Abstract
In this article, we propose a wellposedness theory for a class of second order backward doubly stochastic differential equation (2BDSDE). We prove existence and uniqueness of the solution under a Lipschitz type assumption on the generator, and we investigate the links between the 2BDSDEs and a class of parabolic fully nonlinear Stochastic PDEs. Precisely, we show that the Markovian solution of 2BDSDEs provide a probabilistic interpretation of the classical and stochastic viscosity solution of fully nonlinear SPDEs.
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Notes
Notice that this assumption will be solely used to obtain an aggregated version of the family of processes \(K^\mathbb {P}\) which usually appears in (2.1) instead of the process K. This is cosmetically nicer, but all our other results still hold without this assumption.
References
Aman, A., Mrhardy, N.: Obstacle problem for SPDE with onlinear Neumann boundary condition via reflected generalized backward doubly SDEs. Stat. Probab. Lett. 83(3), 863–874 (2013)
Avellaneda, M., Levy, A., Paras, A.: Pricing and hedging derivative securities in markets with uncertain volatility. Appl. Math. Finance 2(2), 73–88 (1995)
Bachouch, A., Gobet, E., Matoussi, A.: Empirical regression method for backward doubly stochastic differential equations. SIAM/ASA J. Uncertain. Quantif. 4(1), 358–379 (2016)
Bachouch, A., Lasmar, A.B., Matoussi, A., Mnif, M.: Numerical scheme for semilinear SPDEs via backward doubly SDEs. Stoch. Partial Differ. Equ.: Anal. Comput. 1, 1–43 (2016)
Bally, V., Matoussi, A.: Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theor. Probab. 14(1), 125–164 (2001)
Bertsekas, D., Shreve, S.: Stochastic Optimal Control: The Discrete-time Case. Academic Press, New York (1978)
Bichteler, K.: Stochastic integration and \(L^{p}-\)theory of semimartingales. Ann. Probab. 9(1), 49–89 (1981)
Buckdahn, R., Bulla, I., Ma, J.: Pathwise Taylor expansions for Itō random fields. Math. Control Relat. Fields 1(4), 437–468 (2011)
Buckdahn, R., Ma, J.: Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I. Stoch. Process. Appl. 93(2), 181–204 (2001)
Buckdahn, R., Ma, J.: Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II. Stoch. Process. Appl. 93(2), 205–228 (2001)
Buckdahn, R., Ma, J.: Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs. Ann. Probab. 30(3), 1131–1171 (2002)
Buckdahn, R., Ma, J.: Pathwise stochastic control problems and stochastic HJB equations. SIAM J. Control Optim. 45(6), 2224–2256 (2007)
Buckdahn, R., Ma, J., Zhang, J.: Pathwise Taylor expansions for random fields on multiple dimensional paths. Stoch. Process. Appl. 125(7), 2820–2855 (2015)
Buckdahn, R., Ma, J., Zhang, J.: Pathwise viscosity solutions of stochastic PDEs and forward path-dependent PDEs. (2015). arXiv:1501.06978
Caruana, M., Friz, P., Oberhauser, H.: A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Ann. de l’institut Henri Poincaré Anal. Non Linéaire (C) 28(1), 27–46 (2011)
Chen, Z., Peng, S.: A general downcrossing inequality for \(g\)-martingales. Stat. Probab. Lett. 46(2), 169–175 (2000)
Cheridito, P., Soner, H., Touzi, N., Victoir, N.: Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Commun. Pure Appl. Math. 60(7), 1081–1110 (2007)
Crandall, M., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)
Dalang, R., Khoshnevisan, D., Nualart, E.: Hitting probabilities for systems of non-linear stochastic heat equations with additive noise. ALEA Latin Am. J. Probab. Math. Stat. 3, 231–271 (2007)
Dawson, D.: Stochastic evolution equations. Math. Biosci. 15(3), 287–316 (1972)
Dellacherie, C., Meyer, P.: Probabilités et Potentiel, Chapitres XII à XVI, Théorie du Potentiel. Hermann, Paris (1980)
Denis, L., Martini, C.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16(2), 827–852 (2006)
Diehl, J., Friz, P.: Backward stochastic differential equations with rough drivers. Ann. Probab. 40(4), 1715–1758 (2012)
Doob, J.L.: Classical potential theory and its probabilistic counterpart. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1984 edition
El Karoui, N., Hamadène, S., Matoussi, A.: Backward stochastic differential equations and applications. Chapter 8 in the book Indifference Pricing: Theory and Applications, pp. 267–320. Springer, New York (2008)
