Abstract
The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations in \({\mathcal{R}^p}\) (\({p\in [1,\infty)}\)) and backward stochastic differential equations (BSDEs) in \({\mathcal{R}^p\times \mathcal{H}^p}\) (\({p\in (1, \infty)}\)) and in \({\mathcal{R}^\infty\times\overline{L^\infty}^{\rm BMO}}\) , with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman’s inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear cases for stochastic differential equations (SDEs) and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse Hölder inequality for some suitable exponent p ≥ 1. Finally, some relations are established between Kazamaki’s quadratic critical exponent b(M) of a BMO martingale M and the spectral radius of the stochastic integral operator with respect to M, which lead to a characterization of Kazamaki’s quadratic critical exponent of BMO martingales being infinite.
References
Bañuelos R., Bennett A.G.: Paraproducts and commutators of martingale transforms. Proc. Am. Math. Soc. 103, 1226–1234 (1988)
Barrieu, P., El Karoui, N.: Optimal derivatives design under dynamic risk measures, In: Mathematics of finance. Contemp. Math., vol. 351, pp. 13–25. Amer. Math. Soc., Providence, RI (2004)
Bismut J.M.: Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384–404 (1973)
Bismut J.M.: Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control 14, 419–444 (1976)
Bismut, J.M.: Contrôle des systèmes linéaires quadratiques: applications de l’intégrale stochastique, Séminaire de Probabilités XII. In: Dellacherie, C., Meyer, P.A., Weil, M. (eds.) Lecture Notes in Mathematics, vol. 649, pp. 180–264. Springer, Berlin (1978)
Briand P., Hu Y.: Ying BSDE with quadratic growth and unbounded terminal value. Probab. Theory Relat. Fields 136(4), 604–618 (2006)
Briand P., Hu Y.: Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Relat. Fields 141(3-4), 543–567 (2008)
Buckdahn, R.L.: Backward stochastic differential equations driven by a martingale (preprint, 1993)
Dellacherie, C., Meyer, P.A.: Probabilités et Potentiels. Théorie des Martingales. Hermann (1980)
Delbaen, F., Monat, P., Schachermayer, W., Schweizer, M., Stricker, C.: Inéqualités de normes avec poids et fermeture d’un espace d’intégrales stochastiques. Comptes Rendus Acad. Sci. Paris, vol. 319(I), pp. 1079–1081 (1994)
Delbaen F., Monat P., Schachermayer W., Schweizer M., Stricker C.: Weighted norm inequalities and hedging in incomplete markets. Fin. Stochast. 1, 181–227 (1997)
Doléans-Dade C.: On the existence and unicity of solutions of stochastic differential equations. Z. Wahrscheinlichkeitstheorie verw. Gebiete 36, 93–101 (1976)
Doléans-Dade, C., Meyer, P.A.: Équations différentielles stochastiques. Séminaire de Probabilités XI. Lecture Notes in Mathematics, vol. 581, pp. 376–382. Springer, Berlin (1977)
El Karoui, N., Huang, S.: A general result of existence and uniqueness of backward stochastic differential equations. In: El Karoui, N., Mazliak, A. (eds.) Backward Stochastic Differential Equations, Pitman research Notes in Math. Series, 364, pp. 27–36 (1997)
El Karoui N., Peng S., Quenez M.C.: Backward stochastic differential equations in finance. Math. Fin. 7, 1–71 (1997)
Emery, M.: Sur l’exponentielle d’une martingale de BMO. In: Azéma, J., Yor, M. (eds.) Séminaire de Probabilités XVIII. Lecture Notes in Mathematics, vol. 1059, p. 500. Springer, Berlin (1984)
Emery M.: Stabilité des solutions des équations différentielles stochastiques; applications aux intégrales multiplicatives stochastiques. Z. Wahrscheinlichkeitstheorie verw. Gebiete 41, 241–262 (1978)
Emery, M.: Équations différentielles stochastiques lipschitziennes: étude de la stabilité. Séminaire de Probabilités XIII. Lecture Notes in Mathematics 721, pp. 281–293. Springer, Berlin (1979)
Emery, M.: Le théorème de Garnett-Jones d’après Varopoulos, In: Séminaire de Probabilités XV. In: Azéma, J., Yor, M. (eds.) Université de Strasbourg, Lecture Notes in Mathematics, vol. 721, pp. 278–284. Springer, Berlin (1985)
Föllmer H., Schweizer, M.: Hedging of contingent claims under incomplete information. In: Davis, M.H.A., Elliott, R.J. (eds.) Applied Stochastic Analysis, Stochastics Monograph, pp. 389–414. Gordon and Breach, London (1991)
Garnett J., Jones P.: The distance in BMO to L ∞. Ann. Math. 108, 373–393 (1978)
Grandits, P.: On a conjecture of Kazamaki, Séminaire de Probabilités, XXX, 357–360, Lecture Notes in Mathematics, vol. 1626. Springer, Berlin, (1996)
Grandits, P., Krawczyk, L.: Closedness of some spaces of stochastic integrals, Séminaire de Probabilités, XXXII, Lecture Notes in Mathematics, 1686, pp. 73–85. Springer, Berlin (1998)
Grandits, P.: Some remarks on L ∞, H ∞ and BMO. Séminaire de Probabilités, XXXIII. Lecture Notes in Mathematics, 1709, 342–348. Springer, Berlin (1999)
He S., Wang J., Yan J.: Semimartingale Theory and Stochastic Calculus. Science Press and CRC Press Inc, Beijing (1992)
