stochastic differential equations with quadratic growth. Stochastic Process. Appl. 123 3770–3799.
• [80] Ren, Z., Touzi, N. and Zhang, J. (2014). Comparison of viscosity solutions of semi–linear path–dependent PDEs. Preprint. Available at arXiv:1410.7281.
• [81] Ren, Z., Touzi, N. and Zhang, J. (2014). An overview of viscosity solutions of path–dependent PDEs. In Stochastic Analysis and Applications 2014: In Honour of Terry Lyons (D. Crisan, B. Hambly and T. Zariphopoulou, eds.). Springer Proceedings in Mathematics and Statistics 100 397–453. Springer, Berlin.
• [82] Ren, Z., Touzi, N. and Zhang, J. (2015). Comparison of viscosity solutions of fully nonlinear degenerate parabolic path–dependent PDEs. Preprint. Available at arXiv:1511.05910.
• [83] Soner, H. M., Touzi, N. and Zhang, J. (2011). Martingale representation theorem for the ${G}$-expectation. Stochastic Process. Appl. 121 265–287.
• [84] Soner, H. M., Touzi, N. and Zhang, J. (2012). Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153 149–190.
• [85] Soner, H. M., Touzi, N. and Zhang, J. (2013). Dual formulation of second order target problems. Ann. Appl. Probab. 23 308–347.
• [86] Song, Y. (2011). Some properties on ${G}$-evaluation and its applications to ${G}$-martingale decomposition. Sci. China Math. 54 287–300.
• [87] Stroock, D. W. and Varadhan, S. R. S. (1997). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften 233. Springer, Berlin.