El Karoui, N., Tan, X.: Capacities, measurable selection and dynamic programming part I: abstract framework. (2013). arXiv:1310.3363
El Karoui, N., Tan, X.: Capacities, measurable selection and dynamic programming part II: application in stochastic control problems. (2013). arXiv:1310.3364
Fremlin, D.H.: Consequences of Martin’s Axiom, vol. 84 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1984)
Friz, P., Gassiat, P., Lions, P.-L., Souganidis, P.: Eikonal equations and pathwise solutions to fully non-linear SPDEs. (2016). arXiv:1602.04746
Friz, P.K., Gassiat, P., Lions, P.-L., Souganidis, P.E.: Eikonal equations and pathwise solutions to fully non-linear spdes. Stoch. Partial Differ. Equ.: Anal. Comput. 5(2), 256–277 (2017)
Gerencsér, M., Gyöngy, I., Krylov, N.: On the solvability of degenerate stochastic partial differential equations in Sobolev spaces. Stoch. Partial Differ. Equ. Anal. Comput. 3(1), 52–83 (2015)
Gubinelli, M., Tindel, S., Torrecilla, I.: Controlled viscosity solutions of fully nonlinear rough PDEs. (2014). arXiv:1403.2832
Gyöngy, I., Krylov, N.: Accelerated finite difference schemes for linear stochastic partial differential equations in the whole space. SIAM J. Math. Anal. 42(5), 2275–2296 (2010)
Gyöngy, I., Krylov, N.: Accelerated numerical schemes for PDEs and SPDEs. In: Stochastic Analysis 2010. Springer, Heidelberg, pp. 131–168 (2011)
Hamadène, S., Ouknine, Y.: Reflected backward sdes with general jumps. Theory Probab. Appl. 60(2), 357–376 (2015)
Ichikawa, A.: Linear stochastic evolution equations in Hilbert space. J. Differ. Equ. 28(2), 266–277 (1978)
Karandikar, R.: On pathwise stochastic integration. Stoch. Process. Appl. 57, 11–18 (1995)
Kazi-Tani, N., Possamaï, D., Zhou, C.: Second order BSDEs with jumps: existence and probabilistic representation for fully-nonlinear PIDEs. Electron. J. Probab. 20 (2015)
Krylov, N., Rozovskiĭ, B.: On the Cauchy problem for linear stochastic partial differential equations. Izv.: Math. 11(6), 1267–1284 (1977)
Krylov, N., Rozovskiĭ, B.: Stochastic evolution equations. J. Soviet Math. 16(4), 1233–1277 (1981)
Kunita, H.: Stochastic Flows and Stochastic Differential Equations, vol. 24 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)
Lin, Y.: A new existence result for second-order BSDEs with quadratic growth and their applications. Stoch.: Int. J. Probab. Stoch. Process. 88(1), 128–146 (2016)
Lions, P.-L., Souganidis, P.E.: Fully nonlinear stochastic partial differential equations: non-smooth equations and applications. Comptes Rendus de l’Acad. des Sci.-Ser. I-Math. 327(8), 735–741 (1998)
Lions, P.-L., Souganidis, P.E.: Fully nonlinear viscosity stochastic partial differential equations: non-smooth equations and applications. CR Acad. Sci. Paris 327(1), 735–741 (1998)
Lions, P.-L., Souganidis, P.E.: Équations aux dérivées partielles stochastiques nonlinéaires et solutions de viscosité. Séminaire équations aux dérivées partielles 1998–1999(1), 1–13 (2000)
Lions, P.-L., Souganidis, P.E.: Viscosity solutions of fully nonlinear stochastic partial differential equations. Sūrikaisekikenkyūsho Kōkyūroku, 1287, 58–65. (2002). Viscosity solutions of differential equations and related topics (Japanese) (Kyoto, 2001)
Lyons, T.J.: Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2, 117–133 (1995)
Ma, J., Wu, Z., Zhang, D., Zhang, J., et al.: On well-posedness of forward-backward sdes: a unified approach. Ann. Appl. Probab. 25(4), 2168–2214 (2015)
Matoussi, A., Sheutzow, M.: Semilinear stochastic PDE’s with nonlinear noise and backward doubly SDE’s. J. Theor. Probab. 15, 1–39 (2002)
Nutz, M.: Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17(24), 1–7 (2012)
Nutz, M.: A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17(23), 1–23 (2012)
Ocone, D., Pardoux, E.: A generalized itô–ventzell formula. application to a class of anticipating stochastic differential equations. Ann. de l’institut Henri Poincaré Probab. et Stat. (B) 25(1), 39–71 (1989)
Pardoux, É.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3(1–4), 127–167 (1980)
Pardoux, É., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)
