# American Institute of Mathematical Sciences

April  2011, 29(2): 595-613. doi: 10.3934/dcds.2011.29.595

## Generalized solutions to nonlinear stochastic differential equations with vector--valued impulsive controls

 1 Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63, 35121 Padova, Italy 2 Dipartimento di Metodi e Modelli Matematici, per le Scienze Applicate, Via Trieste, 63, 35121 Padova

Received  September 2009 Revised  March 2010 Published  October 2010

We develop a notion of generalized solution to a stochastic differential equation depending in a nonlinear way on a vector--valued stochastic control process $\{U_t\},$ merely of bounded variation, and on its derivative. Our results rely on the concept of Lipschitz continuous graph completion of $\{U_t\}$ and the generalized solution turns out to coincide a.e. with the limit of classical solutions to (1). In the linear case our notion of solution is equivalent to the usual one in distributional sense. We prove that the generalized solution does not depend on the particular graph-completion of the control process $\{U_t\}$ both for vector-valued controls under a suitable commutativity condition and for scalar controls.
Citation: Monica Motta, Caterina Sartori. Generalized solutions to nonlinear stochastic differential equations with vector--valued impulsive controls. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 595-613. doi: 10.3934/dcds.2011.29.595
##### References:
 [1] L. Alvarez, Singular stochastic control, linear diffusions, and optimal stopping: A class of solvable problems, SIAM J. Control Optim., 39 (2001), 1697-1710. doi: doi:10.1137/S0363012900367825. [2] L. Alvarez, Singular stochastic control in the presence of a state-dependent yield structure, Stochastic Process. Appl., 86 (2000), 323-343. doi: doi:10.1016/S0304-4149(99)00102-7. [3] A. Bressan, On differential systems with impulsive controls, Rend. Sem. Mat.Univ. Padova, 78 (1987), 227-235. [4] A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B, 7 (1988), 641-656. [5] A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields, Jour. of Optim. Theory and Appl., 7 (1991), 67-83. doi: doi:10.1007/BF00940040. [6] J. R. Dorroh, G. Ferreyra and P. Sundar, A technique for stochastic control problems with unbounded control set, Jour. of Theoretical Probability, 12 (1999), 255-270 doi: doi:10.1023/A:1021761030407. [7] F. Dufour and B. M. Miller, Generalized solutions in nonlinear stochastic control problems, SIAM J. Control Optim., 40 (2002), 1724-1745. doi: doi:10.1137/S0363012900374221. [8] F. Dufour and B. M. Miller, Singular stochastic control problems, SIAM J. Control Optim., 43 (2004), 708-730. doi: doi:10.1137/S0363012902412719. [9] R. J. Elliott, "Stochastic Calculus and Applications," Applications of Mathematics (New York), 18, Springer-Verlag, New York, 1982. [10] O. Hájek, Book review: Differential systems involving impulses, Bull. Americ. Math. Soc., 12 (1985), 272-279. doi: doi:10.1090/S0273-0979-1985-15377-7. [11] S. He, J. Wang and J. Yan, "Semimartingale Theory and Stochastic Calculus," Science Press, New York, 1992. [12] J. Jacod, "Calculus Stochastique et Problémes de Martingales," Lecture notes in Math., 714, Springer-Verlag, Berlin, 1979. [13] J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288, Springer-Verlag, Berlin, 2003. [14] I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991. [15] J. M. Lasry and P. L. Lions, Une classe nouvelle de problèmes singuliers de contrôle stochastique, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 879-885. doi: doi:10.1016/S0764-4442(00)01740-7. [16] J. M. Lasry and P. L. Lions, Towards a self-consistent theory of volatility, J. Math. Pures Appl., 86 (2006), 541-551. doi: doi:10.1016/j.matpur.2006.04.006. [17] M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288. [18] M. Motta and C. Sartori, Finite fuel problem in nonlinear singular stochastic control, SIAM J. Control Optim., 46 (2007), 1180-1210. doi: doi:10.1137/050637236. [19] P. Protter, "Stochastic Integration and Differential Equations. Second Edition," in "Applications of Mathematics" (New York), 21; republished as "Stochastic Modelling and Applied Probability," Springer-Verlag, Berlin, 2004. [20] F. Rampazzo, Lie brackets and impulsive controls: An unavoidable connection, in "Differential Geometry and Control" (Boulder, CO, 1997), 279-296, Proc. Sympos. Pure Math., 64, Amer. Math. Soc., Providence, RI, (1999). [21] D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1994. [22] H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. of Probability, 6 (1978), 19-41. doi: doi:10.1214/aop/1176995608. [23] H. J. Sussmann, Lie brackets, real analyticity and geometric control, in "Differential Geometric Control Theory" (Houghton, Mich., (1982), 1-116, Progr. Math., 27, Birkhäuser Boston, Boston, MA, 1983.

