1. Zero-sum optimal switching game
Optimal switching is a generalisation of optimal stopping which has various applications in economics and mathematical finance. It consists of one or more agents and a system which they control by successively switching the system’s operational mode according to a discrete set of choices. A typical example is the economic valuation of a power plant which manages its fuel mix for electricity production. One can also think of a trader who changes the composition of his/her portfolio according to the returns of the assets. There are several works on optimal switching problems in continuous time, and a survey of the literature identifies two main approaches used to analyse these problems: an analytical approach using partial differential equations (PDEs) and a probabilistic one.
Methods based on PDEs and associated variational inequalities appeared as early as the 1970s, under the topic of impulsive control for diffusion processes; see Reference Robin and Wets[28] and the references therein. A viscosity solutions approach to this type of PDE appeared in the late 1980s to early 1990s (for instance, Reference Tang and Yong[31]) and is still the topic of active research Reference Lundström, Nyström and Olofsson[23]. For example, the Hamilton–Jacobi–Bellman equation corresponding to the optimal switching problem for a diffusion process 
${({X_t})_{t \ge 0}}$ on a finite-time horizon [0, T] is the following system of PDEs with obstacles depending on the solution 
${({v^i})_{i {\in} {\Gamma ^1}}}$, with 
$\Gamma^{1} :\!=
\{1,\ldots,m_{1}\}$ for all 
$i {\in} \Gamma^{1}$:
\begin{equation}
\begin{gathered}
\min\Bigl\{v^{i}(s,x)-\max_{k {\in} \Gamma^{1}-\{i\}}[v^{k}(s,x)-
{g}^{i,k}(s,x)]({-}\,\partial_s-\mbox{L}^{X})(v^{i})(s,x)-f^{i}(s,x) \Bigr\}
= 0,
\\
v^{i}(T,x)=h^{i}(x).
\end{gathered}
\label{eqn1}
\end{equation}
(1)Here
•
$\Gamma^{1}$ is the set of available switching modes;•
$g^{i,k}$ is the function which gives the cost of switching from mode i to mode k;•
$f^i$ and $h^i$ are the functions which give respectively the instantaneous and terminal yield of the system when it is working in mode i; and•
${\mbox{L}^X}$ is the infinitesimal generator associated with the diffusion process X.
If a sufficiently regular solution of the system (1) exists then 
$v^{i}(s,x)$ is nothing else but the optimal profit that can be generated by switching, with initial conditions i for the system’s mode and x for the process X at time 
$s {\in} [0,T]$.
Probabilistic solution methods for optimal switching problems have been investigated since the 1970s and 1980s in various degrees of generality (see, for instance, Reference Bismut[4], Reference Morimoto[26], Reference Robin and Wets[28], Reference Zabczyk[34], and Reference Zvonkin[35]), and most of the recent research in this area has been a combination of the martingale approach via Snell envelopes Reference Djehiche, Hamadène and Popier[11], Reference Martyr[24], and the theory of backward stochastic differential equations (BSDE) Reference Chassagneux, Elie and Kharroubi[7], Reference Elie and Kharroubi[14], Reference Hu and Tang[20]. The latter methods lead to the study of the following system of reflected BSDEs with lower interconnected obstacles. Find adapted stochastic processes 
${({Y^i},{Z^i},{K^i})_{i {\in} {\Gamma ^1}}}$, with 
$K^i$ an increasing process, satisfying, for all 
$i {\in} \Gamma^{1}$ and 
$s
{\in} [0,T]$,
\begin{equation}
\begin{gathered}
Y^{i}_s = h^{i}+\int^T_s f^{i}_{t}\,\textrm{d} t-\int^T_s Z^{i}_t \,\textrm{d} B_t +
K^{i}_T - K^{i}_s,
\\
Y^{i}_s \ge L^{i}_s({\boldsymbol{Y}}) :\!=
\max_{k {\in}\Gamma^{1}-\{i\}}\{Y^{k}_s-{g}^{i,k}_{s}\}, \qquad
\int^T_s(Y^{i}_t-L^{i}_t({\boldsymbol{Y}}))\,\textrm{d} K^{i}_t=0,
\end{gathered}
\end{equation}
(2)
where 
$h^{i}$, 
$f^{i}$, and 
${g}^{i,k}$ play the same role as above, but here they are not formulated as functions, instead 
$h^{i}$ is a random variable and 
$f^{i}$ and 
${g}^{i,k}$ are stochastic processes. Similar to the PDE system (1), the BSDE system (2) can concisely represent solutions to the optimal switching problem and is investigated in several papers including Reference Chassagneux, Elie and Kharroubi[7], Reference Djehiche, Hamadène and Popier[11], Reference Hamadène and Morlais[17], Reference Hamadène and Zhang[19], and Reference Hu and Tang[20]. For any 
$i {\in} \Gamma^{1}$, 
$Y^i_s$ is then the system’s value (that is, its optimal yield in the sense of expectation) when it starts from mode i at time s. On the other hand, system (2) allows for the construction of the optimal strategy; see, e.g. Reference Chassagneux, Elie and Kharroubi[7], Reference Djehiche, Hamadène and Popier[11], Reference Hamadène and Morlais[17], Reference Hamadène and Zhang[19], and Reference Hu and Tang[20] for more details). All of the aforementioned references are concerned with single-person optimisation problems. Multiple-person optimal switching problems in a continuous-time stochastic setting, the topic under which the present work falls, have been studied less frequently in the literature (there is related work for deterministic systems such as Reference Yong[32] and Reference Yong[33]). In the two-player zero-sum game of optimal switching, which is the setting of the present work, there are works such as Reference Djehiche, Hamadène, Morlais and Zhao[12], Reference Hamadène and Mu[18], Reference Hu and Tang[21], and Reference Tang and Hou[30], and related studies for impulse control games Reference Cosso[10], Reference Stettner[29].
Let the finite, discrete sets 
$\Gamma^{k} = \{1,\ldots,m_{k}\}$, 
$k {\in}
\{1,2\}$ represent the operating modes that player k can choose. Letting 
$\Gamma = \Gamma^{1} \times \Gamma^{2}$ denote the product space of operating modes 
$\gamma\,{=}\,{(\gamma^{(1)},\gamma^{(2)})}$, having cardinality 
$|\Gamma| = m = m_{1} \times m_{2}$, the following costs and rewards are taken into account.
• For
$(i,j) \in \Gamma $, $f^{i,j} $ defines a running reward paid by player 2 to player 1 and $h^{i,j}$ a terminal reward paid by player 2 to player 1, when player 1’s (respectively player 2’s) active mode is i (respectively j).• For
$i_{1},i_{2} {\in} \Gamma^{1}$, $\hat{g}^{i_{1},i_{2}}$ defines a nonnegative payment from player 1 to player 2 when the former switches from mode $i_{1}$ to $i_{2}$.• For
$j_{1},j_{2} {\in} \Gamma^{2}$, $\check{g}^{\kern1.2pt j_{1},j_{2}} $ defines a nonnegative payment from player 2 to player 1 when the former switches from mode $j_{1}$ to $j_{2}$.
For all 
$(i,j) {\in} \Gamma$ and 
$t {\in} [0,T],$ we set 
${\hat{g}^{i,i}_{t} = \check{g}^{\kern1.2pt j,j}_{t}} = 0$.
From the probabilistic point of view, since there are two players and their advantages are antagonistic, the zero-sum switching game leads to the study of a system of reflected BSDEs with interconnected bilateral obstacles which is an extension of (2): For each 
$(i,j) {\in} \Gamma$ and 
$s {\in} [0,T],$ we should have
\begin{gather}
Y^{i,j}_{s} = h^{i,j} + \int_{s}^{T}f^{i,j}_{t}{\,\textrm{d}}t + K^{i,j}_{T} -
K^{i,j}_{s} - \int_{s}^{T}Z^{i,j}_{t}
{\,\textrm{d}}B_{t},\label{eqn3}\end{gather}
(3a)
\begin{gather}
Y^{i,j}_{s} \le U^{i,j}_{s}({\boldsymbol{Y}})\quad\text{and}\quad
Y^{i,j}_{s} \ge L^{i,j}_{s} ({\boldsymbol{Y}}),\label{eqn4}\end{gather}
(3b)
\begin{gather}
\int_{s}^{T}(Y^{i,j}_{t} - U^{i,j}_{t}({\boldsymbol{Y}})){\,\textrm{d}}K^{i,j,-}_{t}
= \int_{s}^{T}(L^{i,j}_{t}({\boldsymbol{Y}}) -
Y^{i,j}_{t}){\,\textrm{d}}K^{i,j,+}_{t} = 0,
\label{eqn5}\end{gather}
(3c)
where
•
${L^{i,j}_s({\boldsymbol{Y}})} :\!= {\max_{k \in \Gamma^1
- \{i\}}\{Y^{k,j}_s-\hat{g}^{i,k}_{s}\}};$•
${U^{i,j}_s({\boldsymbol{Y}})}:\!= {\min_{\ell \in
\Gamma^2 - \{j\}}\{Y^{i,\ell}_s+\check{g}^{ j,\ell}_{s}\}};$ and•
$K^{i,j}= K^{i,j,+} - K^{i,j,-}$ ($K^{i,j,\pm}$ are increasing processes).
While this system is formulated precisely in Section 2, an intuitive interpretation is provided here.
