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 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.

1. Introduction

Consider a company with 𝑁 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 𝑗 in mode 𝑖, let 𝑌_𝑡^+,𝑖,𝑗 denote the optimal expected profit yield at time 𝑡 and let the corresponding optimal expected cost yield be denoted by 𝑌_𝑡^{−,𝑖,𝑗}. 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 𝑗, which can be described by the following system of Snell envelopes: 𝑌_𝑡^+,𝑖,𝑗=esssup_𝜏≥𝑡𝔼^𝜏_𝑡𝜓_𝑖^+,𝑗(𝑠)𝑑𝑠+𝑆_𝜏^+,𝑖,𝑗𝟏_{𝜏<𝑇}+𝜉⁺_𝑖𝟏_{𝜏=𝑇}∣ℱ_𝑡,𝑌_𝑡^{−,𝑖,𝑗}=essinf_𝜏≥𝑡𝔼^𝜏_𝑡𝜓_𝑖^−,𝑗(𝑠)𝑑𝑠+𝑆_𝜏^{−,𝑖,𝑗}𝟏_{𝜏<𝑇}+𝜉⁻_𝑖𝟏_{𝜏=𝑇}∣ℱ_𝑡, (1) for each production line 𝑗=1,…,𝑁, where 𝜓_𝑖^±,𝑗(𝑠) are the profit and cost rates per unit time (the generators) and where 𝑆_𝑡^+,1,𝑗=𝑌_𝑡^+,2,𝑗−ℓ₁(𝑡)∨𝑌_𝑡^−,1,𝑗−𝑎₁(𝑡),𝑆_𝑡^+,2,𝑗=𝑌_𝑡^+,1,𝑗−ℓ₂(𝑡)∨𝑌_𝑡^−,2,𝑗−𝑎₂(𝑡,𝑆_{𝑡−,1,𝑗})=𝑌_𝑡^−,2,𝑗+ℓ₁(𝑡)∧𝑌_𝑡^+,1,𝑗+𝑏₁(𝑡),𝑆_𝑡^−,2,𝑗=𝑌_𝑡^−,1,𝑗+ℓ₂(𝑡)∧𝑌_𝑡^+,2,𝑗+𝑏₂(𝑡) (2) are the obstacles of the switching problem. Here, 𝜉^±_𝑖 are the final profits and costs of each mode 𝑖 at some fixed time 𝑇 and the functions ℓ_𝑖, 𝑎_𝑖, and 𝑏_𝑖 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, 1𝑁^𝑁_𝑗=1𝑌_𝑡^±,𝑖,𝑗, (3) 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 𝜓_𝑖^+,𝑗(𝑡)=𝜓⁺_𝑖𝑡,𝑌_𝑡^+,𝑖,𝑗,1𝑁^𝑁_𝑘=1𝑌_𝑡^+,𝑖,𝑘,𝜓_𝑖^−,𝑗(𝑡)=𝜓⁻_𝑖𝑡,𝑌_𝑡^{−,𝑖,𝑗},1𝑁^𝑁_𝑘=1𝑌_𝑡^{−,𝑖,𝑘}. (4) With this assumption, solving the system (1) becomes a highly complex task for large 𝑁, 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 𝔼𝑌_𝑡^±,𝑖 for the coupling term (3). The corresponding system of Snell envelopes becomes 𝑌_𝑡^+,𝑖=esssup_𝜏≥𝑡𝔼^𝜏_𝑡𝜓⁺_𝑖𝑠,𝑌_𝑠^+,𝑖,𝔼𝑌_𝑠^+,𝑖𝑑𝑠+𝑆_𝑠^+,𝑖𝟏_{𝜏<𝑇}+𝜉⁺_𝑖𝟏_{𝜏=𝑇}∣ℱ_𝑡,𝑌_𝑡^−,𝑖=essinf_𝜏≥𝑡𝔼^𝜏_𝑡𝜓⁺_𝑖𝑠,𝑌_𝑠^−,𝑖,𝔼𝑌_𝑠^−,𝑖𝑑𝑠+𝑆_𝑠^−,𝑖𝟏_{𝜏<𝑇}+𝜉⁻_𝑖𝟏_{𝜏=𝑇}∣ℱ_𝑡, (5) for 𝑖=1,2, where 𝑆_𝑡^+,1=𝑌_𝑡^+,2−ℓ₁(𝑡)∨𝑌_𝑡^−,1−𝑎₁(𝑡,𝑆_𝑡+,2)=𝑌_𝑡^+,1−ℓ₂(𝑡)∨𝑌_𝑡^−,2−𝑎₂(𝑡),𝑆_𝑡^−,1=𝑌_𝑡^−,2+ℓ₁(𝑡)∧𝑌_𝑡^+,1+𝑏₁(𝑡),𝑆_𝑡^−,2=𝑌_𝑡^−,1+ℓ₂(𝑡)∧𝑌_𝑡^+,2+𝑏₂(𝑡). (6) 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 [1] 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): 𝑌_𝑡^+,𝑖=𝜉⁺_𝑖+^𝜏_𝑡𝜓⁺_𝑖𝑠,𝑌_𝑠^+,𝑖,𝔼𝑌_𝑠^+,𝑖𝑑𝑠+𝐾_𝑇^+,𝑖−𝐾_𝑡^+,𝑖−^𝑇_𝑡𝑍_𝑠^+,𝑖𝑑𝐵_𝑠,𝑌_𝑡^−,𝑖=𝜉⁻_𝑖+^𝜏_𝑡𝜓⁻_𝑖𝑠,𝑌_𝑠^−,𝑖,𝔼𝑌_𝑠^−,𝑖𝑑𝑠−𝐾_𝑡^−,𝑖−𝐾_𝑡^−,𝑖−^𝑇_𝑡𝑍_𝑠^−,𝑖𝑑𝐵_𝑠,𝑌_𝑡^+,𝑖≥𝑆_𝑡^+,𝑖,𝑌_𝑡^−,𝑖≤𝑆_𝑡^−,𝑖,_𝑇0𝑌_𝑡^+,𝑖−𝑆_𝑡^+,𝑖𝑑𝐾_𝑡^+,𝑖=0,_𝑇0𝑆_𝑡^−,𝑖−𝑌_𝑡^−,𝑖𝑑𝐾_𝑡^−,𝑖=0. (7) For details, see, for example, [2].

In this paper we consider the following slightly more general system of MF-RBSDEs: 𝑌_𝑡^+,𝑖=𝜉⁺_𝑖+^𝜏_𝑡𝜓⁺_𝑖𝑠,𝑌_𝑠^+,𝑖,𝔼𝑌_𝑠^+,𝑖,𝑍_𝑠^+,𝑖𝑑𝑠+𝐾_𝑇^+,𝑖−𝐾_𝑡^+,𝑖−^𝑇_𝑡𝑍_𝑠^+,𝑖𝑑𝐵_𝑠,𝑌_𝑡^−,𝑖=𝜉⁻_𝑖+^𝜏_𝑡𝜓⁻_𝑖𝑠,𝑌_𝑠^−,𝑖,𝔼𝑌_𝑠^−,𝑖,𝑍_𝑠^−,𝑖𝑑𝑠−𝐾_𝑡^−,𝑖−𝐾_𝑡^−,𝑖−^𝑇_𝑡𝑍_𝑠^−,𝑖𝑑𝐵_𝑠,𝑌_𝑡^+,𝑖≥𝑆_𝑡^+,𝑖,𝑌_𝑡^−,𝑖≤𝑆_𝑡^−,𝑖,_𝑇0𝑌_𝑡^+,𝑖−𝑆_𝑡^+,𝑖𝑑𝐾_𝑡^+,𝑖=0,_𝑇0𝑆_𝑡^−,𝑖−𝑌_𝑡^−,𝑖𝑑𝐾_𝑡^−,𝑖=0. (8) We follow a procedure similar to the one used in [1]; 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, [3], 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 [4], 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 [5] 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. [6] is the first paper to tackle this class of problems. They study an equation of the form 𝑌_𝑡=𝜉+^𝑇_𝑡𝔼^′𝑓𝑠,𝜔^′,𝜔,𝑌_𝑠𝜔^′,𝑍_𝑠𝜔^′,𝑌_𝑠(𝜔),𝑍_𝑠(𝜔)−𝑑𝑠_𝑇𝑡𝑍_𝑠𝑑𝑊_𝑠, (9) 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 [6] 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 𝑌_𝑡=𝜉+^𝑇_𝑡𝔼^′𝑓𝑠,𝜔^′,𝜔,𝑌_𝑠𝜔^′,𝑍_𝑠𝜔^′,𝑌_𝑠(𝜔),𝑍_𝑠(𝜔)𝑑𝑠+𝐾_𝑇−𝐾_𝑡−^𝑇_𝑡𝑍_𝑠𝑑𝑊_𝑠,𝑌_𝑡≥𝑆_𝑡,^𝑇₀𝑌_𝑡−𝑆_𝑡𝑑𝐾_𝑡=0. (10) 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.

The outline of this paper is as follows. Section 2 states the necessary notation and preliminaries. In Section 3 we state and prove the main results of this paper.

2. Notation and Preliminaries

For the rest of the paper we fix a probability space, denoted by (Ω,ℱ,ℙ), on which is defined as a standard 𝑑-dimensional Brownian motion 𝐵=(𝐵_𝑡)_{0≤𝑡≤𝑇} whose natural filtration is (ℱ⁰_𝑡∶=𝜎{𝐵_𝑠,𝑠≤𝑡})_{0≤𝑡≤𝑇}. Let 𝔽=(ℱ_𝑡)_{0≤𝑡≤𝑇} be the filtration (ℱ⁰_𝑡)_{0≤𝑡≤𝑇} completed with the ℙ-null sets of ℱ. This implies that 𝔽 satisfies the usual conditions; that is, it is right continuous and complete.

For future reference, we introduce the following spaces:(i)𝒫is the 𝜎-algebra on [0,𝑇]×Ω of 𝔽-progressively measurable processes,(ii)ℳ^𝑑,2 is the set of 𝒫-measurable and ℝ^𝑑-valued processes 𝑤=(𝑤_𝑡)_{0≤𝑡≤𝑇} such that ‖𝑤‖_ℳ^𝑑,2∶=𝔼^𝑇₀𝑤_𝑠²𝑑𝑠^1/2<∞, (11)(iii)𝒮² (resp., 𝒮²_𝑐) is the set of 𝒫-measurable and càdlàg (resp., continuous) ℝ-valued processes 𝑤=(𝑤_𝑡)_{0≤𝑡≤𝑇} such that ‖𝑤‖_𝑆²∶=𝔼sup_{0≤𝑡≤𝑇}𝑤_𝑡^21/2<∞, (12)(iv)𝒦² (resp., 𝒦²_𝑐) is a subset of 𝒮² (resp., 𝒮²_𝑐) on nondecreasing càdlàg (resp., continuous) processes (𝐾_𝑡)_{0≤𝑡≤𝑇} such that 𝐾₀=0.

