July 25, 2024
bd m88
bd m88

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. equation M24. Their generators are of the form equation M25, equation M26 and their barriers of type equation M27 and equation M28, where equation M29 and equation M30 are switching costs and equation M31. Recently, the BSOSP multi-modes case was solved by Eddahbi et al. [5] when the barriers are of the form equation M32 and equation M33 (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 equation M34 be a given real number, and equation M35 is a fixed probability space endowed with a d-dimensional Brownian motion equation M36. equation M37 is the natural filtration of the Brownian motion augmented by the equation M38-null sets of equation M39. All the measurability notion will refer to this filtration. The euclidean norm of a vector equation M40 is denoted |z|. Furthermore, we introduce the following spaces of processes. equation M41 is the space of equation M42-valued processes equation M43, such that equation M44. equation M45 (resp. equation M46) is the set of equation M47-valued adapted and continuous (resp. càdlàg ) processes equation M48 such that equation M49 equation M50 is the set of equation M51-valued, progressively measurable processes equation M52 such that equation M53 equation M54 (resp. equation M55) is the set of non-decreasing processes K, satisfying equation M56 and that belong to equation M57 equation M58.

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 equation M59, 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 equation M60, one have to switch at time t to another mode equation M61, once we have that the expected profit equation M62 in this mode falls below the following barrier

where equation M64 is nonlinear random function (a special case is when equation M65, where equation M66 is a switching cost from mode i to mode j), equation M67 is the expected cost in mode i, and equation M68 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 equation M69, as soon as the expected cost in mode i, equation M70 rises above the following barrier

where equation M72 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 equation M73 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 equation M76 are equation M77-stopping times which represent the exit times from the production in mode i, equation M78 and equation M79 denote respectively the running profit and cost per unit time dt and equation M80 and equation M81 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

equation M82 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 equation M84 equation M85 where equation M86, equation M87) system of BSDEs with mixed reflections: for equation M88

where T is called the time horizon, equation M89 and equation M90 are called the terminal conditions, the random functions equation M91 and equation M92 are respectively equation M93-progressively measurable for each equation M94, called the generators. equation M95 is a real nonlinear random function, and equation M96, equation M97, and equation M98 are previously given equation M99-adapted processes with some suitable regularity. The unknowns are the processes equation M100 which are required to be equation M101-adapted. Moreover, equation M102 and equation M103 are non-decreasing processes. The second condition in (5) (resp. (6)) says that the first component equation M104 (resp. equation M105) of the solution of RBSDE (5) (resp. (6)) is forced to stay above (resp. below) the barrier equation M106 (resp. equation M107). The role of equation M108 (resp. equation M109) is to push equation M110 (resp. equation M111) 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. equation M112 (resp. equation M113) increases only when equation M114 (resp. equation M115) touches the respective barrier.

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