Pardoux, É., Peng, S.: Backward doubly sde’s and systems of quasilinear spdes. Probab. Theory Relat. Field 98, 209–227 (1994)
Pardoux, É., Protter, P.: A two-sided stochastic integral and its calculus. Probab. Theory Relat. Field 76(1), 15–49 (1987)
Peng, S.: Backward SDE and related \(g-\)expectation. In: El Karoui, N., Mazliak, L. (eds.) Backward Stochastic Differential Equations, vol. 364 of Pitman Research Notes in Mathematics, pp. 141–159. Longman, Harlow (1997)
Peng, S.: Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Probab. Theory Relat. Fields 113(4), 473–499 (1999)
Peng, S., Shi, Y.: A type of time-symmetric forward-backward stochastic differential equations. C.R. Math. 336(9), 773–778 (2003)
Possamaï, D.: Second order backward stochastic differential equations under a monotonicity condition. Stoch. Process. Appl. 123(5), 1521–1545 (2013)
Possamaï, D., Tan, X.: Weak approximation of second-order BSDEs. Ann. Appl. Probab. 25(5), 2535–2562 (2015)
Possamaï, D., Tan, X., Zhou, C.: Stochastic control for a class of non–linear stochastic kernels and applications. (2015). arXiv:1510.08439
Possamaï, D., Zhou, C.: Second order backward stochastic differential equations with quadratic growth. Stoch. Process. Appl. 123(10), 3770–3799 (2013)
Ren, Z., Tan, X.: On the convergence of monotone schemes for path—dependent PDE. (2015). arXiv:1504.01872
Shi, Y., Gu, Y., Liu, K.: Comparison theorems of backward doubly stochastic differential equations and applications. Stoch. Anal. Appl. 23(1), 97–110 (2005)
Soner, H., Touzi, N., Zhang, J.: Martingale representation theorem for the \(G\)-expectation. Stoch. Process. Appl. 121(2), 265–287 (2011)
Soner, H., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67), 1844–1879 (2011)
Soner, H., Touzi, N., Zhang, J.: Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308–347 (2013)
Soner, H.M., Touzi, N., Zhang, J.: Wellposedness of second order backward SDEs. Probab. Theory Relat. Fields 153(1–2), 149–190 (2012)
Stricker, C., Yor, M.: Calcul stochastique dépendant d’un paramètre. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 45(2), 109–133 (1978)
Stroock, D., Varadhan, S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)
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A. MATOUSSI: Research partly supported by the Chair Financial Risks of the Risk Foundation sponsored by Société Générale. The authors research is part of the ANR project CAESARS (ANR-15-CE05-0024) and PANORisk project of Région Pays de la Loire.
W. Sabbagh: The research of Wissal Sabbagh, benefited from the support of the “Chair Markets in Transition”, Fédération Bancaire Française, and of the ANR 11-LABX-0019
Appendices
Appendix A: Technical results
1.1 A.1. Itô and Itô–Ventzell formulae
The following Itô’s formula is a mix between the classical forward and backward Itô’s formulas and is similar to Lemma 1.3 in [55]. We give it here for ease of reference and completeness. The proof being standard, we omit it.
Lemma A.1
Let \(X^1\) and \(X^2\) be defined, for \(i=1,2\), by
for some càdlàg bounded variation and \(\mathbb {G}\)-progressively measurable processes \(K^i\), such that one of them is continuous. We then have
We now give a generalized version of Itô–Ventzell formula that combines the generalized Itô formula of Pardoux and Peng [55] and the Itô–Ventzell formula of Ocone and Pardoux [52].
Lemma A.2
(Generalized Itô–Ventzell formula)
Suppose that \(F\in C^{0,2}(\mathbb {F},[0,T]\times \mathbb {R}^k)\) is a semimartingale with spatial parameter \(x\in \mathbb {R}^k\):
where \(G\in C^{0,2}(\mathbb {F}^B,[0,T]\times \mathbb {R}^k)\), \(H\in C^{0,2}(\mathbb {F}^B,[0,T]\times \mathbb {R}^k;\mathbb {R}^d)\) and \(K\in C^{0,2}(\mathbb {F}^W,[0,T]\times \mathbb {R}^k;\mathbb {R}^l)\). Let \(\phi \in C(\mathbb {F},[0,T];\mathbb {R}^k)\) be a process of the form
where \(\gamma \in \mathbb {H}^2_{k\times d}\), \(\delta \in \mathbb {H}^2_{k\times l}\) and A is a continuous \(\mathbb {F}\)-adapted process with paths of locally bounded variation. Then, \(\mathbb {P}\)-almost surely, it holds for all \(0\le t\le T\) that
1.2 A.2. Proof of Lemma 4.1
We divide the proof in two steps.