Let 𝜉 be an ℱ_𝑇-measurable 𝐿²-random variable, let 𝑓 be an ℝ-valued function, and let 𝑆 be an 𝔽-adapted process. To streamline the presentation of the results, we introduce the following notation.(i)If there exist a pair of processes (𝑌,𝑍) such that 𝑌∈𝒮²resp.,𝒮²_𝑐,𝑍∈ℳ^𝑑,2,𝑌_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝜔,𝑌_𝑠,𝔼𝑌_𝑠,𝑍_𝑠𝑑𝑠−^𝑇_𝑡𝑍_𝑠𝑑𝐵_𝑠, (13) then we say that (𝑌,𝑍)=𝐹𝜉,𝑓resp.,(𝑌,𝑍)=𝐹^𝑐𝜉,𝑓. (14)(ii)If there exist a triple of processes (𝑌,𝑍,𝐾) such that 𝑌∈𝒮²resp.,𝒮²_𝑐,𝑍∈ℳ^𝑑,2,𝐾∈𝒦²resp.,𝒦²_𝑐𝑌_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝜔,𝑌_𝑠,𝔼𝑌_𝑠,𝑍_𝑠𝑑𝑠+𝐾_𝑇−𝐾_𝑡−^𝑇_𝑡𝑍_𝑠𝑑𝐵_𝑠,𝑌_𝑡≥𝑆_𝑡,0≤𝑡≤𝑇,_𝑇0𝑌_𝑡−−𝑆_𝑡−𝑑𝐾_𝑡=0resp.,^𝑇₀𝑌_𝑡−𝑆_𝑡𝑑𝐾_𝑡=0, (15) then we say that (𝑌,𝑍,𝐾)=𝐹₊𝜉,𝑓,𝑆resp.,(𝑌,𝑍,𝐾)=𝐹^𝑐₊𝜉,𝑓,𝑆. (16)(iii)If there exist a triple of processes (𝑌,𝑍,𝐾) such that 𝑌∈𝒮²resp.,𝒮²_𝑐,𝑍∈ℳ^𝑑,2,𝐾∈𝒦²resp.,𝒦²_𝑐𝑌_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝜔,𝑌_𝑠,𝔼𝑌_𝑠,𝑍_𝑠𝑑𝑠−𝐾_𝑇−𝐾_𝑡−^𝑇_𝑡𝑍_𝑠𝑑𝐵_𝑠,𝑌_𝑡≤𝑆_𝑡,0≤𝑡≤𝑇,_𝑇0𝑆_𝑡−−𝑌_𝑡−𝑑𝐾_𝑡=0resp.,^𝑇₀𝑌_𝑡−𝑆_𝑡𝑑𝐾_𝑡=0, (17) then we say that (𝑌,𝑍,𝐾)=𝐹₋𝜉,𝑓,𝑆resp.,(𝑌,𝑍,𝐾)=𝐹^𝑐₋𝜉,𝑓,𝑆. (18)

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 𝑓 is Lipschitz with respect to (𝑦₁,𝑦₂,𝑧) uniformly in (𝑡,𝜔).(H2)The process (𝑓(𝑡,𝜔,0,0,0))_{0≤𝑡≤𝑇} is 𝔽-progressively measurable and 𝑑𝑡⊗𝑑ℙ-square integrable.(H3)The random variable 𝜉 is in 𝐿²(Ω,ℱ_𝑇,ℙ).(H4)The barrier 𝑆 is càdlàg, 𝔽-adapted and satisfies 𝔼sup_{0≤𝑠≤𝑇}𝑆⁺_𝑠²<∞ (19) and 𝑆_𝑇≤𝜉, ℙ-a.s.

More on MF-BSDEs can be found in [5, 6]. For further reference on MF-RBSDEs, see [7, 8].

Finally, a key tool used in this paper is the notion of the Snell envelope. Let 𝒯_𝜃 denote the class of 𝔽-stopping times 𝜏 such that 𝜏≥𝜃 for some 𝔽-stopping time 𝜃.

Proposition 1. Let 𝑈=(𝑈_𝑡)_{0≤𝑡≤𝑇} be an 𝔽-adapted ℝ-valued càdlàg process such that the set of random variables {𝑈_𝜏,𝜏∈𝒯₀} is uniformly integrable. Then there exists an 𝔽-adapted ℝ-valued càdlàg process 𝑍∶=(𝑍_𝑡)_{0≤𝑡≤𝑇} such that 𝑍 is the smallest supermartingale which dominates 𝑈. The process 𝑍 is called the Snell envelope of  𝑈 and it has the following properties. (i)For any 𝔽-stopping time 𝜃 it holds that 𝑍_𝜃=esssup_{𝜏∈𝒯𝜃}𝔼𝑈_𝜏∣ℱ_𝜃𝑎𝑛𝑑ℎ𝑒𝑛𝑐𝑒𝑍_𝑇=𝑈_𝑇. (20)(ii)The Doob-Meyer decomposition of 𝑍 implies the existence of a continuous martingale (𝑀_𝑡)_{0≤𝑡≤𝑇} and two nondecreasing predictable processes (𝐴_𝑡)_{0≤𝑡≤𝑇} and (𝐵_𝑡)_{0≤𝑡≤𝑇} which are, respectively, continuous and purely discontinuous such that for all 0≤𝑡≤𝑇 one has 𝑍_𝑡=𝑀_𝑡−𝐴_𝑡−𝐵_𝑡𝑤𝑖𝑡ℎ𝐴₀=𝐵₀=0. (21)(iii)For any 0≤𝑡≤𝑇, Δ𝐵_𝑡>0⊆Δ𝑈_𝑡<0∩𝑍_𝑡−=𝑈_𝑡−. (22)Hence, if 𝑈 only has positive jumps, then 𝑍 is a continuous process. (iv)If 𝜃 is an 𝔽-stopping time, then 𝜏^∗_𝜃∶=inf𝑡≥𝜃∶𝑍_𝑡=𝑈_𝑠∧𝑇 (23)is optimal after 𝜃; that is, 𝑍_𝜃=esssup_{𝜏∈𝒯𝜃}𝔼𝑈_𝜏∣ℱ_𝜃=𝔼𝑈_𝜏^∗_𝜃∣ℱ_𝜃=𝔼𝑍_𝜏^∗_𝜃∣ℱ_𝜃. (24)

For further reference on the Snell envelope we refer to [9, 10] or [11].

We finally collect results regarding existence, uniqueness, bounds, and comparison for MF-RBSDEs. These are adaptations of results in [7, 8] to our case. Proofs are deferred to the Appendices.

Proposition 2. Let (𝜉,𝑓,𝑆) be some in-data which satisfies (H1)–(H4). Then there exists a unique triplet (𝑌,𝑍,𝐾) which solves (𝑌,𝑍,𝐾)=𝐹₊𝜉,𝑓,𝑆. (25) A similar result holds for (𝑌,𝑍,𝐾)=𝐹₋(𝜉,𝑓,𝑆).

Proof. See Appendix A.

Proposition 3. Let (𝜉,𝑓,𝑆) be a set of data satisfying assumptions (H1)–(H4) and let (𝑌,𝑍,𝐾)=𝐹_±(𝜉,𝑓,𝑆). Then there exists a constant 𝐶 such that 𝔼sup_{0≤𝑡≤𝑇}𝑌_𝑡²+^𝑇₀𝑍_𝑡²𝑑𝑡+𝐾_𝑇²≤𝐶𝔼𝜉²+^𝑇₀𝑓(𝑡,0,0,0)𝑑𝑡+sup_{0≤𝑡≤𝑇}𝑆⁺_𝑡. (26)

Proof. See Appendix B.

Next, we display a comparison result for solutions of (𝑌,𝑍,𝐾)=𝐹₊(𝜉,𝑓,𝑆). A similar result holds for (𝑌,𝑍,𝐾)=𝐹₋(𝜉,𝑓,𝑆).

Proposition 4. Let (𝜉,𝑓,𝑆) and (𝜉^′,𝑓^′,𝑆^′) be two sets of data, each one satisfying assumptions (H1)–(H4) and let (𝑌,𝑍,𝐾)=𝐹₊,𝑌𝜉,𝑓,𝑆_′,𝑍^′,𝐾^′=𝐹₊𝜉^′,𝑓^′,𝑆^′. (27) If the following conditions hold: (i)𝜉≤𝜉^′, a.s., (ii)𝑓(𝑡,𝑦₁,𝑦₂,𝑧)≤𝑓^′(𝑡,𝑦₁,𝑦₂,𝑧), 𝑑ℙ×𝑑𝑡-a.e., and ∀(𝑦₁,𝑦₂,𝑧)∈ℝ×ℝ×ℝ^𝑑, (iii)𝑆_𝑡≤𝑆^′_𝑡, 0≤𝑡≤𝑇, a.s., (iv)𝑆 and 𝑆^′ are continuous, (v)at least one of the two generators 𝑓 and 𝑓^′ is nondecreasing in 𝑦₂,then 𝑌_𝑡≤𝑌^′_𝑡,0≤𝑡≤𝑇, (28)ℙ-a.s.

Proof. See Appendix C.

Proposition 5. Proposition 4 holds true even when 𝑆=𝑆^′ a.s. and 𝑆 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: 𝑌^+,𝑖,𝑍^+,𝑖,𝐾^+,𝑖=𝐹₊𝜉⁺_𝑖,𝜓⁺_𝑖,𝑆^+,𝑖,𝑌^−,𝑖,𝑍^−,𝑖,𝐾^−,𝑖=𝐹₋𝜉⁻_𝑖,𝜓⁻_𝑖,𝑆^−,𝑖, (29) for 𝑖=1,2, where 𝑆_𝑡^+,1=𝑌_𝑡^+,2−ℓ₁(𝑡)∨𝑌_𝑡^−,1−𝑎₁(𝑡,𝑆_𝑡+,2)=𝑌_𝑡^+,1−ℓ₂(𝑡)∨𝑌_𝑡^−,2−𝑎₂(𝑡),𝑆_𝑡^−,1=𝑌_𝑡^−,2+ℓ₁(𝑡)∧𝑌_𝑡^+,1+𝑏₁(𝑡),𝑆_𝑡^−,2=𝑌_𝑡^−,1+ℓ₂(𝑡)∧𝑌_𝑡^+,2+𝑏₂(𝑡). (30)

Further assume the following.(A1)𝜓^±_𝑖 are Lipschitz in (𝑦₁,𝑦₂,𝑧) uniformly in (𝑡,𝜔); that is, there exists a 𝐶>0 such that, for any (𝑡,𝜔)∈[0,𝑇]×Ω, 𝜓^±_𝑖𝑡,𝜔,𝑦₁,𝑦₂,𝑧−𝜓^±_𝑖𝑡,𝜔,𝑦^′₁,𝑦^′₂,𝑧≤𝐶𝑦−𝑦^′+𝑧−𝑧^′. (31) In addition, the processes 𝜓_𝑖^±,0(𝑡)∶=𝜓^±_𝑖(𝑡,𝜔,0,0) are ℱ-progressively measurable and 𝑑𝑡⊗𝑑ℙ-square integrable.(A2)The processes (𝑎_𝑖(𝑡,𝜔))_{0≤𝑡≤𝑇}, (𝑏_𝑖(𝑡,𝜔))_{0≤𝑡≤𝑇}, and (ℓ_𝑖(𝑡,𝜔))_{0≤𝑡≤𝑇} belong to 𝒮²_𝑐. In addition, ℓ_𝑖(𝑡)>0ℙ-a.s.(A3)The random variables 𝜉^±_𝑖 are ℱ_𝑇-measurable and square integrable. Furthermore, ℙ-a.s. it holds that 𝜉⁺₁≥𝜉⁺₂−ℓ₁(𝑇)∨𝜉⁻₁−𝑎₁(𝑇),𝜉⁺₂≥𝜉⁺₁−ℓ₂(𝑇)∨𝜉⁻₂−𝑎₂(𝑇),𝜉⁻₁≤𝜉⁻₂+ℓ₁(𝑇)∧𝜉⁺₁+𝑏₁(𝑇),𝜉⁻₂≤𝜉⁻₁+ℓ₂(𝑇)∧𝜉⁺₂+𝑏₂(𝑇). (32)(A4)The processes (𝑏_𝑖(𝑡))_{0≤𝑡≤𝑇} and (ℓ_𝑖(𝑡))_{0≤𝑡≤𝑇} are of Itô-type; that is, 𝑏_𝑖(𝑡)=𝑏_𝑖(0)+^𝑡₀𝑈_𝑖(𝑠)𝑑𝑠+^𝑡₀𝑉_𝑖(𝑠)𝑑𝐵_𝑠,ℓ_𝑖(𝑡)=ℓ_𝑖(0)+^𝑡₀𝑈_𝑖𝑑𝑠+^𝑡₀𝑉_𝑖𝑑𝐵_𝑠, (33) where (𝑈,𝑈) and (𝑉,𝑉) are some 𝔽-progressively measurable processes which are 𝑑𝑡⊗ℙ-square integrable.

