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Euler-Lagrange equations for composition functionals in calculus of variations on time scales
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 |
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. |
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. |
[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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