Step 1: We start by showing the result in the case where F and g do not depend on (y, z). In this case, we can solve directly the BDSDEs to find that \(\text {for }\mathbb {P}-a.e. \ (\omega ^B,\omega ^W)\in \Omega \)
Then, since \(\xi \) is actually \(\mathcal {F}_T^B\)-measurable, we deduce immediately, using the definition of the r.c.p.d. that for \(\mathbb {P}_0^W\)-a.e. \(\omega ^W\in \Omega ^W\)
Next, we know from the results of Stricker and Yor [70] that we can define a measurable map from \((\Omega ^W\times [0,T],\mathcal {F}_T\otimes \mathcal {B}([0,T]))\) to \((\mathbb {R},\mathcal {B}(\mathbb {R}))\) which coincides \(\mathbb {P}^0_W\otimes dt\)-a.e. with the conditional expectation of \(g_s\), under \(\mathbb {P}^B(\cdot ;\omega ^W)\) (remember that this is a stochastic kernel, and thus measurable), with respect to the \(\sigma \)-algebra \(\mathcal {F}_t^B\). For notational simplicity, we still denote this map as
In other words, the above map does indeed define a stochastic process. That being said, we claim that for \(\mathbb {P}\)-a.e. \(\omega \in \Omega \)
To prove the claim, let us first show it in the case where g is a simple process with the following decomposition
Then, we have by definition of backward stochastic integrals, for \(\mathbb {P}\)-a.e. \((\omega ^B,\omega ^W)\in \Omega \)
Notice next that for \(\mathbb {P}-a.e.\)\((\omega ^B,\omega ^W)\in \Omega \)
Indeed, for any X which is \(\mathcal {G}_t\)-measurable, we have
where we have used the fact that since for every \(\omega ^W\in \Omega ^W\), \(\omega ^B\longmapsto X(\omega ^B,\omega ^W)\) is \(\mathcal {F}_t^B\)-measurable, we have by definition of the conditional expectation that
Hence, we deduce finally that
By a simple density argument, we deduce that the same holds for general processes g. Next, notice that by definition of r.p.c.d., we have for \(\mathbb {P}_0^W\)-a.e. \(\omega ^W\in \Omega ^W\)
By definition of \(\mathbb {P}\), we are exactly saying that the above holds for \(\mathbb {P}\)-a.e. \(\omega \in \Omega \). This finally proves (A.3).
Using similar argument, we show that we also have for \(\mathbb {P}\)-a.e. \(\omega \in \Omega \)
To sum up, we have obtained that for \(\mathbb {P}\)-a.e. \(\omega \in \Omega \)
But, we also have (remember that by the Blumenthal \(0-1\) law \(y_t^{\mathbb {P}^{t,\omega ^B}_B(\cdot )\otimes \mathbb {P}^0_W,t,\omega ^B}\) only depends on \(\omega ^W\)) for any \(\omega ^B\in \Omega ^B\) and for \(\mathbb {P}_0^W\)-a.e. \(\omega ^W\in \Omega ^W\)
Using the same arguments as above, we obtain
which proves the desired result.
Step 2: Since we are in a Lipschitz setting, solutions to BDSDEs can be constructed via Picard iterations. Hence, using Step 1, the results holds at each step of the iteration and therefore also when passing to the limit. We emphasize that this step crucially relies on (4.1). \(\square \)
1.3 A.3 Proof of Lemma 4.4
For each \(\mathbb {P}\in \mathcal {P}\), let \((\overline{\mathcal {Y}}^\mathbb {P}(T,\xi ), \overline{\mathcal {Z}}^\mathbb {P}(T,\xi ))\) be the solution of the BDSDE with generators \(\widehat{F}\) and g, and terminal condition \(\xi \) at time T. We define \(\widetilde{V}^\mathbb {P}:= V- \overline{\mathcal {Y}}^\mathbb {P}(T,\xi )\). Then, \(\widetilde{V}^\mathbb {P}\ge 0, \ \mathbb {P}- a.s.\) For any \(0\le t_1\le t_2\le T\), let \((y^{\mathbb {P},t_2},z^{\mathbb {P},t_2}):= (\mathcal {Y}^\mathbb {P}(t_2,V_{t_2}), \mathcal {Z}^\mathbb {P}(t_2,V_{t_2}))\). Note that
Then by the dynamic programming principle (Theorem 4.1) we get
Denote \( \widetilde{y}^{\mathbb {P},t_2}_t:= y^{\mathbb {P},t_2}_{t}- \overline{\mathcal {Y}}^\mathbb {P}_t ,~ \widetilde{z}^{\mathbb {P},t_2}_t:= \widehat{a}_t^{-1/2}(z^{\mathbb {P},t_2}_{t}- \overline{\mathcal {Z}}^\mathbb {P}_t).\) Then \((\widetilde{y}^{\mathbb {P},t_2},\widetilde{z}^{\mathbb {P},t_2})\) is solution of the following BDSDE on \([0,t_2]\)
where
Then \(\widetilde{V}^\mathbb {P}_{t_1}\ge \widetilde{y}^{\mathbb {P},t_2}_{t_1}\). Therefore, \(\widetilde{V}^\mathbb {P}\) is a positive weak doubly \(f^\mathbb {P}\)-super-martingale under \(\mathbb {P}\) by Definition B.2 (given below in the Appendix).