It is worth noting a few things here. First, the obstacle processes 𝑎_𝑖 need not be of Itô-type. Second, the set of solutions to the system (32) is nonempty. An example of solution is to set 𝜉⁺_𝑖=𝜉⁻_𝑖=1, and let ℓ_𝑖(𝑇)=𝑒^−4𝑇,𝑎_𝑖(𝑇)=𝑏_𝑖(𝑇)=0, (34)

for 𝑖=1,2. Lastly, while 𝑌^+,𝑖 is a supermartingale, 𝑌^−,𝑖 is a submartingale. The assumption (A4) is needed to prove the continuity of the increasing process 𝐾^−,𝑖. This is in turn used to prove continuity of 𝑌^−,𝑖, which finally is used to derive continuity of 𝑌^+,𝑖.

Theorem 6. Let the generators in the system (29) be nondecreasing in the third argument; that is, 𝑦₂⟶𝜓^±_𝑖𝑡,𝑦₁,𝑦₂,𝑧,𝑖=1,2, (35) are nondecreasing functions. Then, under the assumptions (A1)–(A4), the system (29) admits a minimal solution such that 𝑡→𝑌_𝑡^±,𝑖,𝑖=1,2, is continuous. The solution is minimal in the sense that if (𝑌^±,𝑖,𝑍^±,𝑖,𝐾^±,𝑖), 𝑖=1,2, is another solution, then 𝑌_𝑡^±,𝑖≤𝑌_𝑡^±,𝑖, 0≤𝑡≤𝑇, a.s. The solution to the system (32) is not unique in general.

