Dynamic convex duality in constrained utility maximization

Abstract

In this paper, we study a constrained utility maximization problem following the convex duality approach. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of FBSDEs plus additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. Moreover, we also find that the optimal primal wealth process coincides with the adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems, which contrasts the simplicity of the duality approach we propose and the technical complexity of solving the primal problems directly.

Deep Learning for Constrained Utility Maximisation

“…Theorem 3.7 (Li and Zheng (2018), Theorem 12). Suppose that ŷ and v are optimal dual controls, with the corresponding dual state process Ŷ and 2BSDE solutions ( V2 , Ẑ2 , Γ2 ) .…”

“…We next construct the dual problem, see Li and Zheng (2018) [Section 2] for more details. The basis for duality is the Legendre-Fenchel transformation Ũ ∶ (0, ∞) → ℝ of the utility function U, defined by…”

“…Since XY is a super-martingale for any choice of , v and y, we have the following weak duality relation [p. 5] Li and Zheng (2018) Solving the dual problem would give us an upper bound of the value function, that can be combined with a lower bound by the primal problem to produce a confidence interval for the value function. If we have an equality in (4), then we can find the value function by alternatively solving the dual problem.…”

“…We treat this more general case in Sect. 4, where the adjoint BSDE is solved using deep learning, and the optimality conditions derived in Li and Zheng (2018) are used to optimise the control.…”

“…Suppose that r(s) = r, (s) = , (s) = and (s) = are constant. It is shown in Li and Zheng (2018) that the optimiser ŷ and processes Ŷ and Ẑ2 are given by where (t) . Furthermore, the dual value function is given by From these analytical formulae, the primal processes can be derived using Theorem 3.7.…”