Since the game is zero-sum, we suppose that player 1 is the maximiser while player 2 is the minimiser. A system of processes 
${(Y^{i,j},Z^{i,j},K^{i,j})_{(i,j) {\in} \Gamma}}$ is sought which may be understood as follows. Firstly, 
${Y^{i,j}_t}$ is the value of the game played on the time interval [t, T] when the players begin at time t in mode (i, j). Secondly, 
${Z^{i,j}_t}$ is the volatility of 
${Y^{i,j}_t}$. Thirdly, the term 
${K^{i,j}_{t} -
K^{i,j}_{T}}$ captures, in the sense of expectation, the loss of yield which would be caused by remaining in the state (i, j) over the time interval [t, T]. Set 
${\boldsymbol{Y}}={(Y^{i,j})_{(i,j) {\in}
\Gamma}}$. Since player 1 may switch mode at any time, the process 
$L^{i,j}({\boldsymbol{Y}})$ provides a lower bound for 
$Y^{i,j}$. Furthermore, consider any time interval 
$[t_1,t_2] \subset [0,T]$ in which 
$Y^{i,j}$ strictly exceeds this bound, since there is no incentive for player 1 to switch from mode i to any other mode, 
$K^{i,j}$ should not increase on 
$[t_1,t_2]$. Symmetric arguments apply to player 2. Therefore, the processes 
${(Y^{i,j},Z^{i,j},K^{i,j})_{(i,j) {\in}
\Gamma}}$ should verify (3).
System (3) is studied in only a few papers including Reference Djehiche, Hamadène, Morlais and Zhao[12], Reference Hamadène and Mu[18], and Reference Hu and Tang[21]. Hu and Tang Reference Hu and Tang[21] showed that a solution exists when 
${\check{g}^{ j,\ell}}$ and 
${\hat{g}^{i,k}}$ are constant. Djehiche et al. Reference Djehiche, Hamadène, Morlais and Zhao[12] also studied this system in a Markovian framework where, by combining techniques of PDEs and ones of backward equations, proved that system (3) has a unique solution (see Theorem 2). Finally, more recently, Hamadène and Mu Reference Hamadène and Mu[18] studied this system in a more general framework (see Theorem 1) and showed the existence of a solution when the switching costs 
$\check {g}^{ j,\ell}$ and 
$\hat{g}^{i,k}$ are only Itô processes.
Concerning system (3), it is expected that 
${Y^{i,j}_s}$ is the value of the zero-sum switching game when the system starts from 
$(i,j) {\in} \Gamma$ in the same way as 
${Y^i_s}$ of system (2) is the value of the control problem when the system starts from 
$i {\in}
\Gamma^{1}$. However, this fact is not obvious at all and proved only in some specific cases. Actually in Reference Djehiche, Hamadène, Morlais and Zhao[12], the authors showed that when 
$f^{i,j}$ and 
$h^{i,j}$ are separated in the sense that, for any 
$(i,j) {\in} \Gamma$, 
$f^{i,j} = \hat{f}^{i} + \check{f}^j,$ and 
$h^{i,j} = \hat{h}^{i} + \check{h}^{\kern1.3pt j}$, then 
${Y^{i,j}_s}$ is the value of the zero-sum switching game. This latter assumption allows the game to be decomposed into two optimal switching problems which are simpler and decoupled, but limits the potential for practical applications. The case of arbitrary 
$f^{i,j}$, 
$\check {g}^{ j,\ell}$, 
$\hat{g}^{i,k},$ and 
$h^{i,j}$ is still open and this is the main objective of the present work. We prove without the separability assumption, and this is the novelty of this paper, that a solution of system (3) provides the game’s value. Furthermore, we derive equilibrium strategies for the players which adapt to the opponent’s switching decisions and provide robust performance guarantees.
This paper is organised as follows. In Section 2 we introduce the zero-sum switching game and provide some results related to the existence of a solution of system (3). In Section 3 we show the main result, that 
$Y^{i,j}$ coincides with the value of the zero-sum game. Moreover, we provide results on the existence of optimal strategies in the game. For completeness, we also interpret our findings in the diffusion framework.
2. Probabilistic setup and notation
We follow closely the setup in Reference Djehiche, Hamadène, Morlais and Zhao[12], working on a finite horizon [0, T] and filtered probability space 
$(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P}),$ where 
$\mathbb{F} =(\mathcal{F}_{t})_{0 \le t \le T}$ is the usual completion of the natural filtration of 
$B = (B_{t})_{0 \le t \le T}$, a d-dimensional standard Brownian motion, where 
$d \ge 1$. Since our method of proof does not depend on the dimension d, without loss of generality, we assume henceforth that 
$d = 1$.
• Let
$\mathcal{T}$ be the set of $\mathbb{F}$-stopping times bounded above by T and, for a given $\nu {\in} \mathcal{T}$, $\mathcal{T}_{\nu}$ the set of all $\tau {\in} \mathcal{T}$ satisfying $\tau \ge \nu$ almost surely (a.s).• For any sub-
$\sigma$-algebra $\mathcal{F}^{o}$ of $\mathcal{F}$, let $L^{p}(\mathcal{F}^{o})$, $1 \le p \lt \infty$, denote the set of p-integrable $\mathcal{F}^{o}$-measurable random variables, and set $L^{p} :\!= L^{p}(\mathcal{F})$.• Let
$\mathcal{H}^{2}$ be the set of $\mathbb{F}$-progressively measurable processes $w = (w_{t})_{0 \le t \le T}$ which are $dt\otimes d
\mathbb{P}$-square integrable.• Let
$\mathcal{S}^{2}$ be the set of $\mathbb{F}$-adapted processes $w =
(w_{t})_{0 \le t \le T}$ with paths that are right-continuous with left limits satisfying Let
\[
\sup_{0 \le t \le T}|w_{t}| \in L^{2}.
\]
$\mathcal{S}^{2}_{c} \subset \mathcal{S}^{2}$ denote the subset of processes $w \in \mathcal{S}^{2}$ with continuous paths.• Let
$\mathcal{K}^{2}$ denote the set of $\mathbb{F}$-adapted right-continuous with left limits processes K of finite variation satisfying $K_{0} = 0$ and where
\[
\int_{0}^{T}|\textrm{d} K_{t}| \in L^{2},
\]
$|\textrm{d} K_{t}(\omega)|$ is the total variation measure on [0, T]. Let ${\mathcal{K}^{2}_{c}}$ denote the subset of processes $K \in
\mathcal{K}^{2}$ with continuous paths.
Definition 1. Let Y be a right-continuous with left limits semi-martingale having decomposition 
$Y_{t} = Y_{0} + M_{t} + K_{t}$ where M is a local martingale, K has finite variation, and 
$M_{0} = K_{0} = 0$. Note that M is continuous due to the choice of filtration 
$\mathbb{F}$ (see Lemma 14.5.2 of Reference Cohen and Elliott[9]). We say that Y is square integrable and write 
$Y \in \mathcal{W}^{2}$ if
$$
Y_{0} \in L^{2}, \quad M \in \mathcal{S}^{2}, \quad\text{and}\quad K \in
\mathcal{K}^{2}.
$$
If Y is continuous then we write 
${Y \in \mathcal{W}^{2}_{c}}$.
Let 
${\mathcal{S}^{2,m}_{c}}$ denote the m-product of 
${\mathcal{S}^{2}_{c}}$. Similarly, we define 
$\mathcal{H}^{2,m}$, 
$L^{2,m}$, 
$\mathcal{S}^{2,m}$, 
$\mathcal{K}^{2,m}, \ldots,$ for the m-products of the spaces 
$\mathcal{H}^{2}$, 
$L^{2}$, 
$\mathcal{S}^{2}$, 
$\mathcal{K}^{2}$, and so on.
2.1. Costs, rewards, switching controls, and solutions of system (3)
Let 
$\Gamma^{k} = \{1,\ldots,m_{k}\}$, 
$k \in \{1,2\}$, be a finite, discrete set representing the operating modes that player k can choose. Let 
$\Gamma = \Gamma^{1} \times \Gamma^{2}$ denote the product space of operating modes 
$\gamma={(\gamma^{(1)},\gamma^{(2)})}$, having cardinality 
$|\Gamma| = m = m_{1} \times m_{2}$. Next, for 
$(i,j) {\in}
\Gamma$, 
$i_{1},i_{2} \,{\in}\, \Gamma^{1}$ and 
$j_{1},j_{2} \in \Gamma^{2}$, let 
$f^{i,j}$ be a process of 
$\mathcal{H}^{2}$, 
$\hat{g}^{i_{1},i_{2}}$, 
$\check{g}^{ j_{1},j_{2}}$ are nonnegative processes of 
$\mathcal{S}^{2}_{c}$ and finally 
$h^{i,j}$ a random variable belonging to 
$L^{2}(\mathcal {F}_{T})$. We also assume that, for all 
$(i,j) \in
\Gamma$ and 
$t \in [0,T]$, 
${\hat{g}^{i,i}_{t} =
\check{g}^{ j,j}_{t}} = 0$. As a matter of convention, for players 1 and 2, we use the respective ‘hat’ and ‘check’ notation to distinguish objects that are similar otherwise.
Definition 2. For 
$N \ge 2,$ a loop in 
$\Gamma$ of length 
$N-1$ is a sequence 
$\{(i_{1},j_{1}),\ldots,(i_{N},j_{N})\}$ of elements in 
$\Gamma$ with 
$N-1$ distinct members such that 
$(i_{N},j_{N}) = (i_{1},j_{1})$ and either 
$i_{q+1} = i_{q}$ or 
$j_{q+1} = j_{q}$ for any 
$q = 1,\ldots,N-1$.
We now present conditions under which a solution for system (3) exists. To begin with, throughout this paper we make the following assumption.
Assumption 1. We impose the following conditions on the switching costs.
