A Two-Mode Mean-Field Optimal Switching Problem for the Full Balance Sheet

Abstract

We consider the problem of switching a large number of production lines between two modes, high production and low production. The switching is based on the optimal expected profit and cost yields of the respective production lines and considers both sides of the balance sheet. Furthermore, the production lines are all assumed to be interconnected through a coupling term, which is the average of all optimal expected yields. Intuitively, this means that each individual production line is compared to the average of all its peers which acts as a benchmark. Due to the complexity of the problem, we consider the aggregated optimal expected yields, where the coupling term is approximated with the mean of the optimal expected yields. This turns the problem into a twomode optimal switching problem of mean-field type, which can be described by a system of Snell envelopes where the obstacles are interconnected and nonlinear. The main result of the paper is a proof of a continuous minimal solution to the system of Snell envelopes, as well as the full characterization of the optimal switching strategy.

A full balance sheet two-mode optimal switching problem

“…An extension of this model which includes a "mean-field" type interaction, prepared several months after the present paper, has appeared in [11], while this paper was still under review. To minimize eventual repetition, we will refer to [11] whenever similar arguments and proof are called upon.…”

“…We will only account for the construction of the approximating sequences and their properties, the needed estimates and the absolute continuity of the increasing processes dK −,i,n . The proofs of convergence to a continuous solution of (4.1) and the minimality of the solution of (4.1) are similar to the ones in the proof of Theorem 6 in [11], we therefore omit them. and, for n ≥ 1 and any t ≤ T , we consider the following system…”