Proof. The theorem is proved using an approximating scheme. Let 𝑌^+,𝑖,0,𝑍^+,𝑖,0=𝐹𝜉⁺_𝑖,𝜓⁺_𝑖, (36) and denote 𝐿^𝑖_𝑡∶=𝑌_𝑡^+,𝑖,0+𝑏_𝑖(𝑡). (37) Then 𝐿^𝑖_𝑡=𝜉⁺_𝑖+^𝑇_𝑡𝜓⁺_𝑖𝑠,𝑌_𝑠^+,𝑖,0,𝔼𝑌_𝑠^+,𝑖,0,𝑍_𝑠^+,𝑖,0𝑑𝑠−^𝑇_𝑡𝑍_𝑠^+,𝑖,0𝑑𝐵_𝑠+𝑏_𝑖(𝑇)−^𝑇_𝑡𝑈_𝑖(𝑠)𝑑𝑠−^𝑇_𝑡𝑉_𝑖(𝑠)𝑑𝐵_𝑠=𝐿^𝑖_𝑇+^𝑇_𝑡𝜓_𝑖𝑠,𝐿^𝑖_𝑠,𝔼𝐿^𝑖_𝑠,𝑍^𝑖_𝑠𝑑𝑠−^𝑇_𝑡𝑍^𝑖_𝑠𝑑𝐵_𝑠, (38) where 𝑍^𝑖_𝑡∶=𝑍_𝑡^+,𝑖,0+𝑉_𝑖(𝑡),𝜓_𝑖𝑠,𝐿^𝑖_𝑠,𝔼𝐿^𝑖_𝑠,𝑍^𝑖_𝑠∶=𝜓⁺_𝑖𝑠,𝐿^𝑖_𝑠−𝑏_𝑖(𝑠),𝔼𝐿^𝑖_𝑠−𝔼𝑏_𝑖(𝑠),𝑍^𝑖_𝑠−𝑉_𝑖(𝑠)−𝑈_𝑖(𝑠). (39)
Now let (𝑌̇,𝑍̇) be the unique solution to the BSDE 𝑌̇_𝑡=𝑌̇_𝑇+^𝑇_𝑡𝛼𝑠,𝑌̇_𝑠,𝔼𝑌̇_𝑠,𝑍̇_𝑠𝑑𝑠−^𝑇_𝑡𝑍̇_𝑠𝑑𝐵_𝑆, (40) where 𝛼𝑡,𝜔,𝑦₁,𝑦₂𝜓,𝑧∶=₁∧𝜓₂∧𝜓⁻₁∧𝜓⁻₂𝑡,𝜔,𝑦₁,𝑦₂,𝑌̇,𝑧_𝑇∶=𝜉⁺₁+𝑏₁(𝑇)∧𝜉⁺₂+𝑏₂(𝑇)∧𝜉⁻₁∧𝜉⁻₂. (41) By Proposition 4 it holds that 𝑌̇_𝑡≤𝐿^𝑖_𝑡=𝑌_𝑡^+,𝑖,0+𝑏_𝑖(𝑡). (42) Hence, 𝑌̇_𝑡≤𝑌_𝑡^+,𝑖,0+𝑏_𝑖(𝑡)∧𝑌̇_𝑡+ℓ_𝑖(𝑡), (43) since ℓ_𝑖(𝑡)>0 a.s.
Consider now the processes 𝑌^−,𝑖,1,𝑍^−,𝑖,1,𝐾^−,𝑖,1=𝐹₋𝑌̇_𝑇,𝜓⁻_𝑖,𝑆^−,𝑖,1,𝑌^+,𝑖,1,𝑍^+,𝑖,1,𝐾^+,𝑖,1=𝐹₊𝜉⁺_𝑖,𝜓⁺_𝑖,𝑆^+,𝑖,1, (44) where 𝑆_𝑡^−,𝑖,1=𝑌_𝑡^+,𝑖,0+𝑏_𝑖(𝑡)∧𝑌̇_𝑡+ℓ_𝑖(𝑡),𝑆_𝑡^+,1,1=𝑌_𝑡^+,2,0−ℓ₁(𝑡)∨𝑌_𝑡^−,1,1−𝑎₁(𝑡),𝑆_𝑡^+,2,1=𝑌_𝑡^+,1,0−ℓ₂(𝑡)∨𝑌_𝑡^−,2,1−𝑎₂(𝑡), (45) and, for 𝑛≥1, 𝑌^{−,𝑖,𝑛+1},𝑍^{−,𝑖,𝑛+1},𝐾^{−,𝑖,𝑛+1}=𝐹₋𝜉⁻_𝑖,𝜓⁻_𝑖,𝑆^{−,𝑖,𝑛+1},𝑌^{+,𝑖,𝑛+1},𝑍^{+,𝑖,𝑛+1},𝐾^{+,𝑖,𝑛+1}=𝐹₊𝜉⁺_𝑖,𝜓⁺_𝑖,𝑆^{+,𝑖,𝑛+1}, (46) where 𝑆_𝑡^−,1,𝑛+1=𝑌_𝑡^−,2,𝑛+ℓ₁(𝑡)∧𝑌_𝑡^+,1,𝑛+𝑏₁(𝑡,𝑆_{𝑡−,2,𝑛+1})=𝑌_𝑡^−,1,𝑛+ℓ₂(𝑡)∧𝑌_𝑡^+,2,𝑛+𝑏₂(𝑡),𝑆_𝑡^+,1,𝑛+1=𝑌_𝑡^+,2,𝑛−ℓ₁(𝑡)∨𝑌_𝑡^−,1,𝑛+1−𝑎₁(𝑡),𝑆_𝑡^+,2,𝑛+1=𝑌_𝑡^+,1,𝑛−ℓ₂(𝑡)∨𝑌_𝑡^−,2,𝑛+1−𝑎₂(𝑡). (47) Since (𝑌^+,𝑖,0,𝑍^+,𝑖,0) is the solution of a standard MF-BSDE, the existence and uniqueness have been established in [6], and the existence and uniqueness of the processes (𝑌^−,𝑖,1,𝑍^−,𝑖,1,𝐾^−,𝑖,1) were established in Proposition 2. With this in mind, it is easily shown by the use of induction that for any 𝑛≥1 the triples (𝑌^+,𝑖,𝑛,𝑍^+,𝑖,𝑛,𝐾^+,𝑖,𝑛) exist, are unique, and belong to the appropriate spaces.
From Proposition 4 and the fact that 𝐾_𝑇^+,𝑖,1−𝐾_𝑡^+,𝑖,1≥0 it follows that 𝑌_𝑡^+,𝑖,0≤𝑌_𝑡^+,𝑖,1. Moreover, by Proposition 4 again, it follows that 𝑌̇_𝑡≤𝑌_𝑡^−,𝑖,1. Hence 𝑌̇_𝑡+ℓ₁(𝑡)∧𝑌_𝑡^+,1,0+𝑏₁(𝑡≤𝑌_𝑡−,2,1)+ℓ₁(𝑡)∧𝑌_𝑡^+,1,1+𝑏₁(𝑡),𝑌̇_𝑡+ℓ₂(𝑡)∧𝑌_𝑡^+,2,0+𝑏₂(𝑡)≤𝑌_𝑡^−,1,1+ℓ₂(𝑡)∧𝑌_𝑡^+,2,1+𝑏₂(𝑡). (48) Proposition 4 then yields 𝑌_𝑡^−,𝑖,1≤𝑌_𝑡^−,𝑖,2.
Now assume that 𝑌_𝑡^+,𝑖,𝑛≤𝑌_𝑡^{+,𝑖,𝑛+1},𝑌_𝑡^{−,𝑖,𝑛+1}≤𝑌_𝑡^{−,𝑖,𝑛+2}. (49) Then 𝑆_𝑡^+,1,𝑛+1=𝑌_𝑡^+,2,𝑛−ℓ₁(𝑡)∨𝑌_𝑡^−,1,𝑛+1−𝑎₁(𝑡≤𝑌_{𝑡+,2,𝑛+1})−ℓ₁(𝑡)∨𝑌_𝑡^−,1,𝑛+2−𝑎₁(𝑡)=𝑆_𝑡^+,1,𝑛+2,𝑆_𝑡^+,2,𝑛+1=𝑌_𝑡^+,1,𝑛−ℓ₂(𝑡)∨𝑌_𝑡^−,2,𝑛+1−𝑎₂(𝑡)≤𝑌_𝑡^+,1,𝑛+1−ℓ₂(𝑡)∨𝑌_𝑡^−,2,𝑛+2−𝑎₂(𝑡)=𝑆_𝑡^+,2,𝑛+2, (50) from which it follows, by Proposition 4 again, that 𝑌_𝑡^{+,𝑖,𝑛+1}≤𝑌_𝑡^{+,𝑖,𝑛+2}. Hence, it also holds that 𝑆_𝑡^−,1,𝑛+2=𝑌_𝑡^−,2,𝑛+1+ℓ₁(𝑡)∧𝑌_𝑡^+,1,𝑛+1+𝑏₁(𝑡≤𝑌_{𝑡−,2,𝑛+2})+ℓ₁(𝑡)∧𝑌_𝑡^+,1,𝑛+2+𝑏₁(𝑡)=𝑆_𝑡^−,1,𝑛+3,𝑆_𝑡^−,2,𝑛+2=𝑌_𝑡^−,1,𝑛+1+ℓ₂(𝑡)∧𝑌_𝑡^+,2,𝑛+1+𝑏₂(𝑡)≤𝑌_𝑡^−,1,𝑛+2+ℓ₂(𝑡)∧𝑌_𝑡^+,2,𝑛+2+𝑏₂(𝑡)=𝑆_𝑡^−,2,𝑛+3. (51) Thus Proposition 4 yields that 𝑌_𝑡^{−,𝑖,𝑛+2}≤𝑌_𝑡^{−,𝑖,𝑛+3}. By induction, 𝑌_𝑡^+,𝑖,𝑛≤𝑌_𝑡^{+,𝑖,𝑛+1},𝑌_𝑡^{−,𝑖,𝑛+1}≤𝑌_𝑡^{−,𝑖,𝑛+2}, (52) for 𝑛≥1.
By Proposition 3 these sequences are bounded, and since they are also increasing the limits exist. Denote these limits by 𝑌_𝑡^+,𝑖∶=lim_𝑛→∞𝑌_𝑡^+,𝑖,𝑛,𝑌_𝑡^−,𝑖∶=lim_𝑛→∞𝑌_𝑡^{−,𝑖,𝑛}. (53) 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 𝐶 such that, for all 𝑛≥1, 𝔼^𝑇₀𝑑𝐾_𝑡^{−,𝑖,𝑛}𝑑𝑡²𝑑𝑡≤𝐶. (54) The first step towards proving this claim is to prove the absolute continuity of 𝑑𝐾_𝑡^−,𝑖,1 with respect to 𝑑𝑡. Noting that 𝑆_𝑡^−,𝑖,1=𝐿^𝑖_𝑡∧𝑌̇_𝑡+ℓ_𝑖(𝑡)=𝐿^𝑖_𝑡−𝐿^𝑖_𝑡−𝑌̇_𝑡−ℓ_𝑖(𝑡)⁺, (55) then in view of (A4) and the Itô-Tanaka formula, we get 𝑆_𝑡^−,𝑖,1=𝑆₀^−,𝑖,1+^𝑡₀𝑓_𝑖(𝑠)𝑑𝑠+^𝑡₀𝑔_𝑖(𝑠)𝑑𝐵_𝑠−12ℒ^𝑖_𝑡, (56) where ℒ^𝑖 is the local time at zero of the continuous semimartingale 𝐿^𝑖−𝑌̇−ℓ_𝑖 and where 𝑓_𝑖(𝑡)∶=−𝜓_𝑖𝑡,𝐿^𝑖_𝑡,𝔼𝐿^𝑖_𝑡,𝑍^𝑖_𝑡+𝟏_{𝐿^𝑖_{𝑡>̇𝑌𝑡+ℓ𝑖(𝑡)}}×𝜓_𝑖𝑡,𝐿^𝑖_𝑡,𝔼𝐿^𝑖_𝑡,𝑍^𝑖_𝑡−𝛼𝑡,𝑌̇_𝑡,𝔼𝑌̇_𝑡,𝑍̇_𝑡+𝑈_𝑖(𝑡),𝑔_𝑖(𝑡)∶=𝑍^𝑖_𝑡−𝟏_{𝐿^𝑖_{𝑡>̇𝑌𝑡+ℓ𝑖(𝑡)}}𝑍^𝑖_𝑡−𝑍̇_𝑡−𝑉_𝑖(𝑡). (57) It follows that 𝑑𝑆_𝑡^−,𝑖,1−𝑌_𝑡^−,𝑖,1=𝑓_𝑖(𝑡)+𝜓⁻_𝑖𝑡,𝑌_𝑡^−,𝑖,1,𝔼𝑌_𝑡^−,𝑖,1,𝑍_𝑡^−,𝑖,1+𝑔𝑑𝑡_𝑖(𝑡)−𝑍_𝑡^−,𝑖,1𝑑𝐵_𝑡−12𝑑ℒ^𝑖_𝑡−𝑑𝐾_𝑡^−,𝑖,1. (58) From Itô-Tanaka again we get that 𝑑𝑆_𝑡^−,𝑖,1−𝑌_𝑡^−,𝑖,1+=𝟏_{𝑆𝑡^−,𝑖,1_>𝑌𝑡^−,𝑖,1_}𝑓_𝑖(𝑡)+𝜓⁻_𝑖𝑡,𝑌_𝑡^−,𝑖,1,𝔼𝑌_𝑡^−,𝑖,1,𝑍_𝑡^−,𝑖,1𝑑𝑡+𝟏_{𝑆𝑡^−,𝑖,1_>𝑌𝑡^−,𝑖,1_}𝑔_𝑖(𝑡)−𝑍_𝑡^−,𝑖,1𝑑𝐵_𝑡−𝟏_{𝑆𝑡^−,𝑖,1_>𝑌𝑡^−,𝑖,1_}𝑑𝐾_𝑡^−,𝑖,1+12𝑑ℒ^𝑖_𝑡+12𝑑Λ^𝑖_𝑡, (59) where Λ^𝑖_𝑡 is the local time at 0 for 𝑆^−,𝑖,1−𝑌^−,𝑖,1. Since (𝑆_𝑡^−,𝑖,1−𝑌_𝑡^−,𝑖,1)⁺≡𝑆_𝑡^−,𝑖,1−𝑌_𝑡^−,𝑖,1 the differentials must coincide, which yields that12𝑑Λ^𝑖_𝑡=𝟏_{𝑌𝑡^−,𝑖,1_=𝑆𝑡^−,𝑖,1_}𝑓_𝑖(𝑡)+𝜓⁻_𝑖𝑡,𝑌_𝑡^−,𝑖,1,𝔼𝑌_𝑡^−,𝑖,1,𝑍_𝑡^−,𝑖,1−1𝑑𝑡2𝑑ℒ^𝑖_𝑡−𝑑𝐾_𝑡^−,𝑖,1=𝟏_{𝑌𝑡^−,𝑖,1_=𝑆𝑡^−,𝑖,1_}𝑓_𝑖(𝑡)+𝜓⁻_𝑖𝑡,𝑌_𝑡^−,𝑖,1,𝔼𝑌_𝑡^−,𝑖,1,𝑍_𝑡^−,𝑖,1−1𝑑𝑡2𝑑ℒ^𝑖_𝑡−𝑑𝐾_𝑡^−,𝑖,1. (60)Hence 𝑑𝐾_𝑡^−,𝑖,1+12𝑑Λ^𝑖_𝑡+𝟏_{𝑌𝑡^−,𝑖,1_=𝑆𝑡^−,𝑖,1_}𝑑ℒ^𝑖_𝑡=𝟏_{𝑌𝑡^−,𝑖,1_=𝑆𝑡^−,𝑖,1_}𝑓_𝑖(𝑡)+𝜓⁻_𝑖𝑡,𝑌_𝑡^−,𝑖,1,𝔼𝑌_𝑡^−,𝑖,1,𝑍_𝑡^−,𝑖,1𝑑𝑡, (61) from which it follows that 0≤𝑑𝐾_𝑡^−,𝑖,1≤𝟏_{𝑌𝑡^−,𝑖,1_=𝑆𝑡^−,𝑖,1_}𝑓_𝑖(𝑡)+𝜓⁻_𝑖𝑡,𝑌_𝑡^−,𝑖,1,𝔼𝑌_𝑡^−,𝑖,1,𝑍_𝑡^−,𝑖,1≤𝑓𝑑𝑡_𝑖(𝑡)+𝜓⁻_𝑖𝑡,𝑌_𝑡^−,𝑖,1,𝔼𝑌_𝑡^−,𝑖,1,𝑍_𝑡^−,𝑖,1𝑑𝑡. (62) Now, in view of (A1)–(A4) and Proposition 3 there exists a constant 𝐶 such that 𝔼^𝑇₀𝑓_𝑖(𝑡)²+𝜓⁻_𝑖𝑡,𝑌_𝑡^−,𝑖,1,𝔼𝑌_𝑡^−,𝑖,1,𝑍_𝑡^−,𝑖,12+𝑔_𝑖(𝑡)²𝑑𝑡≤𝐶, (63) which together with (62) proves (54) for 𝑛=1.
For the induction step, we only consider the case 𝑖=1 since the other case follows in a similar fashion. Assume that 𝑑𝐾_𝑡^{−,𝑖,𝑛}/𝑑𝑡 satisfies (54) and consider the obstacle process 𝑆_𝑡^−,1,𝑛+1=𝑌_𝑡^−,2,𝑛+ℓ₁(𝑡)∧𝑌_𝑡^+,1,𝑛+𝑏₁(𝑡)=𝑌_𝑡^+,1,𝑛+𝑏₁(𝑡)−𝑌_𝑡^+,1,𝑛+𝑏₁(𝑡)−𝑌_𝑡^−,2,𝑛−ℓ₁(𝑡)⁺. (64) By (A4) and the Itô-Tanaka formula, we get 𝑆_𝑡^−,1,𝑛+1=𝑆₀^−,1,𝑛+1+^𝑡₀𝑓_𝑛+1(𝑠)𝑑𝑠+^𝑡₀𝑔_𝑛+1(𝑠)𝑑𝐵_𝑠−12ℒ_𝑡^1,𝑛−𝟏_{𝑌𝑡^+,1,𝑛_{+𝑏1(𝑡)≤𝑌𝑡}^−,2,𝑛_+ℓ1(𝑡)}𝑑𝐾_𝑡^+,1,𝑛, (65) where ℒ^1,𝑛 is the local time at 0 for the continuous semimartingale 𝑌_𝑡^+,1,𝑛+𝑏₁(𝑡)−𝑌_𝑡^−,2,𝑛−ℓ₁(𝑡) and where 𝑓_𝑛+1(𝑡)∶=−𝜓⁺₁𝑡,𝑌_𝑡^+,1,𝑛,𝔼𝑌_𝑡^+,1,𝑛,𝑍_𝑡^+,1,𝑛+𝑈₁(𝑡)−𝟏_{𝑌𝑡^+,1,𝑛_{+𝑏1(𝑡)>𝑌𝑡}^−,2,𝑛_+ℓ1(𝑡)}×−𝜓⁺₁𝑡,𝑌_𝑡^+,1,𝑛,𝔼𝑌_𝑡^+,1,𝑛,𝑍_𝑡^+,1,𝑛+𝑈₁(𝑡)+𝜓⁻₂𝑡,𝑌_𝑡^−,2,𝑛,𝔼𝑌_𝑡^−,2,𝑛,𝑍_𝑡^−,2,𝑛−𝑈₁(𝑡)−𝑑𝐾_𝑡^−,2,𝑛,𝑓𝑑𝑡_𝑛+1(𝑡)∶=𝑍_𝑡^+,1,𝑛+𝑉₁(𝑡)−𝟏_{𝑌𝑡^+,1,𝑛_{+𝑏1(𝑡)>𝑌𝑡}^−,2,𝑛_+ℓ1(𝑡)}×𝑍_𝑡^+,1,𝑛+𝑉₁(𝑡)−𝑍_𝑡^−,2,𝑛−𝑉₁(𝑡). (66) In view of (A1)–(A4) and Proposition 3, together with the induction assumption, there exists a constant 𝐶>0, independent of 𝑛, such that 𝔼^𝑇₀𝑓_𝑛+1(𝑡)²+𝜓⁻₁𝑡,𝑌_𝑡^−,1,𝑛+1,𝔼𝑌_𝑡^−,1,𝑛+1,𝑍_𝑡^{−,1,𝑛+12}+𝑔_𝑛+1(𝑡)²𝑑𝑡≤𝐶. (67) Following the same steps as we did earlier for 𝑑𝐾_𝑡^−,𝑖,1 it can be shown that 0≤𝑑𝐾_𝑡^−,1,𝑛+1≤𝑓_𝑛+1(𝑡)+𝜓⁻₁𝑡,𝑌_𝑡^−,1,𝑛+1,𝔼𝑌_𝑡^−,1,𝑛+1,𝑍_𝑡^−,1,𝑛+1𝑑𝑡, (68) which yields that there exists a constant 𝐶>0 independent of 𝑛 such that 𝔼^𝑇₀𝑑𝐾_𝑡^−,1,𝑛+1𝑑𝑡²𝑑𝑡≤𝐶. (69) Hence claim (54) is true for all 𝑛≥1.
Proposition 3 and estimate (54) tell us that there is a subsequence along which the sequences of processes (𝑑𝐾_𝑡^{−,𝑖,𝑛}/𝑑𝑡), (𝜓⁻_𝑖(𝑡,𝑌_𝑡^{−,𝑖,𝑛},𝔼𝑌_𝑡^{−,𝑖,𝑛},𝑍_𝑡^{−,𝑖,𝑛})), and (𝑍_𝑡^{−,𝑖,𝑛}) converge weakly in their respective spaces ℳ^1,2, ℳ^1,2, and ℳ^𝑑,2 to the processes (𝑘_𝑡^−,𝑖,0≤𝑡≤𝑇), (𝜑_𝑡^−,𝑖,0≤𝑡≤𝑇), and (𝑍_𝑡^−,𝑖,0≤𝑡≤𝑇).
