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International Journal of Stochastic Analysis
Volume 2014 (2014), Article ID 159519, 16 pages
A Two-Mode Mean-Field Optimal Switching Problem for the Full Balance Sheet
Department of Mathematics, KTH-Royal Institute of Technology, 100 44 Stockholm, Sweden
Received 20 February 2014; Accepted 4 May 2014; Published 25 May 2014
Academic Editor: Qing Zhang
Copyright © 2014 Boualem Djehiche and Ali Hamdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 two-mode 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.
Consider a company with different production lines which all have two modes of production, high mode and low mode, where each mode of production has its own balance sheet of expected profits and costs. For each production line in mode , let denote the optimal expected profit yield at time and let the corresponding optimal expected cost yield be denoted by . Assume that we want to switch between the two modes of production either if the current mode is unprofitable or if we can expect better profits in the other mode. Assume further that the switching is based on both sides of the balance sheet, so that we, for example, switch if we can expect lower costs in the other mode. Then this problem can be modeled as a two-mode optimal switching problem for each production line , which can be described by the following system of Snell envelopes: for each production line , where are the profit and cost rates per unit time (the generators) and where are the obstacles of the switching problem. Here, are the final profits and costs of each mode at some fixed time and the functions , , and represent switching costs.
Now assume that the production lines all have the same generators but they are interconnected through a coupling term. If the coupling term is the average of the profit and cost yields of all the projects, then this intuitively means that each production line is compared to a benchmark constituted of the average of its peers. In this case, the corresponding generators become With this assumption, solving the system (1) becomes a highly complex task for large , since the Snell envelopes are all interconnected through the coupling term (3). But instead of solving (1) we can consider the expected profit and cost yields on an aggregated level, where we use the mean-field approximation for the coupling term (3). The corresponding system of Snell envelopes becomes for , where In this paper we will show the existence of a continuous minimal solution of this system. In a forthcoming paper we will show convergence of the system (1) to our system of Snell envelops of mean-field type. The set of counterexamples derived in  can be used to argue that uniqueness may not hold in general.
In terms of BSDEs, the system is equivalent to the following system of mean-field reflected BSDEs (MF-RBSDEs): For details, see, for example, .
In this paper we consider the following slightly more general system of MF-RBSDEs: We follow a procedure similar to the one used in ; that is, we use an increasing sequence of approximating mean-field reflected stochastic differential equations (MF-RBSDEs) to show the existence of a continuous minimal solution of the system. However, due to the added generality in the problem, we have to prove a comparison result and an upper bound for this type of MF-RBSDE given the in-data.
Mean-field related problems have been studied not only in the setting of backward stochastic differential equations, but in many other fields as well. Examples of areas where mean-field approximations have been successful include statistical mechanics, quantum mechanics, quantum chemistry, economics, finance, and game theory. Recent work includes, for example, , where the authors consider the problem of sector-wise allocation in a portfolio consisting of a very large number of stocks. Another paper on mean-field approximation is the seminal work by Lasry and Lions , which concerns applications of mean-field approximations to problems in economics and finance. For an account on recent work related to our paper we refer to  and the references therein.
Backward stochastic differential equations of the mean-field type have been studied by several authors including [5–8]. To the best of our knowledge, the work of Buckdahn et al.  is the first paper to tackle this class of problems. They study an equation of the form where the mean-field interaction is linear in the generator, obtained as a mean-field limit of BSDE equation driven by SDEs of mean-field type. This type of mean-field backward stochastic differential equation (MF-BSDE) is further studied in  where the authors obtain existence and uniqueness for a general driver, under Lipschitz conditions. Extensions of this work to reflected BSDEs include [7, 8]. The equations they study are of the form The authors prove existence and uniqueness of the MF-BSDEs as well as a comparison theorem under some additional conditions for the generator. These results easily extend to our case where the mean-field interaction of the MF-RBSDEs is nonlinear in the generator. For completeness, we display in the Appendices an adaptation of the proof of [7, 8] to our setting.