Hu Y., Imkeller P., Müller M.: Utility maximization in incomplete markets. Ann. Appl. Probab. 15, 1691–1712 (2005)
Hu, Y., Ma, J., Peng, S., Yao, S.: Representation theorems for quadratic \({\mathcal{F}}\) -consistent nonlinear expectations, Prépublication 07-26, April 2007; see also arXiv:0704.1796v1 [math.PR], April 13, 2007
Hu Y., Zhou X.: Constrained stochastic LQ control with random coefficients, and application to portfolio selection. SIAM J. Control Optim. 44, 444–466 (2005)
Itô K.: Differential equations determining Markov processes (in Japanese). Zenkoku Shijõ Sũgaku Danwakai 1077, 1352–1400 (1942)
Itô K.: On a stochastic integral equation. Proc. Imp. Acad. Tokyo 22, 32–35 (1946)
Itô K.: On stochastic differential equations. Mem. Am. Math. Soc. 4, 1–51 (1951)
Kazamaki, N.: Continuous exponential martingales and BMO. Lecture Notes in Mathematics 1579, Springer, Berlin (1994)
Kobylansky M.: Existence and uniqueness results for backward stochastic differential equations when the generator has a quadratic growth. C. R. Acad. Sci. Ser. I Math. 324, 81–86 (1997)
Kobylansky M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28, 558–602 (2000)
Kohlmann, M., Tang, S.: New developments in backward stochastic Riccati equations and their applications. Mathematical Finance (Konstanz, 2000), pp. 194–214. Trends Math., Birkhäuser, Basel (2001)
Kohlmann M., Tang S.: Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging. Stochast. Process. Appl. 97(2), 255–288 (2002)
Kohlmann M., Tang S.: Minimization of risk and linear quadratic optimal control theory. SIAM J. Control Optim. 42(3), 1118–1142 (2003)
Ma, J., Yong, J.: Forward–backward differential equations and their applications. Lecture Notes in Mathematics, vol. 1702. Berlin, Springer (1999)
Monat P., Stricker C.: Décomposition de Föllmer-Schweizer et fermeture de G T (Θ). C. R. Acad. Sci. Sér. I 318, 573–576 (1994)
Monat P., Stricker C.: Föllmer-Schweizer decomposition and mean-variance hedging for general claims. Ann. Probab. 23, 605–628 (1995)
Monat, P., Stricker, C.: Fermeture de G T (Θ) et de \({\mathcal{L}^2(\mathcal{F}_0)+G_T(\Theta)}\) , |Séminaire de Probabilités XXVIII. Lecture Notes in Mathematics 1583, pp. 189–194. Springer, New York (1994)
Pardoux E., Peng S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55–61 (1990)
Protter P.: On the existence, uniqueness, convergence, and explosions of solutions of systems of stochastic differential equations. Ann. Probab. 5, 243–261 (1977)
Protter P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, New York (2004)
Schachermayer, W.: A characterisation of the closure of H ∞ in BMO, Séminaire de Probabilités, XXX. Lecture Notes in Mathematics, 1626, pp. 344–356. Springer, Berlin (1996)
Schweizer M.: Approximating random variables by stochastic integrals. Ann. Probab. 22, 1536–1575 (1994)
Schweizer M.: A projection result for semimartingales. Stochast. Stochast. Rep. 50, 175–183 (1994)
Tang S.: General linear quadratic optimal stochastic control problems with random coefficients: linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim. 42(1), 53–75 (2003)
Varopoulos N. Th.: A probabilistic proof of the Garnett–Jones theorem on BMO. Proc. J. Math. 90, 201–221 (1980)
Yor, M.: Inégalités de martingales continues arrêtées à un temps quelconque, I: théorèmes géneraux, In: Jeulin, Th., Yor, M. (eds.) Grossissements de filtrations: exemples et applications. Lecture Notes in Mathematics 1118, pp. 110–146. Springer, Berlin (1985)
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Part of this work was done when F. Delbaen was visiting China in the years 2005, 2006 and 2007, Laboratory of Mathematics for Nonlinear Sciences, Fudan University, whose hospitality is greatly appreciated. Part of this work was financed by a grant of Credit-Suisse. The paper only reflects the personal opinion of the author. This work is partially supported by the NSFC under grant 10325101 (distinguished youth foundation), the Basic Research Program of China (973 Program) with grant no. 2007CB814904, and the Chang Jiang Scholars Program. Part of this work was completed when S. Tang was visiting in October, 2007, Department of Mathematics, Eidgenössische Technische Hochschule Zürich, whose hospitality is greatly appreciated.
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Delbaen, F., Tang, S. Harmonic analysis of stochastic equations and backward stochastic differential equations. Probab. Theory Relat. Fields 146, 291 (2010). https://doi.org/10.1007/s00440-008-0191-5
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DOI: https://doi.org/10.1007/s00440-008-0191-5
Keywords
- BMO martingales
- Stochastic equations
- Backward stochastic differential equations
- Fefferman’s inequality
- Reverse Hölder inequalities
- Unbounded coefficients
Mathematics Subject Classification (2000)
- Primary 60H10
- 60H20
- 60H99
- Secondary 60G44
- 60G46