Now for any 𝑛≥0 and any 𝔽-stopping time 𝜏, we have 𝑌_𝜏^{−,𝑖,𝑛}=𝑌₀^{−,𝑖,𝑛}−^𝜏₀𝜓⁻_𝑖𝑠,𝑌_𝑠^{−,𝑖,𝑛},𝔼𝑌_𝑠^{−,𝑖,𝑛},𝑍_𝑠^{−,𝑖,𝑛}𝑑𝑠+𝐾_𝜏^{−,𝑖,𝑛}+^𝜏₀𝑍_𝑠^{−,𝑖,𝑛}𝑑𝐵_𝑠. (70) Taking the weak limits in each side of this equation along the subsequence mentioned earlier yields 𝑌_𝜏^−,𝑖=𝑌₀^−,𝑖−^𝜏₀𝜑_𝑠^−,𝑖𝑑𝑠+^𝜏₀𝑘_𝑠^−,𝑖𝑑𝑠+^𝜏₀𝑍_𝑠^−,𝑖𝑑𝐵_𝑠,ℙ-a.s. (71) 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 𝑌_𝑡^−,𝑖=𝑌₀^−,𝑖−^𝑡₀𝜑_𝑠^−,𝑖𝑑𝑠+^𝑡₀𝑘_𝑠^−,𝑖𝑑𝑠+^𝑡₀𝑍_𝑠^−,𝑖𝑑𝐵_𝑠,∀𝑡≤𝑇,ℙ-a.s. (72) Therefore, the process 𝑌^−,𝑖 is continuous.
Relying on both Dini’s theorem and Lebesgue’s dominated convergence theorem we find that lim_𝑛→∞𝔼sup_{0≤𝑡≤𝑇}𝑌_𝑡^{−,𝑖,𝑛}−𝑌_𝑡^−,𝑖2=0. (73)
In what follows we will characterize the limit processes 𝑌^+,𝑖 of the sequence {𝑌^+,𝑖,𝑛}_𝑛≥0 as Snell envelopes of the processes in the sense that, ℙ-a.s. and for each 𝑡≤𝑇, 𝑌_𝑡^+,𝑖=esssup_𝑇≥𝑡𝔼^𝜏_𝑡𝜓⁺₁𝑠,𝑌_𝑠^+,𝑖,𝔼𝑌_𝑠^+,𝑖,𝑍_𝑠^+,𝑖𝑑𝑠+𝜉⁺_𝑖𝟏_{𝜏=𝑇}+𝑆_𝜏^+,𝑖𝟏_{𝜏<𝑇}∣ℱ_𝑡, (74) and then we derive their time continuity.
In view of Proposition 3 and applying Peng’s monotone limit theorem (see [13]) to the sequence (𝑌^+,𝑖,𝑛,𝑍^+,𝑖,𝑛,𝐾^+,𝑖,𝑛), we obtain that 𝑌^+,𝑖 is càdlàg. Moreover, there exist a càdlàg nondecreasing process 𝐾^+,𝑖∈𝒦² and a process 𝑍^+,𝑖∈ℳ^𝑑,2 such that 𝑌_𝑡^+,𝑖=𝜉_𝑖+^𝑇_𝑡𝜓⁺_𝑖𝑠,𝑌_𝑠^+,𝑖,𝔼𝑌_𝑠^+,𝑖,𝑍_𝑠^+,𝑖𝑑𝑠+𝐾_𝑇^+,𝑖−𝐾_𝑡^+,𝑖−^𝑇_𝑡𝑍_𝑠^+,𝑖𝑑𝐵_𝑠,𝑌_𝑡^+,𝑖≥𝑆_𝑡^+,𝑖. (75) Hence, in view of Proposition 1 we arrive at (74) once we show that ^𝑇₀𝑌_𝑡^+,𝑖₋−𝑆_𝑡^+,𝑖₋𝑑𝐾_𝑡^+,𝑖=0. (76)
Consider the smallest 𝜓⁺_𝑖-supermartingale 𝑌^+,𝑖 with lower obstacle 𝑆^+,𝑖, which solves 𝑌_𝑡^+,𝑖=𝜉⁺_𝑖+^𝑇_𝑡𝜓⁺_𝑖𝑠,𝑌_𝑠^+,𝑖,𝔼𝑌_𝑠^+,𝑖,𝑍_𝑠^+,𝑖+𝐾𝑑𝑠_𝑇+,𝑖−𝐾_𝑡^+,𝑖−^𝑇_𝑡𝑍_𝑠^+,𝑖𝑑𝐵_𝑠,𝑌_𝑡^+,𝑖≥𝑆_𝑡^+,𝑖,_𝑇0𝑌_𝑡^+,𝑖₋−𝑆_𝑡^+,𝑖₋𝑑𝐾_𝑡^+,𝑖=0. (77) By definition, we have that 𝑌_𝑡^+,𝑖≥𝑌_𝑡^+,𝑖. Using Proposition 5, we see that 𝑌_𝑡^+,𝑖≥𝑌_𝑡^+,𝑖,𝑛, 𝑛≥1. Passing to the limit we get that 𝑌^+,𝑖=𝑌^+,𝑖, 𝑖=1,2. Hence, (76) is satisfied.
It remains to show that 𝑌^+,𝑖 are continuous. Noting that 𝑡>𝑡⁻ and that 𝑌^+,𝑖 satisfies (74), we find that 𝑌_𝑡^+,𝑖≤𝑌_𝑡^+,𝑖₋ or equivalently that Δ𝑌_𝑡^+,𝑖≤0. Thus, it is sufficient to prove that the set {Δ𝑌_𝑡^+,𝑖<0} is empty.
By the Doob-Meyer decomposition of the Snell envelope (see Proposition 1) there exists, for each 𝑖, a continuous martingale (𝑀^𝑖_𝑡)_{0≤𝑡≤𝑇}, a continuous nondecreasing process (𝐶^𝑖_𝑡)_{0≤𝑡≤𝑇}, and a purely discontinuous process (𝐷^𝑖_𝑡)_{0≤𝑡≤𝑇}, with 𝐶^𝑖₀=𝐷^𝑖₀=0, such that 𝑌_𝑡^+,𝑖+^𝑡₀𝜓⁺_𝑖𝑠,𝑌_𝑠^+,𝑖,𝔼𝑌_𝑠^+,𝑖,𝑍_𝑠^+,𝑖𝑑𝑠=𝑀^𝑖_𝑡−𝐶^𝑖_𝑡−𝐷^𝑖_𝑡,∀𝑡≤𝑇, (78) which means that Δ𝑌_𝑡^+,𝑖=−Δ𝐷^𝑖_𝑡. Now in view of Proposition 1 we have the following properties of the jumps of 𝐷^𝑖. If there is a jump at time 𝑡 in 𝐷¹, then this means that the process (𝑌^+,2−ℓ₂)∨(𝑌^−,1−𝑎₁) also jumps. Since ℓ₁ and 𝑌^−,1−𝑎₁ are continuous, this can only mean that there is a jump in 𝑌^+,2. This, in turn, means that there is a jump in 𝐷². And conversely, by the same type of reasoning, we can deduce that if there is a jump in 𝐷², there is also a jump in 𝐷¹. Hence, 𝐷¹ and 𝐷² always jump at the same time.
With this in mind, denote 𝑋∶=(𝑌^+,2−ℓ₁) and 𝐹∶=𝑌^−,1−𝑎₁. We have {Δ()}=𝑋𝑋∨𝐹<0_𝑡∨𝐹_𝑡<𝑋_𝑡−∨𝐹_𝑡=𝑋_𝑡∨𝐹_𝑡<𝑋_𝑡−∨𝐹_𝑡∩𝑋_𝑡<𝐹_𝑡<𝑋_𝑡−∪𝑋_𝑡∨𝐹_𝑡<𝑋_𝑡−∨𝐹_𝑡∩𝑋_𝑡<𝑋_𝑡−<𝐹_𝑡∪𝑋_𝑡∨𝐹_𝑡<𝑋_𝑡−∨𝐹_𝑡∩𝐹_𝑡<𝑋_𝑡<𝑋_𝑡−=𝐹_𝑡<𝑋_𝑡−∩𝑋_𝑡<𝐹_𝑡<𝑋_𝑡−∪𝐹_𝑡<𝐹_𝑡∩𝑋_𝑡<𝑋_𝑡−<𝐹_𝑡∪𝑋_𝑡<𝑋_𝑡−∩𝐹_𝑡<𝑋_𝑡<𝑋_𝑡−=𝐹_𝑡<𝑋_𝑡−∩𝑋_𝑡<𝐹_𝑡∩𝐹_𝑡<𝑋_𝑡−∪0/∪𝑋_𝑡<𝑋_𝑡−∩𝐹_𝑡<𝑋_𝑡∩𝑋_𝑡<𝑋_𝑡−=𝐹_𝑡<𝑋_𝑡−∩𝑋_𝑡<𝐹_𝑡∪𝑋_𝑡<𝑋_𝑡−∩𝐹_𝑡<𝑋_𝑡=𝐹_𝑡<𝑋_𝑡−∩𝑋_𝑡<𝐹_𝑡∩𝑋_𝑡<𝑋_𝑡−∪𝑋_𝑡<𝑋_𝑡−∩𝐹_𝑡<𝑋_𝑡∩𝐹_𝑡<𝑋_𝑡−=𝑋_𝑡<𝑋_𝑡−∩𝐹_𝑡<𝑋_𝑡−∩𝑋_𝑡<𝐹_𝑡∪𝐹_𝑡<𝑋_𝑡=𝑋_𝑡<𝑋_𝑡−∩𝐹_𝑡<𝑋_𝑡−=Δ𝑋_𝑡<0∩𝐹_𝑡<𝑋_𝑡−=Δ𝑌_𝑡^+,2<0,𝑌_𝑡^−,1−𝑎₁(𝑡)<𝑌_𝑡^+,2₋−ℓ₁(𝑡). (79) What this tells us is that if 𝑆^+,1 jumps at time 𝑡, then we know two things. One is that 𝑌^+,2 also has a jump at time 𝑡 and the other is that 𝑌_𝑡^−,1−𝑎₁(𝑡)<𝑌_𝑡^+,2₋−ℓ₁(𝑡). (80) Therefore, since Δ𝑌_𝑡^+,𝑖=−Δ𝐷^𝑖_𝑡, in view of Proposition 1, we obtain that Δ𝐷¹_𝑡⊂>0Δ𝐷²_𝑡>0∩𝑌_𝑡^+,1₋=𝑆_𝑡^+,1₋=Δ𝐷²_𝑡>0∩𝑌_𝑡^+,1₋=𝑌_𝑡^+,2₋−ℓ₁(𝑡)∨𝑌_𝑡^−,1−𝑎₁(𝑡)=Δ𝐷²_𝑡>0∩𝑌_𝑡^+,1₋=𝑌_𝑡^+,2₋−ℓ₁(𝑡). (81) Similarly, we have Δ𝐷²_𝑡⊂>0Δ𝐷¹_𝑡>0∩𝑌_𝑡^+,2₋=𝑆_𝑡^+,2₋=Δ𝐷¹_𝑡>0∩𝑌_𝑡^+,2₋=𝑌_𝑡^+,1₋−ℓ₂(𝑡)∨𝑌_𝑡^−,2−𝑎₂(𝑡)=Δ𝐷¹_𝑡>0∩𝑌_𝑡^+,2₋=𝑌_𝑡^+,1₋−ℓ₂(𝑡). (82)
Finally, since we know that 𝐷¹ and 𝐷² always jump at the same time, it holds that Δ𝐷¹_𝑡>0∪Δ𝐷¹_𝑡=>0Δ𝐷¹_𝑡>0∩Δ𝐷²_𝑡=𝑌>0_𝑡+,1−=𝑌_𝑡^+,2₋−ℓ₁(𝑡∩𝑌_𝑡+,2−)=𝑌_𝑡^+,1₋−ℓ₂(𝑡)=0/, (83) since, for any 𝑡≤𝑇, ℓ₁(𝑡)+ℓ₂(𝑡)>0. It follows that the processes 𝐷¹ and 𝐷² are constant and identically equal to 0, since 𝐷¹₀=𝐷²₀=0. Hence, the processes 𝑌^+,1 and 𝑌^+,2 are continuous. This, in turn, given (75), yields the continuity of the increasing processes 𝐾^+,𝑖. Therefore, (𝑌^+,𝑖,𝑍^+,𝑖,𝐾^+,𝑖) 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 (𝑌^+,1,𝑛)_𝑛≥1 to 𝑌^+,1 holds in 𝑆²_𝑐; that is, lim_𝑛→∞𝔼sup_{0≤𝑡≤𝑇}𝑌_𝑡^+,𝑖,𝑛−𝑌_𝑡^+,𝑖2=0. (84) Furthermore, since 𝑌^−,𝑖 is continuous, we can rely on standard arguments, in particular, by applying Itô’s formula to (𝑌^+,𝑖,𝑛−𝑌^+,𝑖,𝑚)²,(𝑚,𝑛≥0), to claim that (𝑍^+,𝑖,𝑛)_≥0 is a Cauchy sequence, and therefore it converges to 𝑍^+,𝑖 in ℳ^𝑑,2; that is, lim_𝑛→∞𝔼^𝑇₀𝑍_𝑡^+,𝑖,𝑛−𝑍_𝑡^+,𝑖2𝑑𝑡=0. (85) Combining this with (84) and the definition of the processes 𝑌^+,𝑖,𝑛 and 𝑌^+,𝑖 yields that lim_𝑛→∞𝔼sup_{0≤𝑡≤𝑇}𝐾_𝑡^+,𝑖,𝑛−𝐾_𝑡^+,𝑖2=0. (86)
In total, we have shown that lim_𝑛→∞𝔼sup_{0≤𝑡≤𝑇}𝑌_𝑡^+,𝑖,𝑛−𝑌_𝑡^+,𝑖2+^𝑇₀𝑍_𝑡^+,𝑖,𝑛−𝑍_𝑡^+,𝑖2𝑑𝑡+sup_{0≤𝑡≤𝑇}𝐾_𝑡^+,𝑖,𝑛−𝐾_𝑡^+,𝑖2=0. (87)
Using (84) and applying Itô’s formula to (𝑌^{−,𝑖,𝑛}−𝑌^{−,𝑖,𝑚})², (𝑚,𝑛≥0), it follows that (𝑍^{−,𝑖,𝑛})_𝑛≥1 is a Cauchy sequence and therefore converges to 𝑍^−,𝑖 in ℳ^𝑑,2; that is, lim_𝑛→∞𝔼^𝑇₀𝑍_𝑡^{−,𝑖,𝑛}−𝑍_𝑡^−,𝑖2𝑑𝑡=0. (88) Using this and taking into account the decomposition obtained in (72), we arrive at 𝑌_𝑡^−,𝑖=𝜉⁻_𝑖+^𝑇_𝑡𝜓⁻_𝑖𝑠,𝑌_𝑠^−,𝑖,𝔼𝑌_𝑠^−,𝑖,𝑍_𝑠^−,𝑖−𝑑𝑠_𝑇𝑡𝑘_𝑠^−,𝑖𝑑𝑠−^𝑇_𝑡𝑍_𝑠^−,𝑖𝑑𝐵_𝑠,𝑌_𝑡^−,1≤𝑌_𝑡^−,2+ℓ₁(𝑡)∧𝑌_𝑡^+,1+𝑏₁(𝑡),𝑌_𝑡^−,2≤𝑌_𝑡^−,1+ℓ₂(𝑡)∧𝑌_𝑡^+,2+𝑏₂(𝑡). (89) As shown above, it also holds that lim_𝑛→∞𝔼sup_{0≤𝑡≤𝑇}𝑌_𝑡^{−,𝑖,𝑛}−𝑌_𝑡^−,𝑖2+^𝑇₀𝑍_𝑡^{−,𝑖,𝑛}−𝑍_𝑡^−,𝑖2𝑑𝑡+sup_{0≤𝑡≤𝑇}𝐾_𝑡^{−,𝑖,𝑛}−^𝑡₀𝑘_𝑠^−,𝑖𝑑𝑠²=0. (90) Moreover, due to the weak convergence of (𝑑𝐾_𝑡^{−,𝑖,𝑛}/𝑑𝑡,0≤𝑡≤𝑇,𝑛≥1) to the process 𝑘^−,𝑖 and the strong convergence of (𝑌^+,𝑖,𝑛) and (𝑌^{−,𝑖,𝑛}) in 𝒮², it follows that 0=^𝑇₀𝑆_𝑡^{−,𝑖,𝑛}−𝑌_𝑡^{−,𝑖,𝑛}𝑑𝐾_𝑡^{−,𝑖,𝑛}⟶^𝑇₀𝑆_𝑡^−,𝑖−𝑌_𝑡^−,𝑖𝑘_𝑡^−,𝑖𝑑𝑠, (91) as 𝑛→∞, which implies that if we define 𝐾_𝑡^−,𝑖∶=^𝑡₀𝑘_𝑠^−,𝑖𝑑𝑠,𝑡≤𝑇, (92) then (𝑌^−,𝑖,𝑍^−,𝑖,𝐾^−,𝑖) is a solution to the second part of (29).
It remains to show that the obtained solution is minimal. To do this, we compare 𝑌^+,𝑖 with the smallest 𝜓⁺_𝑖-supermartingale with lower obstacle 𝑆^+,𝑖, denoted by 𝑌^+,𝑖, and we compare 𝑌^+,𝑖 with the smallest 𝜓⁻_𝑖-supermartingale with upper obstacle 𝑆^−,𝑖, denoted by 𝑌^−,𝑖; that is, let (𝑌^±,𝑖,𝑍^±,𝑖,𝐾^±,𝑖) be the smallest solution to the system 𝑌^+,𝑖,𝑍^+,𝑖,𝐾^+,𝑖=𝐹₊𝜉⁺_𝑖,𝜓⁺_𝑖,𝑆^+,𝑖𝑌,𝑖=1,2,_−,𝑖,𝑍^−,𝑖,𝐾^−,𝑖=𝐹₋𝜉⁻_𝑖,𝜓⁻_𝑖,𝑆^−,𝑖,𝑖=1,2. (93) Using Proposition 4 and the minimality of the solution we can conclude that 𝑌^±,𝑖≤𝑌^±,𝑖. On the other hand, Proposition 4 also tells us that for all 𝑛≥0 we have 𝑌^±,𝑖,𝑛≤𝑌^±,𝑖. Now since 𝑌^±,𝑖,𝑛 are increasing in 𝑛, we can take the limit 𝑛→∞ and obtain that 𝑌^±,𝑖≤𝑌^±,𝑖. Thus, 𝑌^±,𝑖=𝑌^±,𝑖.
Finally to establish that the solution is not unique in general, the counterexample found in [1] is valid here as well.