2. Notation and Preliminaries
For the rest of the paper we fix a probability space, denoted by , on which is defined as a standard -dimensional Brownian motion whose natural filtration is . Let be the filtration completed with the -null sets of . This implies that satisfies the usual conditions; that is, it is right continuous and complete.
For future reference, we introduce the following spaces:(i)is the -algebra on of -progressively measurable processes,(ii) is the set of -measurable and -valued processes such that (iii) (resp., ) is the set of -measurable and càdlàg (resp., continuous) -valued processes such that (iv) (resp., ) is a subset of (resp., ) on nondecreasing càdlàg (resp., continuous) processes such that .
Let be an -measurable -random variable, let be an -valued function, and let be an -adapted process. To streamline the presentation of the results, we introduce the following notation.(i)If there exist a pair of processes such that then we say that (ii)If there exist a triple of processes such that then we say that (iii)If there exist a triple of processes such that then we say that
These mean-field backward stochastic differential equations (MF-BSDEs) and mean-field reflected backward stochastic differential equations (MF-RBSDEs) are said to be standard if the following conditions hold.(H1)The generator is Lipschitz with respect to uniformly in .(H2)The process is -progressively measurable and -square integrable.(H3)The random variable is in .(H4)The barrier is càdlàg, -adapted and satisfies and , -a.s.
Finally, a key tool used in this paper is the notion of the Snell envelope. Let denote the class of -stopping times such that for some -stopping time .
Proposition 1. Let be an -adapted -valued càdlàg process such that the set of random variables is uniformly integrable. Then there exists an -adapted -valued càdlàg process such that is the smallest supermartingale which dominates . The process is called the Snell envelope of and it has the following properties. (i)For any -stopping time it holds that (ii)The Doob-Meyer decomposition of implies the existence of a continuous martingale and two nondecreasing predictable processes and which are, respectively, continuous and purely discontinuous such that for all one has (iii)For any , Hence, if only has positive jumps, then is a continuous process. (iv)If is an -stopping time, then is optimal after ; that is,
Proposition 2. Let be some in-data which satisfies (H1)–(H4). Then there exists a unique triplet which solves A similar result holds for .
Proof. See Appendix A.
Proposition 3. Let be a set of data satisfying assumptions (H1)–(H4) and let . Then there exists a constant such that
Proof. See Appendix B.
Next, we display a comparison result for solutions of . A similar result holds for .
Proposition 4. Let and be two sets of data, each one satisfying assumptions (H1)–(H4) and let If the following conditions hold: (i), a.s., (ii), -a.e., and , (iii), , a.s., (iv) and are continuous, (v)at least one of the two generators and is nondecreasing in ,then -a.s.
Proof. See Appendix C.
Proposition 5. Proposition 4 holds true even when a.s. and only satisfies (H4); that is, it need not be continuous.
Proof. See Appendix D.
3. The System of MF-RBSDEs
Consider the following system of equations: for , where
Further assume the following.(A1) are Lipschitz in uniformly in ; that is, there exists a such that, for any , In addition, the processes are -progressively measurable and -square integrable.(A2)The processes , , and belong to . In addition, -a.s.(A3)The random variables are -measurable and square integrable. Furthermore, -a.s. it holds that (A4)The processes and are of Itô-type; that is, where and are some -progressively measurable processes which are -square integrable.
It is worth noting a few things here. First, the obstacle processes need not be of Itô-type. Second, the set of solutions to the system (32) is nonempty. An example of solution is to set , and let
for . Lastly, while is a supermartingale, is a submartingale. The assumption (A4) is needed to prove the continuity of the increasing process . This is in turn used to prove continuity of , which finally is used to derive continuity of .
Theorem 6. Let the generators in the system (29) be nondecreasing in the third argument; that is, are nondecreasing functions. Then, under the assumptions (A1)–(A4), the system (29) admits a minimal solution such that , is continuous. The solution is minimal in the sense that if , , is another solution, then , , a.s. The solution to the system (32) is not unique in general.
Proof. The theorem is proved using an approximating scheme. Let
Now let be the unique solution to the BSDE where By Proposition 4 it holds that Hence, since a.s.