Now, we assume that the coefficient g does not depend in (y, z), then obviously we have that \(\bar{V}^\mathbb {P}_{t_1} \ge \bar{y}^{\mathbb {P},t_2}_{t_1}\) where
Thanks to this change of variable, we have that \((\bar{y}^{\mathbb {P},t_2},\bar{z}^{\mathbb {P},t_2})\) solves the following standard BSDE on \([0,t_2]\)
where
Now applying the down-crossing inequality for f-martingale Theorem 6 in [16] combined with the result concerning the classical down-crossing inequality for non necessarily positive super-martingales in [24] (chapter III, p. 446), we deduce that for \(\mathbb {P}- a.e.\)\(\omega \), the limit \(\displaystyle \lim \nolimits _{r\in \mathbb {Q}\cap (t,T],r\downarrow t} \, \bar{V}^\mathbb {P}_r\), and consequently the limit \(\displaystyle \lim \nolimits _{r\in \mathbb {Q}\cap (t,T],r\downarrow t}\, \widetilde{V}^\mathbb {P}_r\) exists for all \(t\in [0,T]\). Note that \(y^\mathbb {P}\) is continuous, \(\mathbb {P}-a.s.\), and obviously \(\bar{y}^\mathbb {P}\) is continuous, \(\mathbb {P}-a.s\). Therefore, we get that the \(\overline{\displaystyle \lim }\) in the definition of \(V^+\) is in fact a true limit, which implies that
and thus \(V^{+}\) is càdlàg \(\mathcal {P}-q.s.\) Finally, we can prove the general case when g depend in (y, z) using classically the Banach fixed point theorem. \(\square \)
Appendix B: Doubly f-supersolution and martingales
In this section, we extend some of the results of Peng [58] concerning f-super-solutions of BSDEs to the case of BDSDEs. In the following, we fix a probability measure \(\mathbb {P}\in \mathcal {P}\) and work implicitly with \(\overline{\mathbb {F}^B}^\mathbb {P}\) and \(\mathbb {F}^W\). We introduce the following spaces for a fixed probability \(\mathbb {P}\).
-
\(L^{2}(\mathbb {P})\) denotes the space of all \(\mathcal {F}_T\)-measurable scalar r.v. \(\xi \) with \(\Vert \xi \Vert ^2_{L^{2}}:= \mathbb {E}^{\mathbb {P}}[|\xi |^2] < +\infty \).
-
\(\mathbb {D}^{2}(\mathbb {P})\) denotes the space of \(\mathbb {R}\)-valued processes Y, s.t. \(Y_t\) is \(\mathcal {F}_{t}\) measurable for every \(t\in [0,T]\), with \( \text { c}\grave{{\mathrm{a}}}\text {dl}\grave{{\mathrm{a}}}\text {g paths, and } \ \Vert Y\Vert ^2_{\mathbb {D}^{2}(\mathbb {P})}:= \mathbb {E}^{\mathbb {P}}\left[ \displaystyle \sup \nolimits _{0\le t\le T} \,|Y_t|^2 \right] < +\infty \).
-
\(\mathbb {H}^{2}(\mathbb {P})\) denotes the space of all \(\mathbb {R}^d\)-valued processes Z s.t. \(Z_t\) is \(\mathcal {F}_{t}\) measurable for a.e. \(t\in [0,T]\), with
$$\begin{aligned} \Vert Z\Vert ^2_{\mathbb {H}^{2}(\mathbb {P})}:= \mathbb {E}^{\mathbb {P}}\left[ \left( \displaystyle \int _0^T \Vert \widehat{a}^{1/2}_tZ_t\Vert ^2 dt\right) \right] < +\infty . \end{aligned}$$
Let us be given the following objects
- (i) :
-
a terminal condition \(\xi \) which is \(\mathcal {F}_T\)-measurable and in \(L^2(\mathbb {P})\).
- (ii) :
-
two maps \(f:\Omega \times \mathbb {R}\times \mathbb {R}^d\rightarrow \mathbb {R},~ g:\Omega \times \mathbb {R}\times \mathbb {R}^d\rightarrow \mathbb {R}^l\) verifying
-
\(\mathbb {E}\left[ \displaystyle \int _0^T |f(t,0,0)|^2dt\right] < +\infty \), and \(\mathbb {E}\left[ \displaystyle \int _0^T\Vert g(t,0,0)\Vert ^2dt\right] < +\infty .\)
-
There exist \((\mu ,\alpha )\in \mathbb {R}_+^*\times (0,1)\) s.t. for any \((\omega ,t,y_1,y_2,z_1,z_2)\in \Omega \times [0,T]\times \mathbb {R}^2\times (\mathbb {R}^d)^2\)
$$\begin{aligned} |f(t,\omega , y_1,z_1)-f(t,\omega ,y_2,z_2)|&\le \mu \big (|y_1-y_2|+\Vert z_1-z_2\Vert \big ),\\ \Vert g(t,\omega ,y_1,z_1)-g(t,\omega ,y_2,z_2)\Vert ^2&\le c|y_1-y_2|^2+\alpha \Vert z_1-z_2\Vert ^2. \end{aligned}$$
- (iii) :
-
a real-valued càdlàg, progressively measurable process \(\{V_t, 0\le t\le T\}\) with
$$\begin{aligned} \mathbb {E}\left[ \underset{0\le t\le T}{\displaystyle \sup }|V_t|^2\right] < +\infty . \end{aligned}$$
We want to study the following problem: to find a pair of processes \((y,z)\in \mathbb {D}^2(\mathbb {P}) \times \mathbb {H}^2(\mathbb {P})\) satisfying
We have the following existence and uniqueness theorem
Proposition B.1
Under the above hypothesis there exists a unique pair of processes \((y,z)\in \mathbb {D}^2(\mathbb {P}) \times \mathbb {H}^2(\mathbb {P})\) solution of BDSDE (B.1).