Appendices

A. Proof of Proposition 2

The proof uses techniques found in [2, 7, 8]. Let 𝐒 be the space of progressively measurable {(𝑌_𝑡,𝑍_𝑡)∶0≤𝑡≤𝑇} taking values in ℝ×ℝ^𝑑 such that 𝑍∈ℳ^𝑑,2 and 𝑌_𝑡≥𝑆_𝑡 for 0≤𝑡≤𝑇.

Take any (𝑈,𝑉)∈𝐒 and define 𝑓(𝑡)∶=𝑓(𝑡,𝑈_𝑡,𝔼𝑈_𝑡,𝑉_𝑡). Then, by Proposition  5.1 in [2], there is a unique triplet (𝑌,𝑍,𝐾) such that 𝑌∈𝒮²,𝑍∈ℳ^𝑑,2,𝐾∈𝒦²,𝐾_𝑇∈𝐿²Ω,ℱ_𝑇,𝑌,ℙ_𝑡=𝜉+^𝑇_𝑡𝑓(𝑠)𝑑𝑠+𝐾_𝑇−𝐾_𝑡−^𝑇_𝑡𝑍_𝑠𝑑𝐵_𝑠𝑌,0≤𝑡≤𝑇,_𝑡≥𝑆_𝑡𝐾,0≤𝑡≤𝑇,₀=0,^𝑇₀𝑌_𝑡−𝑆_𝑡𝑑𝐾_𝑡=0. (A.1) We define a mapping Φ from 𝐒 onto itself by saying that (𝑌,𝑍)=Φ(𝑈,𝑉) is the unique element of 𝐒 such that if we define 𝐾_𝑡=𝑌_𝑡−𝑌₀−^𝑡₀𝑓(𝑠)𝑑𝑠+^𝑡₀𝑍_𝑠𝑑𝐵_𝑠,0≤𝑡≤𝑇, (A.2) then (𝑌,𝑍,𝐾) is the unique triplet which solves the system above.

Now let (𝑈^′,𝑉^′) be another element of 𝐒 and let (𝑌^′,𝑍^′)=Φ(𝑈^′,𝑉^′). In addition, let 𝑈=𝑈−𝑈^′,𝑉=𝑉−𝑉^′,𝑌=𝑌−𝑌^′,𝑍=𝑍−𝑍^′,𝑓=𝑓−𝑓^′, (A.3) where 𝑓^′(𝑡)∶=𝑓(𝑡,𝑈^′_𝑡,𝔼𝑈^′_𝑡,𝑉^′_𝑡).

