Mean-Field Maximum Principle for Optimal Control of Forward–Backward Stochastic Systems with Jumps and its Application to Mean-Variance Portfolio Problem
Mokhtar Hafayed , Moufida Tabet , Samira Boukaf
Communications in Mathematics and Statistics ›› 2015, Vol. 3 ›› Issue (2) : 163 -186.
Mean-Field Maximum Principle for Optimal Control of Forward–Backward Stochastic Systems with Jumps and its Application to Mean-Variance Portfolio Problem
We study mean-field type optimal stochastic control problem for systems governed by mean-field controlled forward–backward stochastic differential equations with jump processes, in which the coefficients depend on the marginal law of the state process through its expected value. The control variable is allowed to enter both diffusion and jump coefficients. Moreover, the cost functional is also of mean-field type. Necessary conditions for optimal control for these systems in the form of maximum principle are established by means of convex perturbation techniques. As an application, time-inconsistent mean-variance portfolio selection mixed with a recursive utility functional optimization problem is discussed to illustrate the theoretical results.
Mean-field forward–backward stochastic differential equation with jumps / Optimal stochastic control / Mean-field maximum principle / Mean-variance portfolio selection with recursive utility functional / Time-inconsistent control problem
| [1] |
Kac, M.: Foundations of kinetic theory. In: Berkeley Symposium on Mathematical Statistics and Probability, vol. 3, 171–197 (1956) |
| [2] |
|
| [3] |
|
| [4] |
|
| [5] |
Ni, Y.H., Zhang, J.F., Li, X.: Indefinite mean-field stochastic linear-quadratic optimal control. IEEE Trans. Autom. Control (2014). doi:10.1109/TAC.2014.2385253 |
| [6] |
|
| [7] |
|
| [8] |
|
| [9] |
Shi, J.: Sufficient conditions of optimality for mean-field stochastic control problems. In: 12th International Conference on Control, Automation, Robotics & Vision Guangzhou, 5–7 Dec, ICARCV 2012, 747–752 (2012) |
| [10] |
Meyer-Brandis, T., $\emptyset $ksendal, B., Zhou, X.Y.: A mean-field stochastic maximum principle via malliavin calculus. Stochastics 84, 643–666 (2012) |
| [11] |
|
| [12] |
|
| [13] |
|
| [14] |
|
| [15] |
|
| [16] |
Hafayed, M., Abbas, S.: A general maximum principle for stochastic differential equations of mean-field type with jump processes, arXiv: 1301.7327v4, (2013) |
| [17] |
|
| [18] |
|
| [19] |
|
| [20] |
|
| [21] |
|
| [22] |
|
| [23] |
Xu, R., Wu, T.: Mean-field backward stochastic evolution equations in Hilbert spaces and optimal control for BSPDEs. Abstr. Appl. Anal. 2014, Article ID 839467, pp. 15 (2014) |
| [24] |
|
| [25] |
Shi, J.: Necessary conditions for optimal control of forward–backward stochastic systems with random jumps. Int. J. Stoch. Anal. Article ID 258674, pp. 50 (2012) |
| [26] |
|
| [27] |
|
| [28] |
|
| [29] |
$\emptyset $ksendal, B., Sulem, A.: Maximum principles for optimal control of forward–backward stochastic differential equations with jumps. SIAM J. Control Optim. 48(5), 2845–2976 (2009) |
| [30] |
Framstad, N.C., $\emptyset $ksendal, B., Sulem, A.: Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance. J. Optim. Theory Appl. 121, 77–98 (2004) |
| [31] |
|
| [32] |
|
/
| 〈 |
|
〉 |