• Consistency:
• for all sequences
$\{i_{1},i_{2},i_{3}\} \in \Gamma^{1}$ and $\{j_{1},j_{2},j_{3}\} \in \Gamma^{2}$ with $i_{1} \neq i_{2}$, $i_{2}
\neq i_{3}$ and $j_{1} \,{\neq}\, j_{2}$, $j_{2} \neq j_{3}$, we have, for all $t \in [0,T]$, $ \mathbb{P}\text{-a.s.}$,
\begin{equation}
\hat{g}^{i_1,i_3}_{t} \lt \hat{g}^{i_1,i_2}_{t} + \hat{g}^{i_2,i_3}_{t}
\quad\text{and}\quad \check{g}^{ j_1,j_3}_{t} \lt \check{g}^{ j_1,j_2}_{t} +
\check{g}^{ j_2,j_3}_{t};
\label{eqn6}
\end{equation}(4)
• for all
$(i,j) \in \Gamma,$ we have, $ \mathbb{P}\text{-a.s.}$,
\begin{equation}
\max_{i_{1} \in \Gamma^{1} - \{i\}}\{h^{i_{1},j} -
\hat{g}^{i,i_{1}}_{T}\} \le h^{i,j} \le \min_{j_{1} \in \Gamma^{2} -
\{j\}}\{h^{i,j_{1}} + \check{g}^{ j,j_{1}}_{T}\}.
\label{eqn7}
\end{equation}(5)
• Nonfree loop property: for any loop
$\{(i_{1},j_{1}),\ldots,(i_{N},j_{N})\}$ in $\Gamma,$ we have, for all $t \in [0,T]$, where
\begin{equation}
\sum_{q = 1}^{N-1}\varphi^{q,q+1}_{t} \neq 0, \qquad \mathbb{P}\text{-a.s.},
\end{equation}(6)
${\varphi^{q,q+1}_{t}} = {-\hat{g}^{i_{q},i_{q+1}}_{t}
\mathbf{1}_{\{i_{q} \neq i_{q+1}\}} +
\check{g}^{ j_{q},j_{q+1}}_{t}\mathbf{1}_{\{j_{q} \neq j_{q+1}\}}}$.
In Reference Hamadène and Mu[18] the following result is obtained.
Theorem 1. Suppose that
• Assumption 1 holds;
• for any
${i_1,i_2}\in \Gamma^1$ and $j_1,j_2\in \Gamma^2$, $\hat{g}^{i_1,i_2}$ and $\check{g}^{ j_1,j_2}$ are Itô processes. For any $0 \le t \le T$, and
\[
\hat{g}^{i_1,i_2}(t)=\hat{g}^{i_1,i_2}(0)+ \int_0^t\hat{b}^{i_1,i_2}_s\,\textrm{d}
s +\int_0^t\hat{\phi}^{i_1,i_2}_s\,\textrm{d} B_s
\]
where
\[
\check{g}^{ j_1,j_2}(t)=\check{g}^{ j_1,j_2}(0)+\int_0^t
\check{b}^{ j_1,j_2}_s\,\textrm{d} s +\int_0^t\check{\phi}^{ j_1,j_2}_s\,\textrm{d} B_s,
\]
$\hat{b}^{i_1,i_2}$, $\check{b}^{ j_1,j_2}$, $\hat{\phi}^{i_1,i_2},$ and $\check{\phi}^{ j_1,j_2}$ belong to $\mathcal{H}^2$ with ${\mathbb{E}[\!\sup_{s\leq
T}(|\hat{b}^{i_1,i_2}_s|+|\check{b}^{ j_1,j_2}_s|)^2]} \lt \infty$ moreover.
Then there exists an m-tuple of processes 
$(Y^{i,j},Z^{i,j},K^{i,j,+},K^{i,j,-})_{(i,j)\in \Gamma}$, with 
$K^{i,j,\pm}$ increasing, such that 
$ {Y^{i,j}\in{\mathcal S}_c^2}$, 
$Z^{i,j}\in {\mathcal H}^2$, 
$K^{i,j,\pm}\in \mathcal{K}^{2},$ and 
$(Y^{i,j},Z^{i,j},K^{i,j,+}-K^{i,j,-})_{(i,j)\in \Gamma}$ satisfies (3).
Remark 1. Under the consistency condition (4), it is more economical for the players to switch directly from one state to another, rather than doing so via an intermediary third state. If these conditions are not satisfied then, mathematically, there can be issues related to the well posedness of system (3). Regarding condition (5), note that it is a contradiction to require simultaneously that 
$Y^{ij}$ is continuous on [0, T], 
${Y^{ij}_T=h^{i,j}}$ and 
${\max_{i_1\in
\Gamma^{1}-\{i\}}(Y_s^{i_1,j}-\hat g^{i,i_1}_s)}\leq {Y^{i,j}_s\le
\min_{j_1\in \Gamma^{2}-\{j\}}(Y^{i,j}_s+\check g^{ j,j_1}_s)}$ for any 
$s \lt T$, without assuming (5). If this latter condition is not satisfied, 
$Y^{i,j}$ may not be continuous and then the pair 
$(\alpha^*,\beta^*)$ of (11) may not exist.
We now highlight a result in Reference Djehiche, Hamadène, Morlais and Zhao[12] on the existence of a solution to system (3) when the randomness is modelled by a diffusion process. Let 
$X^{s,x}$ be the solution to the following stochastic differential equation with initial condition 
$(s, x) {\in} [0,T] \times \mathbb{R}^{k}$:
\begin{equation}
\begin{gathered}
{X_t^{s,x} = x + \int_{s}^{t}b(r,X^{s,x}_r)\,\textrm{d}
r+\int_{s}^{t}\phi(r,X^{s,x}_r)\,\textrm{d} B_r}\quad \text{for all }t\in [s,T];
\\
X_r^{s,x}=x, \qquad r\in [0,s].
\end{gathered}
\label{eqn9}
\end{equation}(7)
Here the functions b and 
$\phi$ are both continuous with values in 
$\mathbb{R}^{k}$ and 
$\mathbb{R}^{k \times 1},$ respectively, Lipschitz with respect to x uniformly in t. Next assume that
\begin{equation}
f^{i,j}_{t}\! = \bar f^{i,j}(t,X^{s,x}_t),\qquad h^{i,j}\!=\bar
h^{i,j}(X^{s,x}_T),\qquad \hat g^{i,k}_t = \hat{\bar
g}^{i,k}(t,X^{s,x}_t),\qquad \check g^{ j,\ell}_t\!\! = \check {\bar
g}^{ j,\ell}(t,X^{s,x}_t).
\label{eqn10}
\end{equation}(8)
In this context, Assumption 1 is derived from the following structural conditions on the functions 
$\bar f^{i,j}$, 
$\bar h^{i,j}$, 
$\hat{\bar g}^{i,k}$ and 
$\check {\bar g}^{ j,\ell}$:
Assumption 2. We impose the following conditions.
• Nonnegativity:
$$
{\min_{i_{1} \in \Gamma^{1}}\hat{\bar g}^{i,i_{1}}} \ge 0
\quad\text{and}\quad {\min_{j_{1} \in \Gamma^{2}}\check{\bar
g}^{ j,j_{1}}} \ge 0 \quad\text{for all }i \in \Gamma^{1},\, j \in
\Gamma^{2}.
$$
• Consistency:
• for all sequences
$\{i_{1},i_{2},i_{3}\} \in \Gamma^{1}$ and $\{ j_{1},j_{2},j_{3}\} \in \Gamma^{2}$ with $i_{1} \neq i_{2}$, $i_{2}
\neq i_{3}$ and $j_{1} \,{\neq}\, j_{2}$, $j_{2} \neq j_{3}$, we have, for all t, x, \begin{equation}
\hat{\bar g}^{i_1,i_3}(t,x) \lt \hat{\bar g}^{i_1,i_2}(t,x) + \hat{\bar
g}^{i_2,i_3}(t,x), \!\!\qquad \check{\bar g}^{j_1,j_3}(t,x) \lt \check{\bar
g}^{ j_1,j_2}(t,x) + \check{\bar g}^{j_2,j_3}(t,x);
\end{equation}
• for all
$(i,j) {\in} \Gamma$, we have, for all x,
\begin{equation}
\max_{i_{1} \in \Gamma^{1} - \{i\}}\{\bar h^{i_{1},j} (x) - \hat{\bar
g}^{i,i_{1}}(T,x)\} \le \bar h^{i,j}(x) \le \min_{j_{1} \in \Gamma^{2} -
\{j\}}\{\bar h^{i,j_{1}} (x)+ \check{\bar g}^{ j,j_{1}}(T,x)\}
\end{equation}
• Nonfree loop property: for any loop
$\{(i_{1},j_{1}),\ldots,(i_{N},j_{N})\}$ in $\Gamma,$ we have, for all t, x, where
\begin{equation}
\sum_{q = 1}^{N-1}\bar \varphi^{q,q+1}(t,x) \neq 0,
\end{equation}
${\bar \varphi^{q,q+1}(t,x)}= {-\hat{\bar
g}^{i_{q},i_{q+1}} (t,x)\mathbf{1}_{\{i_{q} \neq i_{q+1}\}} + \check{\bar
g}^{ j_{q},j_{q+1}}(t,x)\mathbf{1}_{\{j_{q} \neq j_{q+1}\}}}$.
Theorem 2. (see Reference Djehiche, Hamadène, Morlais and Zhao[12].) In the Markovian setting (7) and (8), suppose that
• Assumption 2 holds;
• the functions
${\check {\bar g}^{ j,\ell}}$, where $j,\ell\in
\Gamma^2$, or ${\hat {\bar g}^{i,k}}$, where $i,k\in \Gamma^1$, are ${\mathcal{C}}^{1,2}$ and their derivatives are of polynomial growth.