Consider now the processes where and, for , where Since is the solution of a standard MF-BSDE, the existence and uniqueness have been established in , and the existence and uniqueness of the processes were established in Proposition 2. With this in mind, it is easily shown by the use of induction that for any the triples exist, are unique, and belong to the appropriate spaces.
From Proposition 4 and the fact that it follows that . Moreover, by Proposition 4 again, it follows that . Hence Proposition 4 then yields .
Now assume that Then from which it follows, by Proposition 4 again, that . Hence, it also holds that Thus Proposition 4 yields that . By induction, for .
By Proposition 3 these sequences are bounded, and since they are also increasing the limits exist. Denote these limits by In what follows we will prove that these limits are in fact continuous and solve the system (29). To do this we will use the following claim: there exists a positive constant such that, for all , The first step towards proving this claim is to prove the absolute continuity of with respect to . Noting that then in view of (A4) and the Itô-Tanaka formula, we get where is the local time at zero of the continuous semimartingale and where It follows that From Itô-Tanaka again we get that where is the local time at for . Since the differentials must coincide, which yields thatHence from which it follows that Now, in view of (A1)–(A4) and Proposition 3 there exists a constant such that which together with (62) proves (54) for .
For the induction step, we only consider the case since the other case follows in a similar fashion. Assume that satisfies (54) and consider the obstacle process By (A4) and the Itô-Tanaka formula, we get where is the local time at for the continuous semimartingale and where In view of (A1)–(A4) and Proposition 3, together with the induction assumption, there exists a constant , independent of , such that Following the same steps as we did earlier for it can be shown that which yields that there exists a constant independent of such that Hence claim (54) is true for all .
Proposition 3 and estimate (54) tell us that there is a subsequence along which the sequences of processes , , and converge weakly in their respective spaces , , and to the processes , , and .
Now for any and any -stopping time , we have Taking the weak limits in each side of this equation along the subsequence mentioned earlier yields The processes on each side of the equality being optional, we can use the optional section theorem (see, e.g., [12, Theorem 86, page 138]) to conclude that Therefore, the process is continuous.
Relying on both Dini’s theorem and Lebesgue’s dominated convergence theorem we find that
In what follows we will characterize the limit processes of the sequence as Snell envelopes of the processes in the sense that, -a.s. and for each , and then we derive their time continuity.
In view of Proposition 3 and applying Peng’s monotone limit theorem (see ) to the sequence , we obtain that is càdlàg. Moreover, there exist a càdlàg nondecreasing process and a process such that Hence, in view of Proposition 1 we arrive at (74) once we show that
Consider the smallest -supermartingale with lower obstacle , which solves By definition, we have that . Using Proposition 5, we see that , . Passing to the limit we get that , . Hence, (76) is satisfied.
It remains to show that are continuous. Noting that and that satisfies (74), we find that or equivalently that . Thus, it is sufficient to prove that the set is empty.
By the Doob-Meyer decomposition of the Snell envelope (see Proposition 1) there exists, for each , a continuous martingale , a continuous nondecreasing process , and a purely discontinuous process , with , such that which means that . Now in view of Proposition 1 we have the following properties of the jumps of . If there is a jump at time in , then this means that the process also jumps. Since and are continuous, this can only mean that there is a jump in . This, in turn, means that there is a jump in . And conversely, by the same type of reasoning, we can deduce that if there is a jump in , there is also a jump in . Hence, and always jump at the same time.
With this in mind, denote and . We have What this tells us is that if jumps at time , then we know two things. One is that also has a jump at time and the other is that Therefore, since , in view of Proposition 1, we obtain that Similarly, we have
Finally, since we know that and always jump at the same time, it holds that since, for any , . It follows that the processes and are constant and identically equal to , since . Hence, the processes and are continuous. This, in turn, given (75), yields the continuity of the increasing processes . Therefore, is a solution to the first part of the system (29).
By Dini’s theorem and Lebesgue’s dominated convergence theorem again we also conclude that the convergence of to holds in ; that is, Furthermore, since is continuous, we can rely on standard arguments, in particular, by applying Itô’s formula to , to claim that is a Cauchy sequence, and therefore it converges to in ; that is, Combining this with (84) and the definition of the processes and yields that