For any 𝛽>0, applying Itô’s formula to 𝑒^𝛽𝑡𝑌²_𝑡 and then taking expectations we find that 𝔼𝑒^𝛽𝑡𝑌²_𝑡+^𝑇_𝑡𝛽𝑒^𝛽𝑠𝑌²_𝑠𝑑𝑠+^𝑇_𝑡𝑒^𝛽𝑠𝑍_𝑠²𝑑𝑠=2𝔼^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠𝑓(𝑠)𝑑𝑠+^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠𝑑𝐾_𝑠. (A.4) Moreover, since 2𝔼^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠𝑑𝐾_𝑠=2𝔼^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠−𝑌^′_𝑠𝑑𝐾_𝑠−𝐾^′_𝑠=2𝔼^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠−𝑆_𝑠−𝑌^′_𝑠−𝑆^′_𝑠𝑑𝐾_𝑠−𝑑𝐾^′_𝑠=−2𝔼^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠−𝑆_𝑠𝑑𝐾^′_𝑠−2𝔼^𝑇_𝑡𝑒^𝛽𝑠𝑌^′_𝑠−𝑆^′_𝑠𝑑𝐾_𝑠≤0, (A.5) it holds that 𝔼^𝑇₀𝛽𝑒^𝛽𝑠𝑌²_𝑠𝑑𝑠+^𝑇₀𝑒^𝛽𝑠𝑍_𝑠²𝑑𝑠≤2𝔼^𝑇₀𝑒^𝛽𝑠𝑌_𝑠𝑓(𝑠)𝑑𝑠. (A.6) Using the Lipschitz property of 𝑓 and then Young’s inequality yields that 𝔼^𝑇₀𝛽𝑒^𝛽𝑠𝑌²_𝑠𝑑𝑠+^𝑇₀𝑒^𝛽𝑠𝑍_𝑠²𝑑𝑠≤2𝔼^𝑇₀𝑒^𝛽𝑠𝑌_𝑠𝑓(𝑠)𝑑𝑠=2𝔼^𝑇₀𝑒^𝛽𝑠𝑌_𝑠𝑓𝑠,𝑈_𝑠,𝔼𝑈_𝑠,𝑉_𝑠−𝑓𝑠,𝑈^′_𝑠,𝔼𝑈^′_𝑠,𝑉^′_𝑠𝑑𝑠≤2𝐾𝔼^𝑇₀𝑒^𝛽𝑠𝑌_𝑠𝑈_𝑠−𝑈^′_𝑠+𝔼𝑈_𝑠−𝔼𝑈^′_𝑠+𝑉_𝑠−𝑉^′_𝑠𝑑𝑠≤2𝐾𝔼^𝑇₀𝑒^𝛽𝑠𝑌_𝑠𝑈_𝑠−𝑈^′_𝑠+𝔼𝑈_𝑠−𝑈^′_𝑠+𝑉_𝑠−𝑉^′_𝑠𝑑𝑠≤12𝐾²𝔼^𝑇₀𝑒^𝛽𝑠𝑌²_𝑠+1𝑑𝑠𝔼12^𝑇₀𝑒^𝛽𝑠𝑈_𝑠−𝑈^′_𝑠+𝔼𝑈_𝑠−𝑈^′_𝑠+𝑉_𝑠−𝑉^′_𝑠²𝑑𝑠≤12𝐾²𝔼^𝑇₀𝑒^𝛽𝑠𝑌²_𝑠+1𝑑𝑠4𝔼^𝑇₀𝑒^𝛽𝑠𝑈_𝑠−𝑈^′_𝑠²+𝔼𝑈_𝑠−𝑈^′_𝑠²+𝑉_𝑠−𝑉^′_𝑠²𝑑𝑠=12𝐾²𝔼^𝑇₀𝑒^𝛽𝑠𝑌²_𝑠+1𝑑𝑠4^𝑇₀𝑒^𝛽𝑠2𝔼𝑈²_𝑠+𝔼𝑉_𝑠²𝑑𝑠≤12𝐾²𝔼^𝑇₀𝑒^𝛽𝑠𝑌²_𝑠+1𝑑𝑠2𝔼^𝑇₀𝑒^𝛽𝑠𝑈²_𝑠+𝑉_𝑠²𝑑𝑠. (A.7) Setting 𝛽=12𝐾²+1 we find 𝔼^𝑇₀𝑒^𝛽𝑠𝑌²_𝑠+𝑍_𝑠²≤1𝑑𝑠2𝔼^𝑇₀𝑒^𝛽𝑠𝑈²_𝑠+𝑉_𝑠²𝑑𝑠. (A.8) Thus, the mapping Φ is a strict contraction on 𝐒 with the norm ‖(𝑌,𝑍)‖_𝛽=𝔼^𝑇₀𝑒^𝛽𝑠𝑌²_𝑠+𝑍_𝑠²𝑑𝑠^1/2. (A.9) The Banach fixed point theorem then yields that Φ has a unique fixed point, which is the sought solution.

B. Proof of Proposition 3

Applying Itô’s formula to 𝑌²_𝑡 yields that 𝑌²_𝑡+^𝑇_𝑡𝑍_𝑠²𝑑𝑠=𝜉²+2^𝑇_𝑡𝑌_𝑠𝑓𝑠,𝑌_𝑠,𝔼𝑌_𝑠,𝑍_𝑠𝑑𝑠+2^𝑇_𝑡𝑌_𝑠𝑑𝐾_𝑠−2^𝑇_𝑡𝑌_𝑠𝑍_𝑠𝑑𝐵_𝑠=𝜉²+2^𝑇_𝑡𝑌_𝑠𝑓𝑠,𝑌_𝑠,𝔼𝑌_𝑠,𝑍_𝑠𝑑𝑠+2^𝑇_𝑡𝑆_𝑠𝑑𝐾_𝑠−2^𝑇_𝑡𝑌_𝑠𝑍_𝑠𝑑𝐵_𝑠, (B.1) and so, by taking expectations, we get 𝔼𝑌²_𝑡+^𝑇_𝑡𝑍_𝑠²𝜉𝑑𝑠=𝔼²+2^𝑇_𝑡𝑆_𝑠𝑑𝐾_𝑠+𝔼2^𝑇_𝑡𝑌_𝑠𝑓𝑠,𝑌_𝑠,𝔼𝑌_𝑠,𝑍_𝑠𝑑𝑠. (B.2) Using the Lipschitz property of 𝑓 yields that 𝔼2^𝑇_𝑡𝑌_𝑠𝑓𝑠,𝑌_𝑠,𝔼𝑌_𝑠,𝑍_𝑡𝑑𝑠≤𝔼^𝑇_𝑡2𝑌_𝑠𝑓(𝑠,0,0,0)+2𝐾𝑌_𝑠²+2𝐾𝑌_𝑠𝔼𝑌_𝑠+2𝐾𝑌_𝑠𝑍_𝑠≤𝑑𝑠_𝑇𝑡𝔼𝑌_𝑠²+𝔼𝑓(𝑠,0,0,0)²+2𝐾𝔼𝑌_𝑠²+2𝐾𝔼𝑌_𝑠²+2𝐾²𝔼𝑌_𝑠²+12𝔼𝑍_𝑠²≤𝑑𝑠_𝑇𝑡1+4𝐾+2𝐾²𝔼𝑌_𝑠²+𝔼𝑓(𝑠,0,0,0)²+1𝑑𝑠2^𝑇_𝑡𝔼𝑍_𝑠²𝑑𝑠. (B.3) Hence 𝔼𝑌_𝑡²+12𝔼^𝑇_𝑡𝑍_𝑠²𝜉𝑑𝑠≤𝔼²+2^𝑇_𝑡𝑆_𝑠𝑑𝐾_𝑠+^𝑇_𝑡𝑓(𝑠,0,0,0)²+𝑑𝑠1+4𝐾+2𝐾^2𝑇_𝑡𝔼𝑌_𝑠²𝑑𝑠. (B.4) Therefore, 𝔼𝑌_𝑡²≤𝔼𝜉²+2^𝑇_𝑡𝑆_𝑠𝑑𝐾_𝑠+^𝑇_𝑡𝑓(𝑠,0,0,0)²+𝑑𝑠1+4𝐾+2𝐾^2𝑇_𝑡𝔼𝑌_𝑠²𝑑𝑠,12𝔼^𝑇_𝑡𝑍_𝑠²𝜉𝑑𝑠≤𝔼²+2^𝑇_𝑡𝑆_𝑠𝑑𝐾_𝑠+^𝑇_𝑡𝑓(𝑠,0,0,0)²+𝑑𝑠1+4𝐾+2𝐾^2𝑇_𝑡𝔼𝑌_𝑠²𝜉𝑑𝑠≤𝔼²+2^𝑇_𝑡𝑆⁺_𝑠𝑑𝐾_𝑠+^𝑇_𝑡𝑓(𝑠,0,0,0)²+𝑑𝑠1+4𝐾+2𝐾^2𝑇_𝑡𝔼𝑌_𝑠²𝑑𝑠. (B.5)(B.6) Using Gronwall’s inequality on (B.5), we obtain that 𝔼𝑌_𝑡²≤𝐶𝔼𝜉²+2^𝑇_𝑡𝑆_𝑠𝑑𝐾_𝑠+^𝑇_𝑡𝑓(𝑠,0,0,0)²𝜉𝑑𝑠≤𝐶𝔼²+2^𝑇_𝑡𝑆⁺_𝑠𝑑𝐾_𝑠+^𝑇_𝑡𝑓(𝑠,0,0,0)²𝑑𝑠. (B.7) Inserting into (B.6) yields that 12𝔼^𝑇_𝑡𝑍_𝑠²𝜉𝑑𝑠≤𝔼²+2^𝑇_𝑡𝑆⁺_𝑠𝑑𝐾_𝑠+^𝑇_𝑡𝑓(𝑠,0,0,0)²+𝑑𝑠1+4𝐾+2𝐾²×^𝑇_𝑡𝐶𝔼𝜉²+2^𝑇_𝑠𝑆⁺_𝑢𝑑𝐾_𝑢+^𝑇_𝑠𝑓(𝑢,0,0,0)²𝜉𝑑𝑢𝑑𝑠≤𝔼²+2^𝑇_𝑡𝑆⁺_𝑠𝑑𝐾_𝑠+^𝑇_𝑡𝑓(𝑠,0,0,0)²+𝑑𝑠1+4𝐾+2𝐾²×𝐶𝔼𝑇𝜉²+2^𝑇₀𝑆⁺_𝑠𝑑𝐾_𝑠+^𝑇₀𝑓(𝑠,0,0,0)²𝑑𝑠. (B.8) Hence 𝔼^𝑇₀𝑍_𝑠²𝑑𝑠≤𝐶^′𝔼𝜉²+2^𝑇₀𝑆⁺_𝑠𝑑𝐾_𝑠+^𝑇₀𝑓(𝑠,0,0,0)²𝑑𝑠. (B.9) Consider now the equality 𝐾_𝑇=𝑌₀−𝜉−^𝑇₀𝑓𝑡,𝑌_𝑡,𝔼𝑌_𝑡,𝑍_𝑡𝑑𝑡+^𝑇₀𝑍_𝑡𝑑𝐵_𝑡. (B.10) We have 𝔼𝐾²_𝑇=𝔼𝑌₀−𝜉−^𝑇₀𝑓𝑡,𝑌_𝑡,𝔼𝑌_𝑡,𝑍_𝑡+𝑑𝑡_𝑇0𝑍_𝑡𝑑𝐵_𝑡²≤4𝔼𝑌²₀+𝜉²+^𝑇₀𝑓𝑡,𝑌_𝑡,𝔼𝑌_𝑡,𝑍_𝑡𝑑𝑡²+^𝑇₀𝑍_𝑡𝑑𝐵_𝑡²=𝑌Itô'sruleonthesquaredstochasticintegral=4𝔼²₀+𝜉²+^𝑇₀𝑓𝑡,𝑌_𝑡,𝔼𝑌_𝑡,𝑍_𝑡𝑑𝑡²+^𝑇₀𝑍_𝑡²𝑌𝑑𝑡≤4𝔼²₀+𝜉²+^𝑇₀𝑓𝑡,𝑌_𝑡,𝔼𝑌_𝑡,𝑍_𝑡²+𝑑𝑡_𝑇0𝑍_𝑡²𝜉𝑑𝑡≤𝐶𝔼²+2^𝑇₀𝑆⁺_𝑡𝑑𝐾_𝑡+^𝑇₀𝑓(𝑡,0,0,0)²𝜉𝑑𝑡+4𝔼²+^𝑇₀𝑓𝑡,𝑌_𝑡,𝔼𝑌_𝑡,𝑍_𝑡²𝑑𝑡, (B.11) where the last inequality follows from using (B.7) and (B.9). Now from the Lipschitz property of 𝑓 we get 𝔼^𝑇₀𝑓𝑡,𝑌_𝑡,𝔼𝑌_𝑡,𝑍_𝑡²𝑑𝑡≤𝐾^2𝑇₀𝔼𝑓(𝑡,0,0,0)+𝑌_𝑡+𝔼𝑌_𝑡+𝑍_𝑡²𝑑𝑡≤4𝐾^2𝑇₀𝔼𝑓(𝑡,0,0,0)²+𝑌_𝑡²+𝔼𝑌_𝑡²+𝑍_𝑡²𝑑𝑡≤4𝐾^2𝑇₀𝔼𝑓(𝑡,0,0,0)²+𝑌_𝑡²+𝔼𝑌_𝑡²+𝑍_𝑡²𝑑𝑡=4𝐾²𝔼^𝑇₀𝑓(𝑡,0,0,0)²𝑑𝑡+8𝐾²𝔼^𝑇₀𝑌_𝑡²𝑑𝑡+4𝐾²𝔼^𝑇₀𝑍_𝑡²𝑑𝑡, (B.12) which, by inserting into the above and then using (B.7) and (B.9) again, yields that 𝔼𝐾²_𝑇≤𝐶𝔼𝜉²+2^𝑇₀𝑆⁺_𝑠𝑑𝐾_𝑠+^𝑇₀𝑓(𝑠,0,0,0)²𝜉𝑑𝑠+4𝔼²+8𝐾²𝔼^𝑇₀𝑓(𝑡,0,0,0)²𝜉𝑑𝑡≤𝐶𝔼²+2^𝑇₀𝑆⁺_𝑠𝑑𝐾_𝑠+^𝑇₀𝑓(𝑠,0,0,0)²𝑑𝑠+81+𝐾²𝔼𝜉²+^𝑇₀𝑓(𝑡,0,0,0)²𝑑𝑡+2^𝑇₀𝑆⁺_𝑡𝑑𝐾_𝑡=𝐶𝔼𝜉²+^𝑇₀𝑓(𝑠,0,0,0)²𝑑𝑠+2^𝑇₀𝑆⁺_𝑠𝑑𝐾_𝑠≤𝐶𝔼𝜉²+^𝑇₀𝑓(𝑠,0,0,0)²𝑑𝑠+𝔼2𝐶sup_{0≤𝑡≤𝑇}𝑆⁺_𝑠𝐾_𝑇≤𝐶𝔼𝜉²+^𝑇₀𝑓(𝑠,0,0,0)²𝑑𝑠+2𝐶²𝔼sup_{0≤𝑡≤𝑇}𝑆⁺_𝑠²+12𝔼𝐾²_𝑇. (B.13) Thus 𝔼𝐾²_𝑇≤𝐶𝜉²+^𝑇₀𝑓(𝑠,0,0,0)²𝑑𝑠+sup_{0≤𝑡≤𝑇}𝑆⁺_𝑠². (B.14) Using the Burkholder-Davis-Gundy inequality we find that 𝔼sup_{0≤𝑡≤𝑇}𝑌_𝑡²≤𝐶𝔼^𝑇₀𝑍_𝑡²𝑑𝑡. (B.15) Hence, again in view of (B.9), we get 𝔼sup_{0≤𝑡≤𝑇}𝑌_𝑡²+^𝑇₀𝑍_𝑡²𝑑𝑡+𝐾_𝑇²≤𝔼(1+𝐶)^𝑇₀𝑍_𝑡²𝑑𝑡+𝐾_𝑇²≤𝐶𝔼𝜉²+^𝑇₀𝑓(𝑠,0,0,0)²𝑑𝑠+sup_{0≤𝑡≤𝑇}𝑆⁺_𝑠². (B.16)

