A Mean-Field Optimal Control for Fully Coupled Forward-Backward Stochastic Control Systems with Lévy Processes

2021 ◽  
Author(s):  
Zhen Huang ◽  
Ying Wang ◽  
Xiangrong Wang
Keyword(s):  
Optimal Control ◽  
Control Systems ◽  
Stochastic Control ◽  
Lévy Processes ◽  
Mean Field ◽  
Levy Processes ◽  
Fully Coupled

scholarly journals CALIBRATION OF LÉVY PROCESSES USING OPTIMAL CONTROL OF KOLMOGOROV EQUATIONS WITH PERIODIC BOUNDARY CONDITIONS

2018 ◽  
Vol 23 (3) ◽  
pp. 390-413 ◽  
Author(s):  
Mario Annunziato ◽  
Hanno Gottschalk
Keyword(s):  
Optimal Control ◽  
Lévy Processes ◽  
Necessary Optimality Conditions ◽  
Functional Space ◽  
Information Criterion ◽  
Estimation Procedure ◽  
Levy Processes ◽  
Parametric Estimation ◽  
L1 Norm ◽  
Control Approach

We present an optimal control approach to the problem of model calibration for Lévy processes based on an non-parametric estimation procedure of the measure. The optimization problem is related to the maximum likelihood theory of sieves [25] and is formulated with the Fokker-Planck-Kolmogorov approach [3, 4]. We use a generic spline discretization of the Lévy jump measure and select an adequate size of the spline basis using the Akaike Information Criterion (AIC) [12]. The first order necessary optimality conditions are derived based on the Lagrange multiplier technique in a functional space. The resulting Partial Integral-Differential Equations (PIDE) are discretized, numerically solved using a scheme composed of Chang-Cooper, BDF2 and direct quadrature methods, jointly to a non-linear conjugate gradient method. For the numerical solver of the Kolmogorov's forward equation we prove conditions for non-negativity and stability in the L1 norm of the discrete solution.

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