## Introduction

Analyze and Optimize Your Production Strategies with Multidimensional BSDEs and Balance Sheet Optimal Switching

Emerging from the Works of Prominent Researchers, Multidimensional BSDEs with Mixed Reflections Offer Exciting Opportunities for Optimization in OSPs

In this paper, we are interested by Balance sheet OSP (BSOSP) which is a combination between the classical OSP described above and optimal stopping involving the balance sheet. BSOSP incorporates both the action of switching between modes and the action of abandoning a project once it becomes unprofitable. There are only few papers dealing with BSOSPs. Djehiche and Hamdi [4] considered the 2-modes case, i.e. . Their generators are of the form , and their barriers of type and , where and are switching costs and . Recently, the BSOSP multi-modes case was solved by Eddahbi et al. [5] when the barriers are of the form and (see Eddahbi et al. [6] for the mean-field case).

Now, let us describe precisely the problem studied in this paper by introducing some notations. Let be a given real number, and is a fixed probability space endowed with a d-dimensional Brownian motion . is the natural filtration of the Brownian motion augmented by the -null sets of . All the measurability notion will refer to this filtration. The euclidean norm of a vector is denoted |z|. Furthermore, we introduce the following spaces of processes. is the space of -valued processes , such that . (resp. ) is the set of -valued adapted and continuous (resp. càdlàg ) processes such that is the set of -valued, progressively measurable processes such that (resp. ) is the set of non-decreasing processes K, satisfying and that belong to .

Next, to illustrate the BSOSP studied in this paper, let us deal with a concrete example. Consider a company that has m modes of production (if , minimal, average and maximal production modes). The manager of the company has two options. A switch option, i.e. in order to maximize its global profit, she switches the production between the modes depending on their random performances but this switching incorporates a cost called switching cost. The manager has also an abandon option i.e. stop the production once it becomes unprofitable.

More precisely, being in mode , one have to switch at time t to another mode , once we have that the expected profit in this mode falls below the following barrier

where is nonlinear random function (a special case is when , where is a switching cost from mode i to mode j), is the expected cost in mode i, and is the cost incurred when exiting/terminating the production while in mode i. Since we consider both sides of the balance sheet, the manager has to switch at time t to another mode , as soon as the expected cost in mode i, rises above the following barrier

where is a cost of default (i.e. in this case the project is no longer profitable and thus leads to the abandon of this latter even before its maturity), and is the benefit incurred when exiting/terminating the production while in mode i. It is well known that the BSOSP can be formulated using the following system of Snell envelopes

where are -stopping times which represent the exit times from the production in mode i, and denote respectively the running profit and cost per unit time dt and and are respectively the values at time T of the profit and the cost yields.

The BSOSP consists in showing existence and uniqueness of the processes

and also proving that the following stopping times are optimal

Since the Snell envelope is strongly connected to RBSDEs, solving the BSOSP is equivalent to showing existence of continuous solution to the following general (since we take where , ) system of BSDEs with mixed reflections: for

where T is called the time horizon, and are called the terminal conditions, the random functions and are respectively -progressively measurable for each , called the generators. is a real nonlinear random function, and , , and are previously given -adapted processes with some suitable regularity. The unknowns are the processes which are required to be -adapted. Moreover, and are non-decreasing processes. The second condition in (5) (resp. (6)) says that the first component (resp. ) of the solution of RBSDE (5) (resp. (6)) is forced to stay above (resp. below) the barrier (resp. ). The role of (resp. ) is to push (resp. ) upwards (resp. downwards) in order to keep it above (resp. below) the respective barrier in a minimal way in the sense of the third condition of RBSDE (5) (resp. (6)) which is called the minimal boundary condition i.e. (resp. ) increases only when (resp. ) touches the respective barrier.

Let us make precise the notion of a solution of the system of RBSDEs (S).