Proof
In the case where \(V\equiv 0\), the proof can be found in [55]. Otherwise, we can make the change of variable \(\overline{y}_t:= y_t+V_t\) and treat the equivalent BDSDE
\(\square \)
We also have a comparison theorem in this context
Proposition B.2
Let \(\xi _1\) and \(\xi _2\in L^2(\mathbb {P}), V^i, i=1,2\) be two adapted càdlàg processes and \(f_s^i(y,z), g_s^i(y,z)\) four functions verifying the above assumption. Let \((y^i,z^i)\in \mathbb {D}^2(\mathbb {P}) \times \mathbb {H}^2(\mathbb {P})\), \(i=1,2\) be the solution of the following BDSDEs
respectively. If we have \(\mathbb {P}-a.s.\) that \(\xi _1\ge \xi _2 , V^1-V^2\) is non decreasing, and \(f_s^1(y_s^1,z_s^1)\ge f_s^2(y_s^1,z_s^1)\) then it holds that for all \(t\in [0,T]\)
For a given \(\mathbb {G}\)-stopping time, we now consider the following BDSDE
where \(\xi \in L^2(\mathbb {P})\) and \(V\in \mathbb {I}^2(\mathbb {P})\).
Definition B.1
If y is a solution of BDSDE of form (B.3), the we call y a doubly f-super-solution on \([0,\tau ]\). If \(V\equiv 0\) in \([0,\tau ]\), then we call y a doubly f-solution.
We now introduce the notion of doubly f-(super)martingales.
Definition B.2
-
(i)
A doubly f-martingale on [0, T] is a doubly f-solution on [0, T].
-
(ii)
A process \((Y_t)\) is a doubly f-super-martingale in the strong (resp. weak) sense if for all stopping time \(\tau \le t\) (resp. all \(t\le T)\), we have \(\mathbb {E}^\mathbb {P}[|Y_{\tau }|^2]< +\infty \) (resp. \(\mathbb {E}^\mathbb {P}[|Y_{t}|^2]< +\infty )\) and if the doubly f-solution \((y_s)\) on \([0,\tau ]\) (resp. [0, t]) with terminal condition \(Y_{\tau }\) (resp. \(Y_{t})\) verifies \(y_{\sigma }\le Y_{\sigma }\) for every stopping time \(\sigma \le \tau \) (resp. \(y_s\le Y_s\) for every \(s\le t)\).
Appendix C: Reflected backward doubly stochastic differential equations
In this section, we want to study the problem of a reflected backward doubly stochastic differential equation (RBDSDE for short) with one càdlàg barrier. This is an extension of the work of Hamadène and Ouknine [35] for the standard reflected BSDEs to our case. So in addition to the terminal condition and generators that we used in the previous section, we need
- (iv) :
-
a barrier \(\{S_t, 0\le t\le T\}\), which is a real-valued càdlàg \(\mathcal {F}_t\)-measurable process satisfying \(S_T\le \xi \) and
$$\begin{aligned} \mathbb {E}\left[ \underset{0\le t\le T}{\displaystyle \sup }(S_t^+)^2\right] < +\infty . \end{aligned}$$
Now we present the definition of the solution of RBDSDEs with one lower barrier.
Definition C.1
We call (Y, Z, K) a solution of the backward doubly stochastic differential equation with one reflecting lower barrier S(.), terminal condition \(\xi \) and coefficients f and g, if the following holds:
-
(i)
\(Y\in \mathbb {D}^2(\mathbb {P}),~ Z\in \mathbb {H}^2(\mathbb {P})\).
-
(ii)
\(Y_t = \xi +\displaystyle \int _t^T f(s,Y_s,Z_s)ds +\displaystyle \int _t^T g(s,Y_s,Z_s)\cdot d\overleftarrow{W}_s -\displaystyle \int _t^T Z_s\cdot dB_s +K_T-K_t ,~ 0\le t\le T\).
-
(iii)
\(Y_t\ge S_t ~ ,~ 0\le t\le T,~ a.s.\)
-
(iv)
If \(K^c\) (resp. \(K^d)\) is the continuous (resp. purely discontinuous) part of K, then
$$\begin{aligned} \displaystyle \int _0^T (Y_{s}-S_{s})dK^c_s = 0 ,~ a.s.\ \text {and }\forall t\le T,\ \Delta K_t^d = (S_{t^-}-Y_t)^+\mathbf {1}_{[Y_{t^-}=S_{t^-}]}. \end{aligned}$$
Remark C.1
The condition (iv) implies in particular that \(\displaystyle \int _0^T (Y_{s^-}-S_{s^-})dK_s = 0.\) Actually
The last term of the second equality is null since \(K^d\) jumps only when \(Y_{s^-}=S_{s^-}\). \(\square \)
The main objective of this section is to prove the following theorem.