Then there exists an m-tuple of processes 
$(Y^{i,j},Z^{i,j},K^{i,j,+},K^{i,j,-})_{(i,j)\in \Gamma}$ depending on the initial condition (s,x) for 
$X^{s,x}$, such that 
${Y^{i,j}\in
{\mathcal S}_c^2}$, 
$Z^{i,j} \in {\mathcal H}^2$, 
$K^{i,j,\pm}\in
\mathcal{K}^{2}$ (where 
$K^{i,j,\pm}$ are increasing processes), and 
${(Y^{i,j},Z^{i,j},K^{i,j,+}-K^{i,j,-})_{(i,j)\in \Gamma}}$ satisfies (3). Moreover, there also exist deterministic continuous functions with polynomial growth 
$(v^{i,j})_{(i,j)\in \Gamma}$ such that, for any (i, j) and 
$t
\in [0,T-s]$,
\[
Y^{i,j}_{s+t} = v^{i,j}(s+t,X^{s,x}_{s+t})
\]
and 
${(v^{i,j})_{(i,j)\in \Gamma}}$ is the unique solution in viscosity sense of the following Hamilton–Jacobi–Bellman system of PDEs with obstacles: For any 
$(i,j)\in \Gamma$ and 
$(s,x) \in [0,T]\times \mathbb{R}^{k}$,
\begin{gather}
\begin{gathered}\min\Bigl\{v^{i,j}(s,x)-L^{i,j}({\boldsymbol{v}})(s,x),
\max \bigl\{ v^{i,j}(s,x) -U^{i,j}({\boldsymbol{v}})(s,x),
({-}\,\partial_s-\mbox{L}^{X})(v^{i,j})(s,x)-f^{i,j}(s,x) \bigr\}
\Bigr\} = 0;\end{gathered}
\\
v^{i,j}(T,x)=h^{i,j}(x),
\end{gather}
where 
$\mbox{L}^X$ is the generator associated with 
$X^{s,x}$, and 
${\boldsymbol{v}} = {(v^{i,j})_{(i,j) \in \Gamma}}$, 
$L^{i,j}({\boldsymbol{v}}) :\!= {\max_{k \in \Gamma^1 -
\{i\}}\{v^{k,j}-\hat{g}^{i,k}\},}$ and 
$U^{i,j}({\boldsymbol{v}}) :\!=
{\min_{\ell \in \Gamma^2 -
\{j\}}\{v^{i,\ell}+\check{g}^{ j,\ell}\}}$.
2.1.1. Individual switching controls and strategies
Definition 3. (Switching controls.) A control for player 1 is a sequence 
$\alpha = (\sigma_{n},\xi_{n})_{n
\ge 0}$ such that,
• for all
$n \ge 0$, $\sigma_{n} \in \mathcal{T}$ and is such that $\sigma_{n} \le \sigma_{n+1}$, $\mathbb{P}$-a.s., and $\mathbb{P}(\{\sigma_{n} \lt T \text{ for all } n \ge 0\}) = 0$;• for all
$n \ge 0$, $\xi_{n}$ is an ${\mathcal {F}}_{\sigma_{n}}$-measurable $\Gamma^{1}$-valued random variable;• for
$n \ge 1$, on $\{\sigma_{n} \lt T\}$ we have $\sigma_{n} \lt
\sigma_{n+1}$ and $\xi_{n} \neq \xi_{n-1}$, while on $\{\sigma_{n} = T\}$ we have $\xi_{n} = \xi_{n-1}$.
Let 
$\mathsf{A}$ denote the set of controls for player 1. The set 
$\mathsf{B}$ of controls 
$\beta = (\tau_{n}, \zeta_{n})_{n \ge 0}$ for player 2, where the 
$\zeta_{n}$ are 
$\Gamma^{2}$-valued, is defined analogously. Let 
$C^{\alpha}_{N}$ denote the cost of the first 
$N \ge 1$ switches,
\[
C^{\alpha}_{N} :\!= \sum_{n=1}^{N}\hat{g}^{\xi_{n-1},
\xi_{n}}_{\sigma_{n}},
\]
note that the limit 
${\lim_{N \to \infty}C^{\alpha}_{N}}$ is well defined.
Remark 2. Under the consistency condition (5) of Assumption 1, which has also been used previously in Reference Djehiche, Hamadène, Morlais and Zhao[12] and Reference Hu and Tang[21], switching at time T provides no additional benefit for either player. Consequently, it is without loss of generality that the definition of controls excludes switching at time T. Furthermore, as Remark 1 explains, this consistency condition ensures the well posedness of system (3).
Definition 4. A control 
$\alpha \in \mathsf{A}$ for player 1 is said to be square integrable if,
\[
\lim_{N \to \infty}C^{\alpha}_{N} \in L^{2}.
\]
Let 
$\mathcal{A}$ denote the set of such controls. Similarly, the set 
$\mathcal{B}$ of square-integrable controls for player 2 consists of those 
$\beta \in \mathsf{B}$ satisfying
\[
\lim_{N \to \infty}C^{\beta}_{N} \in L^{2},
\]
where
\[
C^{\beta}_{N} :\!= \sum_{n=1}^{N}\check{g}^{\zeta_{n-1},
\zeta_{n}}_{\tau_{n}}.
\]
Definition 5. (Nonanticipative switching strategies.) Let 
$s \in [0,T]$ and 
$\nu \in \mathcal{T}_{s}$. Two controls 
$\alpha^{1},\alpha^{2} \in \mathsf{A}$ with 
$\alpha^{1} =
{(\sigma^{1}_{n},\xi^{1}_{n})_{n \ge 0}}$ and 
$\alpha^{2} =
{(\sigma^{2}_{n},\xi^{2}_{n})_{n \ge 0}}$ are said to be equivalent, denoted by 
$\alpha^{1} \equiv \alpha^{2}$, on 
$[s,\nu]$ if we have a.s.,
\[
\xi^{1}_{0}\mathbf{1}_{[\sigma^{1}_{0},\sigma^{1}_{1}]}(t) + \sum_{n \ge
1}\xi^{1}_{n}\mathbf{1}_{(\sigma^{1}_{n},\sigma^{1}_{n+1}]}(t) =
\xi^{2}_{0}\mathbf{1}_{[\sigma^{2}_{0},\sigma^{2}_{1}]}(t) + \sum_{n \ge
1}\xi^{2}_{n}\mathbf{1}_{(\sigma^{2}_{n},\sigma^{2}_{n+1}]}(t), \qquad s
\le t \le \nu.
\]
A nonanticipative strategy for player 1 is a mapping 
$\overline{\alpha} \colon \mathsf{B} \to \mathsf{A}$ such that
• (nonanticipativity) for any
$s \in [0,T]$, $\nu \in \mathcal{T}_{s}$, and $\beta^{1},\beta^{2} \in \mathsf{B}$ such that $\beta^{1} \equiv
\beta^{2}$ on $[s,\nu]$, we have $\overline{\alpha}(\beta^{1}) \equiv
\overline{\alpha}(\beta^{2})$ on $[s,\nu]$;• (square integrability) for any
$\beta \in \mathcal{B},$ we have $\overline{\alpha}(\beta) \in
\mathcal{A}$.
In a similar manner we define nonanticipative strategies for player 2. Let 
$\mathcal{A}$ and 
$\mathcal{B}$ denote the set of nonanticipative strategies for players 1 and 2, respectively.
Definition 6. For 
$s \in [0,T]$ and 
$i \in \Gamma^{1}$, let 
${\mathsf{A}_{s}^{i}}$ denote the set of controls 
$\alpha \in
\mathsf{A}$ satisfying 
$\xi_{0} = i$ and 
$\sigma_{0} = s$. Similarly, define 
${\mathsf{B}_{s}^{ j}}$ for 
$s \in [0,T]$ and 
$j \in
\Gamma^{2}$. Analogous notation will be used below for other classes of controls, for example, square-integrable controls 
${\mathcal{A}_{s}^{i}}$ and 
${\mathcal{B}_{s}^{ j}}$, and strategies 
${\mathcal{A}^{i}_{s}}$ and 
${\mathcal{B}^{ j}_{s}}$.
2.1.2 Coupling of controls
We now define the coupling of two controls 
$\alpha \in \mathsf{A}$ and 
$\beta \in \mathsf{B}$ under the following assumption: player 1’s switch is implemented first if both players decide to switch at the same instant.
Definition 7. Given controls 
$\alpha \in \mathsf{A}$ and 
$\beta \in \mathsf{B}$, define the coupling 
$\gamma(\alpha,\beta) = (\rho_{n},\gamma_{n})_{n \ge
0},$ where 
$\rho_{n} \in \mathcal{T}$ is defined by
\begin{equation}
\rho_{n} = \sigma_{r_{n}} \wedge \tau_{s_{n}},
\end{equation}
with 
$r_{0} = s_{0} = 0$, 
$r_{1} = s_{1} = 1$ and, for 
$n \ge 2$,
\[
r_{n} = r_{n-1} + \mathbf{1}_{\{\sigma_{r_{n-1}} \le \tau_{s_{n-1}}\}},
\qquad s_{n} = s_{n-1} + \mathbf{1}_{\{\tau_{s_{n-1}} \lt\sigma_{r_{n-1}}\}},
\]
and 
$\gamma_{n}$ is a 
$\Gamma$-valued random variable such that 
$\gamma_{0} = (\xi_{0},\zeta_{0})$ and, for 
$n \ge 1$,
\begin{equation}
\gamma_{n} =
\begin{cases}
(\xi_{r_{n}},\gamma_{n-1}^{(2)}) & \text{on } \{\sigma_{r_{n}} \le
\tau_{s_{n}},\; \sigma_{r_{n}} \lt T\},
\\
(\gamma_{n-1}^{(1)},\zeta_{s_{n}}) & \text{on } \{\tau_{s_{n}} \lt
\sigma_{r_{n}} \},
\\
\gamma_{n-1}, & \text{on } \{\tau_{s_{n}} = \sigma_{r_{n}} = T\}.
