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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

Monica Motta 1, and  Caterina Sartori 2,

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.
Keywords: Nonlinear stochastic control systems, generalized solutions, singular stochastic control, commutative vector fields..
Mathematics Subject Classification: Primary: 49E05, 49N25; Secondary: 93E0.
Citation: Monica Motta, Caterina Sartori. Generalized solutions to nonlinear stochastic differential equations with vector--valued impulsive controls. Discrete & 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.  Google Scholar

[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.  Google Scholar

[3]

A. Bressan, On differential systems with impulsive controls, Rend. Sem. Mat.Univ. Padova, 78 (1987), 227-235.  Google Scholar

[4]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B, 7 (1988), 641-656.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

F. Dufour and B. M. Miller, Singular stochastic control problems, SIAM J. Control Optim., 43 (2004), 708-730. doi: doi:10.1137/S0363012902412719.  Google Scholar

[9]

R. J. Elliott, "Stochastic Calculus and Applications," Applications of Mathematics (New York), 18, Springer-Verlag, New York, 1982.  Google Scholar

[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.  Google Scholar

[11]

S. He, J. Wang and J. Yan, "Semimartingale Theory and Stochastic Calculus," Science Press, New York, 1992.  Google Scholar

[12]

J. Jacod, "Calculus Stochastique et Problémes de Martingales," Lecture notes in Math., 714, Springer-Verlag, Berlin, 1979.  Google Scholar

[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.  Google Scholar

[14]

I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

A. Bressan, On differential systems with impulsive controls, Rend. Sem. Mat.Univ. Padova, 78 (1987), 227-235.  Google Scholar

[4]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B, 7 (1988), 641-656.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

F. Dufour and B. M. Miller, Singular stochastic control problems, SIAM J. Control Optim., 43 (2004), 708-730. doi: doi:10.1137/S0363012902412719.  Google Scholar

[9]

R. J. Elliott, "Stochastic Calculus and Applications," Applications of Mathematics (New York), 18, Springer-Verlag, New York, 1982.  Google Scholar

[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.  Google Scholar

[11]

S. He, J. Wang and J. Yan, "Semimartingale Theory and Stochastic Calculus," Science Press, New York, 1992.  Google Scholar

[12]

J. Jacod, "Calculus Stochastique et Problémes de Martingales," Lecture notes in Math., 714, Springer-Verlag, Berlin, 1979.  Google Scholar

[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.  Google Scholar

[14]

I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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