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##### References:
 [1] L. Alvarez, Singular stochastic control, linear diffusions, and optimal stopping: A class of solvable problems, SIAM J. Control Optim., 39 (2001), 1697-1710. doi: doi:10.1137/S0363012900367825. [2] L. Alvarez, Singular stochastic control in the presence of a state-dependent yield structure, Stochastic Process. Appl., 86 (2000), 323-343. doi: doi:10.1016/S0304-4149(99)00102-7. [3] A. Bressan, On differential systems with impulsive controls, Rend. Sem. Mat.Univ. Padova, 78 (1987), 227-235. [4] A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B, 7 (1988), 641-656. [5] A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields, Jour. of Optim. Theory and Appl., 7 (1991), 67-83. doi: doi:10.1007/BF00940040. [6] J. R. Dorroh, G. Ferreyra and P. Sundar, A technique for stochastic control problems with unbounded control set, Jour. of Theoretical Probability, 12 (1999), 255-270 doi: doi:10.1023/A:1021761030407. [7] F. Dufour and B. M. Miller, Generalized solutions in nonlinear stochastic control problems, SIAM J. Control Optim., 40 (2002), 1724-1745. doi: doi:10.1137/S0363012900374221. [8] F. Dufour and B. M. Miller, Singular stochastic control problems, SIAM J. Control Optim., 43 (2004), 708-730. doi: doi:10.1137/S0363012902412719. [9] R. J. Elliott, "Stochastic Calculus and Applications," Applications of Mathematics (New York), 18, Springer-Verlag, New York, 1982. [10] O. Hájek, Book review: Differential systems involving impulses, Bull. Americ. Math. Soc., 12 (1985), 272-279. doi: doi:10.1090/S0273-0979-1985-15377-7. [11] S. He, J. Wang and J. Yan, "Semimartingale Theory and Stochastic Calculus," Science Press, New York, 1992. [12] J. Jacod, "Calculus Stochastique et Problémes de Martingales," Lecture notes in Math., 714, Springer-Verlag, Berlin, 1979. [13] J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288, Springer-Verlag, Berlin, 2003. [14] I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991. [15] J. M. Lasry and P. L. Lions, Une classe nouvelle de problèmes singuliers de contrôle stochastique, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 879-885. doi: doi:10.1016/S0764-4442(00)01740-7. [16] J. M. Lasry and P. L. Lions, Towards a self-consistent theory of volatility, J. Math. Pures Appl., 86 (2006), 541-551. doi: doi:10.1016/j.matpur.2006.04.006. [17] M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288. [18] M. Motta and C. Sartori, Finite fuel problem in nonlinear singular stochastic control, SIAM J. Control Optim., 46 (2007), 1180-1210. doi: doi:10.1137/050637236. [19] P. Protter, "Stochastic Integration and Differential Equations. Second Edition," in "Applications of Mathematics" (New York), 21; republished as "Stochastic Modelling and Applied Probability," Springer-Verlag, Berlin, 2004. [20] F. Rampazzo, Lie brackets and impulsive controls: An unavoidable connection, in "Differential Geometry and Control" (Boulder, CO, 1997), 279-296, Proc. Sympos. Pure Math., 64, Amer. Math. Soc., Providence, RI, (1999). [21] D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1994. [22] H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. of Probability, 6 (1978), 19-41. doi: doi:10.1214/aop/1176995608. [23] H. J. Sussmann, Lie brackets, real analyticity and geometric control, in "Differential Geometric Control Theory" (Houghton, Mich., (1982), 1-116, Progr. Math., 27, Birkhäuser Boston, Boston, MA, 1983.
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