\end{cases}
\end{equation}
Define, for all 
$0 \le t \le T$,
\begin{equation}
u_{t} = \gamma_{0}\mathbf{1}_{[\rho_{0},\rho_{1}]}(t) + \sum_{n \ge
1}\gamma_{n}\mathbf{1}_{(\rho_{n},\rho_{n+1}]}(t),
\label{eqn11}
\end{equation}(9)
where 
$({\rho _n},{\rho _{n + 1}}] = \emptyset$ on 
$\{\rho_{n} = \rho_{n+1}\}$.
Note that the coupling 
$\gamma(\alpha,\beta) = (\rho_{n},\gamma_{n})_{n
\ge 0}$ of the controls 
${\alpha \in \mathsf{A}^{i}_{s}}$ and 
${\beta \in \mathsf{B}^{ j}_{s}}$ has the following properties:
•
$\rho_{0} = s$ and, for all $n \ge 0,$ we have $\rho_{n} \in \mathcal{T}$ and $\rho_{n} \le \rho_{n+1}$ $\mathbb{P}$-a.s., and $\mathbb{P}(\{\rho_{n} \lt T \text{ for all } n \ge 0\}) = 0$;•
$\gamma_{0} = (i,j)$ and, for all $n \ge 0,$ the random variable $\gamma_{n}$ is ${\mathcal {F}}_{\rho_{n}}$-measurable, $\Gamma$-valued and $\gamma_{n+1} \neq \gamma_{n}$ on $\{\rho_{n+1} \lt T\}$.
Hereafter, we use notation such as 
$f^{\gamma}$, where 
$\gamma$ is an 
$\Gamma$-valued random variable, with the interpretation 
$f^{\gamma} =
f^{i,j}$ on 
$\{\gamma = (i,j)\}$. We write 
${C^{\gamma(\alpha,\beta)}_{N}}$ for the joint cumulative cost of the first N switches,
\[
C^{\gamma(\alpha,\beta)}_{N} = \sum_{n=1}^{N}
[\hat{g}^{\gamma_{n-1}^{(1)}, \gamma_{n}^{(1)}}_{\rho_{n}} -
\check{g}^{\gamma_{n-1}^{(2)},\gamma_{n}^{(2)}}_{\rho_{n}}], \qquad N \ge
1.
\]
Definition 8. The coupling 
$\gamma(\alpha,\beta) = (\rho_{n},\gamma_{n})_{n \ge 0}$ of the controls 
$\alpha \in \mathsf{A}^{i}_{s}$ and 
$\beta \in
\mathsf{B}^{ j}_{s}$ is said to be admissible, writing 
$\gamma(\alpha,\beta) \in {\mathcal{G}^{i,j}_{s}}$ to indicate this, if 
${\sup_{N \ge 1}|C^{\gamma(\alpha,\beta)}_{N}| \in L^{2}}$.
Note that, for every 
$\alpha \in \mathsf{A}$ and 
$\beta \in \mathsf{B},$ we have 
${\lim_{N \to \infty}C^{\gamma(\alpha,\beta)}_{N}} =
{\lim_{N \to \infty}C^{\alpha}_{N}} - {\lim_{N \to
\infty}C^{\beta}_{N}}$. Using the triangle inequality, we see that every pair of square-integrable controls 
$(\alpha,\beta)$, 
$\alpha \in
{\mathcal{A}^{i}_{s}}$ and 
$\beta \in {\mathcal{B}^{ j}_{s}}$, satisfies 
$\gamma(\alpha,\beta) \in {\mathcal{G}^{i,j}_{s}}$.
2.2. The zero-sum switching game
For the zero-sum game, we assume that player 1 is the maximiser and define the total reward from its perspective. Letting 
$(s,i,j) \in [0,T)
\times \Gamma$ be the initial state and recalling (9), we have
\begin{equation}
J_{s}^{i,j}(\gamma(\alpha,\beta)) =
\mathbb{E}\biggl[\int_{s}^{T}f^{u_{t}}_{t}{\,\textrm{d}}t -
\sum_{n=1}^{\infty}[\hat{g}^{\gamma_{n-1}^{(1)},
\gamma_{n}^{(1)}}_{\rho_{n}} -
\check{g}^{\gamma_{n-1}^{(2)},\gamma_{n}^{(2)}}_{\rho_{n}}] + h^{u_{T}}
\biggm| j_{s}\biggr], \qquad \alpha \in \mathsf{A}^{i}_{s},\,
\beta \in \mathsf{B}^{ j}_{s}.
\end{equation}
The lower and upper values for this game, denoted respectively by 
${\check{V}^{i,j}_{s}}$ and 
${\hat{V}^{i,j}_{s}}$, are defined as follows:
\begin{equation}
\check{V}^{i,j}_{s} :\!= \mathop{\textrm{ess sup}}_{\alpha \in
\mathcal{A}_{s}^{i}}\mathop{\textrm{ess inf}}_{\beta \in
\mathcal{B}_{s}^{ j}}J^{i,j}_{s}(\gamma(\alpha,\beta)), \qquad
\hat{V}^{i,j}_{s} :\!= \mathop{\textrm{ess inf}}_{\beta \in
\mathcal{B}_{s}^{ j}}\mathop{\textrm{ess sup}}_{\alpha \in
\mathcal{A}_{s}^{i}}J^{i,j}_{s}(\gamma(\alpha,\beta)).
\end{equation}
Note that 
${\check{V}^{i,j}_{s} \le \hat{V}^{i,j}_{s}}$ a.s.
Definition 9. The game is said to have a value at (s, i, j) if
\begin{equation}
\check{V}^{i,j}_{s} = \hat{V}^{i,j}_{s} \quad\text{a.s.}
\end{equation}
The common value 
${V^{i,j}_{s}}$, when it exists, is referred to as the game’s solution at (s, i, j). When 
$s = T,$ we formally set 
${\check{V}^{i,j}_{T} = \hat{V}^{i,j}_{T}} = h^{i,j}$.
In this paper we construct a pair of controls 
$(\alpha^{*},\beta^{*}) \in
{\mathsf{A}^{i}_{s} \times \mathsf{B}^{ j}_{s}}$ such that 
$\gamma(\alpha^{*},\beta^{*}) \in {\mathcal{G}^{i,j}_{s}}$ and the game has a value 
${V^{i,j}_{s} =
J^{i,j}_{s}(\gamma(\alpha^{*},\beta^{*}))}$ (see Theorem 3 below). Our result is obtained by dynamic programming and the connection between doubly reflected backward stochastic differential equations (DRBSDEs) with implicitly defined barriers and zero-sum optimal stopping games. We also prove the existence of optimal non anticipative strategies 
${\overline{\alpha^{*}} \in
\mathcal{A}^{i}_{s}}$ and 
${\overline{\beta^{*}} \in \mathcal{B}^{ j}_{s}},$ which are robust in the sense that each is a best response to the worst-case opponent.
3. A probabilistic verification theorem for the zero-sum game
In Theorem 3 we use system (3) to prove the existence of a value for the zero-sum game. Recall that 
$m = |\Gamma|$ is the number of joint operating modes 
$(i,j) \in \Gamma $. For 
$(i,j) \in
\Gamma,$ define the lower and upper switching operators 
$L^{i,j}
\colon \mathcal{S}^{2,m}_{c} \to \mathcal{S}^{2}_{c}$ and 
$U^{i,j} \colon
{\mathcal{S}^{2,m}_{c} \to \mathcal{S}^{2}_{c}},$ respectively, as follows: for 
${\boldsymbol{Y}} \in {\mathcal{S}^{2,m}_{c}}$,
\begin{equation}
L^{i,j}({\boldsymbol{Y}}) = \max\limits_{i_{1} \in \Gamma^{1} -
\{i\}}\{Y^{i_{1},j} - \hat{g}^{i,i_{1}}\}, \qquad U^{i,j}({\boldsymbol{Y}}) =
\min\limits_{j_{1} \in \Gamma^{2} - \{j\}}\{Y^{i,j_{1}} +
\check{g}^{ j,j_{1}}\}.
\end{equation}
Let 
${\boldsymbol{L}} \colon {\mathcal{S}^{2,m}_{c} \to
\mathcal{S}^{2,m}_{c}}$ and 
${\boldsymbol{U}} \colon
{\mathcal{S}^{2,m}_{c} \to \mathcal{S}^{2,m}_{c}}$ be the operators defined, using matrix notation, by 
${\boldsymbol{L}} =
{(L^{i,j})_{(i,j) \in \Gamma}}$ and 
${\boldsymbol{U}} =
{(U^{i,j})_{(i,j) \in \Gamma}}$. The following definition formalises the concept of a solution to (3).
Definition 10. A solution to the system of DRBSDEs with terminal value 
${\boldsymbol{h}} \in L^{2,m}(j_{T})$, driver 
${\boldsymbol{f}} \in
\mathcal{H}^{2,m}$, and implicit barriers 
${\boldsymbol{L}}$ and 
${\boldsymbol{U}}$, is a triple 
$({\boldsymbol{Y}},{\boldsymbol{Z}},{\boldsymbol{K}}) \in \mathcal{S}^{2,m}_{c}
\times \mathcal{H}^{2,m} \times \mathcal{K}^{2,m}_{c}$ such that a.s. for all 
$(i,j) \in \Gamma$ and all 
$0 \le s \le T$,
\begin{gather}
{Y^{i,j}_{s}} = h^{i,j} + {\int_{s}^{T}f^{i,j}_{t}{\,\textrm{d}}t + K^{i,j}_{T} -
K^{i,j}_{s} - \int_{s}^{T}Z^{i,j}_{t}{\,\textrm{d}}B_{t}},\label{eqn12}\end{gather}(10a)
\begin{gather}
{Y^{i,j}_{s} \le U^{i,j}_{s}({\boldsymbol{Y}})}\quad\text{and}\quad
{Y^{i,j}_{s} \ge L^{i,j}_{s}({\boldsymbol{Y}})},\label{eqn13}\end{gather}(10b)
\begin{gather}
{\int_{s}^{T}(Y^{i,j}_{t} - U^{i,j}_{t}({\boldsymbol{Y}})){\,\textrm{d}}
K^{i,j,-}_{t}} = {\int_{s}^{T}(L^{i,j}_{t}({\boldsymbol{Y}}) -
Y^{i,j}_{t}){\,\textrm{d}}K^{i,j,+}_{t}} = 0,
\label{eqn14}\end{gather}(10c)
where 
$K^{i,j,+}$ and 
$K^{i,j,-}$ are the increasing processes in the orthogonal decomposition 
$K^{i,j} :\!= K^{i,j,+} - K^{i,j,-}$.