Theorem C.1
Under the above hypotheses, the RBDSDE in Definition C.1 has a unique solution (Y, Z, K).
Before we start proving this theorem, let us establish the same result in the case where f and g do not depend on y and z. More precisely, given f and g such that
and \(\xi \) as above, consider the reflected BDSDE
Proposition C.1
There exists a unique triplet (Y, Z, K) verifies conditions of Definition C.1 and satisfies (C.1).
Proof
a) Existence: The method combines penalization and the Snell envelope method. For each \(n\in \mathbb {N}^*\), we set
and consider the BDSDE
It is well known (see Pardoux and Peng [55]) that BDSDE (C.2) has a unique solution \((Y^n,Z^n)\in \mathbb {D}^2(\mathbb {P})\times \mathbb {H}^2(\mathbb {P})\) such that for each \(n\in \mathbb {N},\)
From now on the proof will be divided into three steps.
Step 1: For all \(n\ge 0\) and \((s,y)\in [0,T]\times \mathbb {R}\),
which provide by the comparison theorem, \(Y_t^n\le Y_t^{n+1},~ t\in [0,T]~ a.s.\) For each \(n\in \mathbb {N},\) denoting
we have
The process \(\bar{Y}_t^n\) satisfies
In fact, for any \(n\in \mathbb {N}\) and \(t\le T\) we have
Therefore for any \(\mathbb {G}\)-stopping time \(\tau \ge t\) we have
since \(\bar{Y}_{\tau }^n\ge (\bar{S}_{\tau }\wedge \bar{Y}_{\tau }^n){\mathbf{1}}_{[\tau < T]}+ \bar{\xi }{\mathbf{1}}_{\{\tau = T\}}\). On the other hand, let \(\tau _t^{*}\) be the stopping time defined as follows:
where \(\bar{K}_t^n=n \displaystyle \int _0^{t} (\bar{Y}_s^n-\bar{S}_s )^-ds\). Let us show that \({\mathbf{1}}_{[\tau _t^{*}< T]}\bar{Y}_{\tau _t^{*}}^n)= (\bar{S}_{\tau _t^{*}}\wedge \bar{Y}_{\tau _t^{*}}^n){\mathbf{1}}_{[\tau _t^{*} < T]}\).
Let \(\omega \) be fixed such that \(\tau _t^{*}(\omega ) < T\). Then there exists a sequence \((t_k)_{k\ge 0}\) of real numbers which decreases to \(\tau _t^{*}(\omega )\) such that \(\bar{Y}_{t_k}^n(\omega )\le \bar{S}_{t_k}(\omega )\). As \(\bar{Y}^n\) and \(\bar{S}\) are RCLL processes then taking the limit as \(k\rightarrow \infty \) we obtain \(\bar{Y}_{\tau _t^{*}}^n\le \bar{S}_{\tau _t^{*}}\) which implies \({\mathbf{1}}_{[\tau _t^{*}< T]}\bar{Y}_{\tau _t^{*}}^n)= (\bar{S}_{\tau _t^{*}}\wedge \bar{Y}_{\tau _t^{*}}^n){\mathbf{1}}_{[\tau _t^{*} < T]}\). Now from (C.5), we deduce that:
Taking the conditional expectation and using inequality (C.6) we obtain: \(\forall n\ge 0\), and \(t\ge T\)
Step 2: There exists a RCLL \((Y_t)_{t\le T}\) of \(\mathbb {D}^2(\mathbb {P})\) such that \(\mathbb {P}-a.s.\)
-
(i)
\(Y=\underset{n \rightarrow \infty }{\displaystyle \lim } Y^n\) in \(\mathbb {H}^2(\mathbb {P})\), \(S\le Y\).
-
(ii)
for any \(t\le T,\)
$$\begin{aligned} Y_t= \underset{\tau \ge t}{{\mathrm{ess}}\,{\mathrm{sup}}}\ \mathbb {E}\left[ \displaystyle \int _t^\tau f(s)ds + \bar{S}_\tau {\mathbf{1}}_{\{\tau < T\}}+ \bar{\xi }{\mathbf{1}}_{\{\tau =T\}}|\mathcal {G}_t\right] - \displaystyle \int _0^t g(s)\cdot d\overleftarrow{W}_s. \end{aligned}$$(C.8)
Actually for \(t\le T\) let us set
since \(\bar{S}\in \mathbb {D}^2(\mathbb {P})\), \(f\in \mathbb {H}^2(\mathbb {P})\) and \(\bar{\xi }\) is square integrable, the process \(\tilde{Y}\) belongs to \(\mathbb {D}^2(\mathbb {P})\). On the other hand for any \(n\ge 0\) and \(t\le T\) we have \(\bar{Y}_t^n\le \tilde{Y}_t\). Thus there exist a \(\mathbb {G}\)-progressively measurable process \(\bar{Y}\) such that \(\mathbb {P}-a.s.\), for any \(t\le T, \ \bar{Y}_t^n\ \nearrow \bar{Y}_t\le \tilde{Y}_t \) and we have \(Y_t^n\nearrow Y_t= \bar{Y}_t-\displaystyle \int _0^t g(s)\cdot d\overleftarrow{W}_s\), then \(Y=\underset{n \rightarrow \infty }{\displaystyle \lim } Y^n\) in \(\mathbb {H}^2(\mathbb {P}).\)
Besides, the process \(\bar{Y}_\cdot ^n+\displaystyle \int _0^\cdot f(s)ds\) is a càdlàg super-martingale as the Snell envelope of
and it converges increasingly to \(\bar{Y}_\cdot +\int _0^\cdot f(s)ds\). It follows that the latter process is a càdlàg super-martingale. Hence, the process Y is also \(\mathbb {G}\)-progressively measurable, càdlàg, and belongs to \(\mathbb {D}^2(\mathbb {P})\). Even more than that, \(Y_t\) is \(\mathcal {F}_t\)-measurable for every \(t\in [0,T]\) as the limit of \(Y_t^n\), which has this property.
