Infinite horizon optimal control of forward–backward stochastic system driven by Teugels martingales with Lévy processes

2017 ◽  
Vol 17 (03) ◽  
pp. 1750020 ◽  
Author(s):  
P. Muthukumar ◽  
R. Deepa
Keyword(s):  
Optimal Control ◽  
Stochastic System ◽  
Lévy Processes ◽  
Infinite Horizon ◽  
Levy Processes ◽  
Necessary Conditions For Optimality ◽  
Independent Brownian Motion ◽  
Teugels Martingales ◽  
Sufficient And Necessary Conditions ◽  
Convexity Assumptions

In this paper, we consider the infinite horizon nonlinear optimal control of forward–backward stochastic system governed by Teugels martingales associated with Lévy processes and one dimensional independent Brownian motion. Our aim is to establish the sufficient and necessary conditions for optimality of the above stochastic system under the convexity assumptions. Finally an application is given to illustrate the problem of optimal control of stochastic system.

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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