Note that, for any solution to (10), the stochastic integral 
${\int_{0}^{t}Z^{i,j}_{s}{\,\textrm{d}}B_{s}}$ is well defined, and is a martingale belonging to 
${\mathcal{S}^{2}_{c}}$ (see Chapter 3 of Reference Chung and Williams[8]). Since we have provided conditions, besides Assumptions 1 or 2, under which such a solution exists (see Theorems 1 or 2), hereafter, we work under Assumption 1 and assume that a solution to (10) exists.
Theorem 3. Suppose that there exists a solution 
$({\boldsymbol{Y}},{\boldsymbol{Z}},
{\boldsymbol{K}})$ to the DRBSDE (10). For every initial state 
$(s,i,j) \in [0,T] \times \Gamma$,
(i) (existence of value and optimal controls) the switching game has a value with
where
\begin{equation}
Y^{i,j}_{s} = V^{i,j}_{s} = J^{i,j}_{s}(\gamma(\alpha^{*},\beta^{*}))
\quad\text{a.s.},
\label{eqn15}
\end{equation}(11)
$(\alpha^{*},\beta^{*}) \in \mathsf{A}^{i}_{s} \times
\mathsf{B}^{ j}_{s}$ are a pair of controls satisfying $\gamma(\alpha^{*},\beta^{*}) \in {\mathcal{G}^{i,j}_{s}}$.(ii) (existence of optimal strategies) there exist nonanticipative strategies
${\overline{\alpha^{*}} \in \mathcal{A}^{i}_{s}}$ and ${\overline{\beta^{*}} \in \mathcal{B}^{ j}_{s}}$ that are optimal in the robust sense, Furthermore, these robust values are equal to the game’s value,
\begin{align}
\mathop{\textrm{ess inf}}\limits_{\beta \in \mathcal{B}^{ j}_{s}} J^{i,j}_{s}
(\gamma(\overline{\alpha^{*}}(\beta),\beta)) &=
\mathop{\textrm{ess sup}}\limits_{\overline{\alpha} \in \mathcal{A}^{i}_{s}}\mathop{\textrm{ess inf}}\limits_{\beta
\in \mathcal{B}^{ j}_{s}}
J^{i,j}_{s}(\gamma(\overline{\alpha}(\beta),\beta)),
\\
\mathop{\textrm{ess sup}}\limits_{\alpha \in \mathcal{A}^{i}_{s}}
J^{i,j}_{s}(\gamma(\alpha,\overline{\beta^{*}}(\alpha))) &=
\mathop{\textrm{ess inf}}\limits_{\overline{\beta} \in \mathcal{B}^{ j}_{s}}\mathop{\textrm{ess sup}}\limits_{\alpha
\in \mathcal{A}^{i}_{s}}
J^{i,j}_{s}(\gamma(\alpha,\overline{\beta}(\alpha))).
\end{align}
\[
\mathop{\textrm{ess sup}}_{\overline{\alpha} \in \mathcal{A}^{i}_{s}}\mathop{\textrm{ess inf}}_{\beta \in
\mathcal{B}^{ j}_{s}} J^{i,j}_{s}(\gamma(\overline{\alpha}(\beta),\beta))
= V^{i,j}_{s} = \mathop{\textrm{ess inf}}_{\overline{\beta} \in
\mathcal{B}^{ j}_{s}}\mathop{\textrm{ess sup}}_{\alpha \in \mathcal{A}^{i}_{s}}
J^{i,j}_{s}(\gamma(\alpha,\overline{\beta}(\alpha))).
\]
This concept of robustness, which is well known in the optimal control and differential games literature Reference Ball, Chudoung and Day[1], Reference Bayraktar, Cosso and Pham[2], Reference Krasovski and Subbotin[22], is natural in the context of zero-sum games Reference Cosso[10].
Remark 3. Since the switching costs are nonnegative, we obtain the following type of Mokobodski’s condition: there exists a system of processes 
${\boldsymbol{w}} = \{w^{i,j}\}_{(i,j) \in \Gamma}$ belonging to 
${\mathcal{W}^{2,m}_{c}}$ such that, for all 
$(i,j) \in \Gamma$: for all 
$0 \le t \le T$ a.s.,
\begin{equation}
\max_{i_{1} \in \Gamma^{1} - \{i\}}\{w^{i_{1},j}_{t} -
\hat{g}^{i,i_{1}}_{t}\} \le w^{i,j}_{t} \le \min_{j_{1} \in \Gamma^{2} -
\{j\}}\{w^{i,j_{1}}_{t} + \check{g}^{ j,j_{1}}_{t}\}.
\label{eqn16}
\end{equation}(12)
Indeed, by taking 
${\boldsymbol{w}}$ to be the m-dimensional null process, 
${\boldsymbol{w}} \equiv \textbf{0}$, it is easily verified that 
${{\boldsymbol{w}} \in \mathcal{W}^{2,m}_{c}}$ and (12) holds. Mokobodski’s condition (12) is an extension of that typically assumed for single-agent switching problems in a variety of settings Reference Bismut[5], Reference Bouchard[6], Reference Elie and Kharroubi[14], Reference Martyr[24], or for two-player Dynkin games or DRBSDEs Reference Bismut[3], Reference Dumitrescu, Quenez and Sulem[13], Reference Hamadène and Hassani[16], Reference Martyr[25], Reference Pham and Zhang[27], both of which are special, somewhat degenerate, cases of the optimal switching game studied here.
Let us point out that, for any solution 
$({\boldsymbol{Y}},{\boldsymbol{Z}},
{\boldsymbol{K}})$ to the DRBSDE (10), 
${\boldsymbol{Y}}$ satisfies Mokobodski’s condition (12) and, a posteriori, also belongs to 
${\mathcal{W}^{2,m}_{c}}$. Condition (12) can therefore be seen as a feasibility check for the inequality constraint (10b): there exists at least one system of processes 
${\boldsymbol{Y}}$ which satisfies (10b) within a suitable class of candidates. Actually, we know from the results in Reference Pham and Zhang[27] that the well posedness of (10) is intricately linked to Mokobodski’s condition (12).
3.1. Proof of Theorem 3
The existence of a solution to the DRBSDE (10) is closely related to the existence of both a value and a Nash equilibrium in the following Dynkin game (see, for example, Reference Dumitrescu, Quenez and Sulem[13], Reference Hamadène and Hassani[15], Reference Hamadène and Hassani[16], and Reference Stettner[29] for the relation to impulse control games with delay).
Proposition 1. Suppose that there exists a solution 
$({\boldsymbol{Y}},{\boldsymbol{Z}},
{\boldsymbol{K}})$ to the DRBSDE (10). Then, for all 
$(s,i,j)
\in [0,T] \times \Gamma$ a.s.,
• we have
where
\begin{equation}
Y^{i,j}_{s} = \mathop{\textrm{ess inf}}_{\tau \in \mathcal{T}_{s}} \mathop{\textrm{ess sup}}_{\sigma \in
\mathcal{T}_{s}}\mathcal{J}^{i,j}_{s}(\sigma,\tau) = \mathop{\textrm{ess sup}}_{\sigma \in
\mathcal{T}_{s}}\mathop{\textrm{ess inf}}_{\tau \in
\mathcal{T}_{s}}\mathcal{J}^{i,j}_{s}(\sigma,\tau),
\end{equation}
and
\begin{align}
\mathcal{J}^{i,j}_{s}(\sigma,\tau) &:\!=
\mathbb{E}\biggl[\int_{s}^{\sigma \wedge \tau}f^{i,j}_{t}{\,\textrm{d}}t +
\mathbf{1}_{\{\tau \lt \sigma\}}{U^{i,j}_{\tau}}({\boldsymbol{Y}}) +
\mathbf{1}_{\{\sigma \le \tau,\; \sigma \lt
T\}}{L^{i,j}_{\sigma}}({\boldsymbol{Y}}) \biggm| j_{s} \biggr]
\\
&\hskip2.9pt\quad\, + \mathbb{E}[h^{i,j}\mathbf{1}_{\{\sigma = \tau = T\}}
\ \,\,|\, \ \, j_{s} ],
\end{align}
${\boldsymbol{h}}$, ${\boldsymbol{f}}$, ${\boldsymbol{L}},$ and ${\boldsymbol{U}}$ are the data for (10) (see Definition 10);• we have
${Y^{i,j}_{s} =
\mathcal{J}^{i,j}_{s}(\sigma^{i,j}_{s},\tau^{i,j}_{s})},$ where ${\sigma^{i,j}_{s} \in \mathcal{T}_{s}}$ and ${\tau^{i,j}_{s}} \in \mathcal{T}_{s}$ are stopping times defined by and we use the convention that
\begin{equation}
\begin{aligned}
\sigma^{i,j}_{s} &= \inf\{s \le t \le T \colon Y^{i,j}_{t} =
L^{i,j}_{t}({\boldsymbol{Y}})\} \wedge T,
\\
\tau^{i,j}_{s} &= \inf\{s \le t \le T \colon Y^{i,j}_{t} =
U^{i,j}_{t}({\boldsymbol{Y}})\} \wedge T,
\end{aligned}
\label{eqn17}
\end{equation}(13)
$\inf \emptyset = + \infty $. Moreover, ${(\sigma^{i,j}_{s},\tau^{i,j}_{s})}$ is a Nash equilibrium for the Dynkin game,
\begin{equation}
\mathcal{J}^{i,j}_{s}(\sigma,\tau^{i,j}_{s}) \le
\mathcal{J}^{i,j}_{s}(\sigma^{i,j}_{s},\tau^{i,j}_{s}) \le
\mathcal{J}^{i,j}_{s}(\sigma^{i,j}_{s},\tau) \quad\text{for all }\sigma
\in \mathcal{T}_{s} \text{ and } \tau \in \mathcal{T}_{s}.
