Annals of Applied Probability

Optimal position targeting via decoupling fields

Stefan Ankirchner, Alexander Fromm, Thomas Kruse, and Alexandre Popier

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We consider a variant of the basic problem of the calculus of variations, where the Lagrangian is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by reducing it, via a limiting argument, to an unconstrained control problem that consists in finding an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation of the terminal state from a given target position. Using the Pontryagin maximum principle, we characterize a solution of the unconstrained control problem in terms of a fully coupled forward–backward stochastic differential equation (FBSDE). We use the method of decoupling fields for proving that the FBSDE has a unique solution. We exploit a monotonicity property of the decoupling field for solving the original constrained problem and characterize its solution in terms of an FBSDE with a free backward part.

Article information

Ann. Appl. Probab., Volume 30, Number 2 (2020), 644-672.

Received: April 2017
Revised: January 2019
First available in Project Euclid: 8 June 2020

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Mathematical Reviews number (MathSciNet)

Primary: 49J05: Free problems in one independent variable 60H99: None of the above, but in this section
Secondary: 60G99: None of the above, but in this section 93E20: Optimal stochastic control

Optimal stochastic control calculus of variations forward backward stochastic differential equation decoupling field


Ankirchner, Stefan; Fromm, Alexander; Kruse, Thomas; Popier, Alexandre. Optimal position targeting via decoupling fields. Ann. Appl. Probab. 30 (2020), no. 2, 644--672. doi:10.1214/19-AAP1511.

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