C. Proof of Proposition 4

Itô’s formula applied to |(𝑌_𝑡−𝑌^′_𝑡)⁺|² yields that 𝑌_𝑡−𝑌^′_𝑡⁺²=2^𝑇_𝑡𝑌_𝑡−𝑌^′_𝑡⁺𝑓𝑡,𝑌_𝑡,𝔼𝑌_𝑡,𝑍_𝑡−𝑓^′𝑡,𝑌^′_𝑡,𝔼𝑌^′_𝑡,𝑍^′_𝑡𝑑𝑡+2^𝑇_𝑡𝑌_𝑡−𝑌^′_𝑡⁺𝑑𝐾_𝑡−𝑑𝐾^′_𝑡−2^𝑇_𝑡𝑌_𝑡−𝑌^′_𝑡⁺𝑍_𝑡−𝑍^′_𝑡𝑑𝐵_𝑡−^𝑇_𝑡𝑌_𝑡−𝑌^′_𝑡⁺𝑑𝐿_𝑡−^𝑇_𝑡𝟏_{{𝑌𝑡>𝑌′𝑡}}𝑍_𝑡−𝑍^′_𝑡²𝑑𝑡≤2^𝑇_𝑡𝑌_𝑡−𝑌^′_𝑡⁺𝑓𝑡,𝑌_𝑡,𝔼𝑌_𝑡,𝑍_𝑡−𝑓^′𝑡,𝑌^′_𝑡,𝔼𝑌^′_𝑡,𝑍^′_𝑡𝑑𝑡+2^𝑇_𝑡𝑌_𝑡−𝑌^′_𝑡⁺𝑑𝐾_𝑡−𝑑𝐾^′_𝑡−2^𝑇_𝑡𝑌_𝑡−𝑌^′_𝑡⁺𝑍_𝑡−𝑍^′_𝑡𝑑𝐵_𝑡−^𝑇_𝑡𝟏_{{𝑌𝑡>𝑌′𝑡}}𝑍_𝑡−𝑍^′_𝑡²𝑑𝑡. (C.1) Taking expectations we find that 𝔼𝑌_𝑡−𝑌^′_𝑡⁺²+𝔼^𝑇_𝑡𝟏_{{𝑌𝑠>𝑌′𝑠}}𝑍_𝑠−𝑍^′_𝑠²𝑑𝑠≤2𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺𝑓𝑠,𝑌_𝑠,𝔼𝑌_𝑠,𝑍_𝑠−𝑓^′𝑠,𝑌^′_𝑠,𝔼𝑌^′_𝑠,𝑍^′_𝑠𝑑𝑠+2𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺𝑑𝐾_𝑠−𝑑𝐾^′_𝑠. (C.2) But on {𝑌_𝑡>𝑌^′_𝑡} it holds that 𝑌_𝑡>𝑌^′_𝑡≥𝑆^′_𝑡≥𝑆_𝑡, from which it follows that ^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺𝑑𝐾_𝑠−𝑑𝐾^′_𝑠=−^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺𝑑𝐾^′_𝑠≤0. (C.3) Hence 𝔼𝑌_𝑡−𝑌^′_𝑡⁺²+𝔼^𝑇_𝑡𝟏_{{𝑌𝑠>𝑌′𝑠}}𝑍_𝑠−𝑍^′_𝑠²𝑑𝑠≤2𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺×𝑓𝑠,𝑌_𝑠,𝔼𝑌_𝑠,𝑍_𝑠−𝑓^′𝑠,𝑌^′_𝑠,𝔼𝑌^′_𝑠,𝑍^′_𝑠𝑑𝑠=2𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺×𝑓𝑠,𝑌_𝑠,𝔼𝑌_𝑠,𝑍_𝑠−𝑓𝑠,𝑌^′_𝑠,𝔼𝑌^′_𝑠,𝑍^′_𝑠+𝑓𝑠,𝑌^′_𝑠,𝔼𝑌^′_𝑠,𝑍^′_𝑠−𝑓^′𝑠,𝑌^′_𝑠,𝔼𝑌^′_𝑠,𝑍^′_𝑠𝑑𝑠≤2𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺×𝑓𝑠,𝑌_𝑠,𝔼𝑌_𝑠,𝑍_𝑠−𝑓𝑠,𝑌^′_𝑠,𝔼𝑌^′_𝑠,𝑍^′_𝑠𝑑𝑠 (C.4) by assumption. Without loss of generality we may assume that 𝑓 is the generator which is nondecreasing in the third argument so that by the Lipschitz property of 𝑓 we get 𝔼𝑌_𝑡−𝑌^′_𝑡⁺²+𝔼^𝑇_𝑡𝟏_{{𝑌𝑠>𝑌′𝑠}}𝑍_𝑠−𝑍^′_𝑠²𝑑𝑠≤2𝐾𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺×𝑌_𝑠−𝑌^′_𝑠+𝔼𝑌_𝑠−𝔼𝑌^′_𝑠⁺+𝑍_𝑠−𝑍^′_𝑠𝑑𝑠≤2𝐾𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺𝑌_𝑠−𝑌^′_𝑠+𝑍_𝑠−𝑍^′_𝑠𝑑𝑠+2𝐾𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺𝔼𝑌_𝑠−𝑌^′_𝑠⁺𝑑𝑠, (C.5) where the last inequality follows from Jensen’s inequality. For the first integral we have 2𝐾𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺𝑌_𝑠−𝑌^′_𝑠+𝑍_𝑠−𝑍^′_𝑠𝑑𝑠=2𝐾𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺²𝑑𝑠+𝔼^𝑇_𝑡2𝐾𝑌_𝑠−𝑌^′_𝑠⁺𝟏_{{𝑌𝑠>𝑌′𝑠}}𝑍_𝑠−𝑍^′_𝑠𝑑𝑠≤2𝐾𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺²𝑑𝑠+2𝐾²𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺²+1𝑑𝑠2𝔼^𝑇_𝑡𝟏_{{𝑌𝑠>𝑌′𝑠}}𝑍_𝑠−𝑍^′_𝑠²=𝑑𝑠2𝐾+2𝐾²𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺²+1𝑑𝑠2𝔼^𝑇_𝑡𝟏_{{𝑌𝑠>𝑌′𝑠}}𝑍_𝑠−𝑍^′_𝑠²𝑑𝑠. (C.6) Inserting into the above and canceling out terms yield that 𝔼𝑌_𝑡−𝑌^′_𝑡⁺²+12𝔼^𝑇_𝑡𝟏_{{𝑌𝑠>𝑌′𝑠}}𝑍_𝑠−𝑍^′_𝑠²≤𝑑𝑠2𝐾+2𝐾²𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺²𝑑𝑠+2𝐾𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺𝔼𝑌_𝑠−𝑌^′_𝑠⁺=𝑑𝑠2𝐾+2𝐾²𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺²𝑑𝑠+2𝐾^𝑇_𝑡𝔼𝑌_𝑠−𝑌^′_𝑠⁺²≤𝑑𝑠2𝐾+2𝐾²𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺²𝑑𝑠+2𝐾^𝑇_𝑡𝔼𝑌_𝑠−𝑌^′_𝑠⁺²=𝑑𝑠4𝐾+2𝐾²𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺²𝑑𝑠. (C.7) Hence, it holds that 𝔼𝑌_𝑡−𝑌^′_𝑡⁺²≤4𝐾+𝐾²𝔼^𝑇_𝑡𝑌_𝑠−𝑌^′_𝑠⁺²𝑑𝑠. (C.8) Using Gronwall’s inequality yields that 𝔼𝑌_𝑡−𝑌^′_𝑡⁺²=0 (C.9) and hence 𝑌_𝑡≤𝑌^′_𝑡 ℙ-a.s.

D. Proof of Proposition 5

Consider the penalized equations of the MF-RBSDE for 𝑛≥1: 𝑌^𝑛_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝑌^𝑛_𝑠,𝔼𝑌^𝑛_𝑠,𝑍^𝑛_𝑠𝑑𝑠+𝑛^𝑇_𝑡𝑌^𝑛_𝑠−𝑆_𝑠⁻𝑑𝑠−^𝑇_𝑡𝑍^𝑛_𝑠𝑑𝐵_𝑠,𝑌_𝑡^′,𝑛=𝜉^′+^𝑇_𝑡𝑓^′𝑠,𝑌_𝑠^′,𝑛,𝔼𝑌_𝑠^′,𝑛,𝑍_𝑠^′,𝑛𝑑𝑠+𝑛^𝑇_𝑡𝑌_𝑠^′,𝑛−𝑆_𝑠⁻𝑑𝑠−^𝑇_𝑡𝑍_𝑠^′,𝑛𝑑𝐵_𝑠. (D.1) Letting 𝑓^𝑛𝑡,𝑦₁,𝑦₂,𝑧∶=𝑓𝑡,𝑦₁,𝑦₂,𝑧+𝑛𝑦₂−𝑧⁻,𝑓^′,𝑛𝑡,𝑦₁,𝑦₂,𝑧∶=𝑓^′𝑡,𝑦₁,𝑦₂,𝑧+𝑛𝑦₂−𝑧⁻, (D.2) we have two sequences of unreflected continuous MF-BSDEs with generators 𝑓^𝑛 and 𝑓^′,𝑛, where 𝑓^𝑛𝑡,𝑦₁,𝑦₂,𝑧≤𝑓^′,𝑛𝑡,𝑦₁,𝑦₂,𝑧. (D.3) Following the steps in the proof of Proposition 4 we see that 𝑌^𝑛_𝑡≤𝑌_𝑡^′,𝑛 a.s. Taking the limit 𝑛→∞ we obtain the result.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The financial support from The Swedish Export Credit Corporation (SEK) is gratefully acknowledged.