\end{equation}
Proof. Recalling the ordering (10b), the result follows from Proposition 2.2.1 of Reference Hamadène and Hassani[15], for example.
We will use Proposition 1 and a dynamic programming argument to first establish claim (i) of Theorem 3, then obtain (ii) as a corollary. Since (11) trivially holds when 
$s = T$, let 
$s
\in [0,T)$ and 
$(i,j) \in \Gamma$ be arbitrary. Define a sequence 
$(\rho_{n},\gamma_{n})_{n \ge 0}$ as
\begin{equation}
\rho_{0} = s, \qquad \gamma_{0} = (i,j)
\end{equation}
and, for 
$n \ge 1,$
\begin{equation}
\rho_{n} = \sigma^{\gamma_{n-1}}_{\rho_{n-1}} \wedge
\tau^{\gamma_{n-1}}_{\rho_{n-1}}, \qquad \gamma_{n} =
\begin{cases}
(\mathcal{L}_{\rho_{n}}^{\gamma_{n-1}}({\boldsymbol{Y}}),
\gamma_{n-1}^{(2)})& \text{on } \mathcal{M}^{+}_{n},
\\[2pt]
(\gamma_{n-1}^{(1)},
\mathcal{U}^{\gamma_{n-1}}_{\rho_{n}}({\boldsymbol{Y}}) )& \text{on }
\mathcal{M}^{-}_{n},
\\
\gamma_{n-1}& \text{otherwise,}
\end{cases}
\end{equation}
where 
${\sigma^{\gamma_{n-1}}_{\rho_{n-1}}}$ and 
${\tau^{\gamma_{n-1}}_{ \rho_{n-1}}}$ are defined using (13) above, 
${\mathcal{L}^{\gamma_{n-1}}_{\rho_{n}}}$ and 
${\mathcal{U}^{\gamma_{n-1}}_{\rho_{n}}}$ are obtained from the switching selectors
\begin{equation}
\mathcal{L}^{i,j}_{t}({\boldsymbol{Y}}) \in \mathop{\textrm{arg max}}\limits_{i_{1} \in
\Gamma^{1} - \{i\}}\{Y^{i_{1},j}_{t} - \hat{g}^{i,i_{1}}_{t}\}, \qquad
\mathcal{U}^{i,j}_{t}({\boldsymbol{Y}}) \in \mathop{\textrm{arg min}}\limits_{j_{1} \in
\Gamma^{2} - \{j\}}\{Y^{i,j_{1}}_{t} + \check{g}^{ j,j_{1}}_{t}\},
\label{eqn18}
\end{equation}(14)
and, for 
$n \ge 1$, 
$\mathcal{M}^{+}_{n}$ and 
$\mathcal{M}^{-}_{n}$ are the events
\[
\mathcal{M}^{+}_{n} = \{\sigma^{\gamma_{n-1}}_{\rho_{n-1}} \le
\tau^{\gamma_{n-1}}_{\rho_{n-1}},\;\sigma^{\gamma_{n-1}}_{\rho_{n-1}} \lt
T\},\qquad \mathcal{M}^{-}_{n} = \{\tau^{\gamma_{n-1}}_{\rho_{n-1}} \lt
\sigma^{\gamma_{n-1}}_{\rho_{n-1}}\}.
\]
Lemma 1. Under the conditions of Theorem 3, we have 
$\gamma(\alpha^{*},\beta^{*}) \in {\mathcal{G}^{i,j}_{s}}$ and 
${Y^{i,j}_{s}} =
{J^{i,j}_{s}}(\gamma(\alpha^{*},\beta^{*}))$ a.s., where 
$\alpha^{*} = (\sigma^{*}_{n},\xi^{*}_{n})_{n \ge 0}$ and 
$\beta^{*} =
(\tau^{*}_{n},\zeta^{*}_{n})_{n \ge 0}$ are sequences defined from 
$(\rho_{n},\gamma_{n})_{n \ge 0}$ as
\begin{equation}
\sigma^{*}_{0} = \tau^{*}_{0} = s, \quad (\xi^{*}_{0},\zeta^{*}_{0}) =
(i,j)
\end{equation}
and, for 
$n \ge 1,$
\begin{gather}
\sigma^{*}_{n} = \inf\{t \ge \sigma^{*}_{n-1} \colon u^{(1)}_{t} \neq
\xi^{*}_{n-1} \} \wedge T, \quad \xi^{*}_{n} = u^{(1)}_{\sigma^{*}_{n}+},\label{eqn19}\end{gather}(15)
\begin{gather}
\tau^{*}_{n} = \inf\{t \ge \tau^{*}_{n-1} \colon u^{(2)}_{t} \neq
\zeta^{*}_{n-1} \} \wedge T, \quad \zeta^{*}_{n} =
u^{(2)}_{\tau^{*}_{n}+},\nonumber
\end{gather}
where u is defined using (9).
Proof. We begin by establishing that 
$\alpha^{*} \in \mathsf{A}^{i}_{s}$. The nonfree loop property (6) prevents accumulation of the switching times 
$\rho^{*}_{n}$, in the sense that 
$\mathbb{P}(\{\rho^{*}_{n} \lt T \text{ for all} n \ge 0\}) = 0$ (see, for example, [Reference Hamadène and Morlais17, pp. 192–193]). Since 
$\sigma^{*}_{n} \ge
\rho_{n}$ for 
$n \ge 0$, it follows that 
$\mathbb{P}(\{\sigma^{*}_{n} \lt T
\text{ for all } n \,{\ge}\, 0\}) = 0$. Also, the consistency property (4) ensures that it is not optimal for a single player to switch twice at the same instant, so we have 
$\sigma^{*}_{n} \lt \sigma^{*}_{n+1}$ on 
$\{\sigma^{*}_{n} \lt T\}$ for 
$n
\ge 1$ (see Reference Hamadène and Morlais[17] or Reference Martyr[24]). By the construction of 
$\alpha^{*}$, noting that 
${u^{(1)}_{\sigma^{*}_{n}+}}$ is 
${\mathcal {F}}_{\sigma^{*}_{n}}$-measurable since 
$\mathbb{F}$ is right-continuous, the remaining parts of Definition 3 are satisfied, and 
${\alpha^{*}
\in \mathsf{A}^{i}_{s}}$. Similarly, 
${\beta^{*} \in
\mathsf{B}^{ j}_{s}}$.
We now prove that 
$\gamma(\alpha^{*},\beta^{*}) \in
{\mathcal{G}^{i,j}_{s}}$ by proceeding in a similar manner to Reference Hamadène and Morlais[17]. Using (10a) and (10c) together with the construction of 
$\rho_1$ gives 
$\mathbb{P}$-a.s.,
\begin{align}
Y^{i,j}_{s} &= \int_{s}^{\rho_{1}}f^{i,j}_{t}{\,\textrm{d}}t +
h^{i,j}\mathbf{1}_{\{\rho_{1} = T\}} +
Y^{i,j}_{\rho_{1}}\mathbf{1}_{\{\rho_{1} \lt T\}} +
\int_{s}^{\rho_{1}}{\,\textrm{d}}K^{i,j,+}_{t} -
\int_{s}^{\rho_{1}}{\,\textrm{d}}K^{i,j,-}_{t}
\\
&\quad\, - \int_{s}^{\rho_{1}}Z^{i,j}_{t}{\,\textrm{d}}B_{t} \nonumber
\\
&= \int_{s}^{\rho_{1}}f^{i,j}_{t}{\,\textrm{d}}t + h^{i,j} \mathbf{1}_{\{\rho_{1}
= T\}} + Y^{i,j}_{\rho_{1}}\mathbf{1}_{\{\rho_{1} \lt T\}} -
\int_{s}^{\rho_{1}}Z^{i,j}_{t}{\,\textrm{d}}B_{t}.