Next let us prove that \(Y\ge S\). We have
Dividing the two sides by n and taking the limit as \(n\rightarrow \infty \), we obtain
Since the processes Y and S are càdlàg, then, \(\mathbb {P}-a.s.\), \(Y_t\ge S_t, \) for \(t< T\). But \(Y_T=\xi \ge S_T\), therefore \(Y\ge S\).
Finally let us show that Y satisfies (C.8). But this is a direct consequence of the continuity of the Snell envelope through sequences of increasing càdlàg processes. In fact on the one hand, the sequence of increasing càdlàg processes \(((\bar{S}_{t}\wedge \bar{Y}_{t}^n){\mathbf{1}}_{[t < T]}+ \bar{\xi }{\mathbf{1}}_{\{t = T\}})_{t\le T})_{t\le T}\) converges increasingly to the càdlàg process \((\bar{S}_{t}{\mathbf{1}}_{[t < T]}+ \bar{\xi }{\mathbf{1}}_{\{t = T\}})_{t\le T}){[t\le T}\) since \(\bar{Y}_{t}\ge \bar{S}_{t}\). Therefore,
which implies that
Step 3: We know from (C.8) that the process \(\int _0^\cdot f(s)ds+\bar{Y}_{\cdot }^n\) is a Snell envelope. Then, there exist a process \(K\in \mathbb {I}^2(\mathbb {P})\) and a \(\mathbb {G}\)-martingale such that
Additionally \(K=K^c+K^d\) where \(K^c\) is continuous, non-decreasing and \(K^d\) non-decreasing purely discontinuous predictable such that for any \(t\le T, \Delta _t K^d= (S_{t^-}-Y_t){\mathbf{1}}_{\{Y_{t^-}=S_{t^-}\}}\). Now the martingale M belongs to \(\mathbb {D}^2(\mathbb {P})\), so that the Itô’s martingale representation theorem implies the existence of a \(\mathbb {G}\)-predictable process \(Z\in \mathbb {H}^2(\mathbb {P})\) such that
Hence
The proof of \(\displaystyle \int _0^T (Y_{s}-S_{s})dK^c_s = 0\) is the same as in [35], so we omit it.
It remains to show that \({Z_t}\) and \({K_t}\) are in fact \(\mathcal {F}_t\)-measurable. For \(K_t\), it is obvious since it is the limit of \(K_t^n= \displaystyle \int _0^t n(Y_s^n-S_s)^- ds\) which is \(\mathcal {F}_t\)-measurable for each \(t\le T\). Now
and the right side is \(\mathcal {F}_T^B\vee \mathcal {F}_{t,T}^{W}\)-measurable. Hence from the Itô’s martingale representation theorem \((Z_s)_{t\le s\le T}\) is \(\mathcal {F}_s^B\vee \mathcal {F}_{t,T}^{W}\) adapted. Consequently \(Z_s\) is \(\mathcal {F}_s^B\vee \mathcal {F}_{t,T}^{W}\)-measurable for any \(t<s\), so it is \(\mathcal {F}_s^B\vee \mathcal {F}_{s,T}^{W}\)-measurable.
b) Uniqueness: Under Lipschitz continuous conditions, the proof of uniqueness is standard in BSDE theory (see e.g. proof of Proposition 2.1. in [1]). \(\square \)
The existence of solution of RBDSDE in Theorem C.1 is obtained via a standard fixed Banach point theorem for reflected BSDEs (see for instance El Karoui, Hamadène and Matoussi [25]).
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Matoussi, A., Possamaï, D. & Sabbagh, W. Probabilistic interpretation for solutions of fully nonlinear stochastic PDEs. Probab. Theory Relat. Fields 174, 177–233 (2019). https://doi.org/10.1007/s00440-018-0859-4
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DOI: https://doi.org/10.1007/s00440-018-0859-4
Mathematics Subject Classification
- Primary 60H15
- 60G46
- Secondary 35H60