\end{align}
By considering the first switch for either player, we have
\begin{align}
Y^{i,j}_{s} &= \int_{s}^{\rho_{1}}f^{i,j}_{t}{\,\textrm{d}}t +
h^{i,j}\mathbf{1}_{\{\rho_{1} = T\}} +
(Y^{\gamma^{(1)}_{1},j}_{\sigma^{i,j}_{s}} -
\hat{g}^{i,\gamma^{(1)}_{1}}_{\sigma^{i,j}_{s}})
\mathbf{1}_{\{\sigma^{i,j}_{s} \lt T\}}\mathbf{1}_{\{\sigma^{i,j}_{s} \le
\tau^{i,j}_{s}\}} \nonumber
\\
&\quad\, + (Y^{i,\gamma^{(2)}_{1}}_{\tau^{i,j}_{s}} +
\check{g}^{ j,\gamma^{(2)}_{1}}_{\tau^{i,j}_{s}})
\mathbf{1}_{\{\tau^{i,j}_{s} \lt\, \sigma^{i,j}_{s}\}} -
\int_{s}^{\rho_{1}}Z^{i,j}_{t}{\,\textrm{d}}B_{t} \nonumber
\\
&= \int_{s}^{\rho_{1}}f^{u_{t}}_{t}{\,\textrm{d}}t +
Y^{\gamma_{1}}_{\rho_{1}}\mathbf{1}_{\{\rho_{1} \lt T\}} +
h^{\gamma_{0}}\mathbf{1}_{\{\rho_{1} = T\}} -
[\hat{g}^{\gamma^{(1)}_{0},\gamma^{(1)}_{1}}_{\rho_{1}} -
\check{g}^{\gamma^{(2)}_{0},\gamma^{(2)}_{1}}_{\rho_{1}}] -
\int_{s}^{\rho_{1}}Z^{u_{t}}_{t}{\,\textrm{d}}B_{t}
\end{align}
(to account for the event 
$\{\rho_{1} = T\}$, recall that 
${\hat{g}^{i,i}_{t} = \check{g}^{ j,j}_{t}} = 0$). Proceeding iteratively, for 
$n \,{=}\, 1,\ldots,N,$ we obtain by substitution,
\begin{align}
Y^{i,j}_{s} &= \int_{s}^{\rho_{N}}f^{u_{t}}_{t}{\,\textrm{d}}t + \sum_{n =
1}^{N}h^{\gamma_{n-1}}\mathbf{1}_{\{\rho_{n} = T, \rho_{n-1} \lt T\}} -
\sum_{n = 1}^{N}[\hat{g}^{\gamma^{(1)}_{n-1},
\gamma^{(1)}_{n}}_{\rho_{n}} - \check{g}^{\gamma^{(2)}_{n-1},
\gamma^{(2)}_{n}}_{\rho_{n}} ] \nonumber
\\
&\quad\, + Y^{\gamma_{N}}_{\rho_{N}}\mathbf{1}_{\{\rho_{N} \lt T\}} -
\int_{s}^{\rho_{N}}Z^{u_{t}}_{t}{\,\textrm{d}}B_{t},
\label{eqn20}\end{align}(16)
from which we obtain
\begin{equation}
C^{\gamma(\alpha^{*},\beta^{*})}_{N} = Y^{\gamma_{N}}_{\rho_{N}}
\mathbf{1}_{\{\rho_{N} \lt T\}} - Y^{i,j}_{s} +
\int_{s}^{\rho_{N}}f^{u_{t}}_{t}{\,\textrm{d}}t + \sum_{n =
1}^{N}h^{\gamma_{n-1}}\mathbf{1}_{\{\rho_{n} = T, \rho_{n-1} \lt T\}} -
\int_{s}^{\rho_{N}}Z^{u_{t}}_{t}{\,\textrm{d}}B_{t}.
\label{eqn21}
\end{equation}(17)
Let 
$M^{u} = {(M^{u}_{t})_{s \le t \le T}}$ denote the stochastic integral 
${M^{u}_{t} = \int_{s}^{t}Z^{u_{r}}_{r}{\,\textrm{d}}B_{r}}$, which is a well-defined square-integrable martingale on [s, T] Reference Chung and Williams[8]. Continuing from (17), we have a.s.,
\begin{equation}
\sup_{N \ge 1}|C^{\gamma(\alpha^{*},\beta^{*})}_{N}| \le
\int_{s}^{T}|f^{u_{t}}_{t}|{\,\textrm{d}}t + \max_{(i,j) \in \Gamma}|h^{i,j}| +
|Y^{i,j}_{s}| + \max_{(i,j) \in \Gamma}\sup_{s \le t \le T}|Y^{i,j}_{s}|
+ \sup_{s \le t \le T}|M^{u}_{t}|.
\label{eqn22}
\end{equation}(18)
The right-hand side of (18) is a square-integrable random variable, thereby proving 
$\gamma(\alpha^{*},\beta^{*}) \in {\mathcal{G}^{i,j}_{s}}$.
It is now straightforward to prove 
${Y^{i,j}_{s} = J^{i,j}_{s}}
(\gamma(\alpha^{*},\beta^{*}))$ a.s. by taking conditional expectations in (16) then passing to the limit 
$N \to \infty$, which is justified since 
$\gamma(\alpha^{*},\beta^{*})
\in {\mathcal{G}^{i,j}_{s}}$,
\begin{equation}
Y^{i,j}_{s} = \mathbb{E}\bigg[\int_{s}^{T}f^{u_{t}}_{t}{\,\textrm{d}}t + h^{u_{T}}
- \sum_{n = 1}^{\infty}[\hat{g}^{\gamma^{(1)}_{n-1},
\gamma^{(1)}_{n}}_{\rho_{n}} -
\check{g}^{\gamma^{(2)}_{n-1},\gamma^{(2)}_{n}}_{\rho_{n}} ]\biggm|
j_{s}\bigg] =
J^{i,j}_{s}(\gamma(\alpha^{*},\beta^{*})).
\end{equation}
For a given 
$\alpha = (\sigma_{n},\xi_{n})_{n \ge 0} \in
{\mathsf{A}_{s}^{i}}$, let 
${\overline{\beta^{*}}(\alpha)} =
(\tau_{n},\zeta_{n})_{n \ge 0}$ be the control for player 2 defined similarly to (15), with the sequence 
${({\rho _n},{\gamma _n})_{n \ge 0}}$ constructed by
\begin{equation}
\rho_{0} = s, \qquad \gamma_{0} = (i,j)
\end{equation}
and, for 
$n \ge 1,$
\begin{equation}
\rho_{n} = \sigma_{r_{n}} \wedge \tau^{\gamma_{n-1}}_{\rho_{n-1}}, \qquad
\gamma_{n} =
\begin{cases}
(\xi_{r_{n}},\gamma_{n-1}^{(2)}) & \text{on } \mathcal{M}^{+}_{n},
\\[3pt]
(\gamma_{n-1}^{(1)}, \mathcal{U}^{\gamma_{n-1}}_{\rho_{n}}
({\boldsymbol{Y}})) & \text{on } \mathcal{M}^{-}_{n},
\\
\gamma_{n-1} & \text{otherwise},
\end{cases}
\end{equation}
where, reusing earlier notation, 
${\mathcal{U}^{\gamma_{n-1}}_{\rho_{n}}}$ is obtained from (14), 
$\{r_{n}\}_{n \ge 0}$ is defined iteratively by 
$r_{0} = 0$, 
$r_{1} = 1$ and, for 
$n \ge 2$,
\[
r_{n} = r_{n-1} + \mathbf{1}_{\{\sigma_{r_{n-1}} \le
\tau^{\gamma_{n-2}}_{\rho_{n-2}}\}},
\]
and, for 
$n \ge 1$, 
$\mathcal{M}^{+}_{n}$ and 
$\mathcal{M}^{-}_{n}$ are the events
\[
\mathcal{M}^{+}_{n} = \{\sigma_{r_{n}} \le
\tau^{\gamma_{n-1}}_{\rho_{n-1}},\,\sigma_{r_{n}} \lt T \},\qquad
\mathcal{M}^{-}_{n} = \{\tau^{\gamma_{n-1}}_{\rho_{n-1}} \lt
\sigma_{r_{n}}\}.
\]
In an analogous manner using the lower switching selector 
$\boldsymbol{\mathcal{L}}({\boldsymbol{Y}})$ in (14), for each 
$\beta \in
{\mathsf{B}_{s}^{ j}}$ we define 
${\overline{\alpha^{*}}(\beta) \in \mathsf{A}_{s}^{ j}}$ for player 1. The following lemma points out key properties of 
$\overline{\alpha^{*}}$ and 
$\overline{\beta^{*}}$ utilised below to complete the proof of Theorem 3.
Lemma 2. We have
(i)
${\overline{\alpha^{*}} \in \mathcal{A}^{i}_{s}}$ and ${\overline{\beta^{*}} \in \mathcal{B}^{ j}_{s}}$;(ii) the claim
\begin{equation}
\mathop{\textrm{ess sup}}_{\alpha \in \mathcal{A}^{i}_{s}} J^{i,j}_{s}(\gamma(\alpha,
\overline{\beta^{*}}(\alpha))) = Y^{i,j}_{s} = \mathop{\textrm{ess inf}}_{\beta \in
\mathcal{B}^{ j}_{s}} J^{i,j}_{s}(\gamma(\overline{\alpha^{*}}
(\beta),\beta)).
\label{eqn23}
\end{equation}(19)
Proof. (i) We only show that 
${\overline{\beta^{*}} \in \mathcal{B}^{ j}_{s}}$ since the proof that 
${\overline{\alpha^{*}} \in \mathcal{A}^{i}_{s}}$ follows by similar arguments. Just as in the proof of Lemma 1, the construction of 
$\overline{\beta^{*}}(\alpha)$ together with the nonfree loop and consistency properties are sufficient to establish that 
$\overline{\beta^{*}}(\alpha) \in {\mathsf{B}_{s}^{ j}}$ for each 
$\alpha \in {\mathsf{A}_{s}^{i}}$. Moreover, 
$\overline{\beta^{*}}$ satisfies the nonanticipative property in Definition 5 by construction. Let 
$\alpha \in {\mathcal{A}^{i}_{s}}$ be given, and let 
$\overline{\beta^{*}}(\alpha) = \beta = (\tau_{n},\zeta_{n})_{n \ge 0}
\in {\mathsf{B}^{ j}_{s}}$. To show that this control is square integrable we will proceed as in the proof of Lemma 1, to obtain a.s.,
\begin{align}
Y^{i,j}_{s} &= \int_{s}^{\rho_{1}}f^{i,j}_{t}{\,\textrm{d}}t +
h^{i,j}\mathbf{1}_{\{\rho_{1} = T\}} +
Y^{i,j}_{\rho_{1}}\mathbf{1}_{\{\rho_{1} \lt T\}} +
\int_{s}^{\rho_{1}}{\,\textrm{d}}K^{i,j,+}_{t} -
\int_{s}^{\rho_{1}}{\,\textrm{d}}K^{i,j,-}_{t} \nonumber
\\
&\quad\, - \int_{s}^{\rho_{1}}Z^{i,j}_{t}{\,\textrm{d}}B_{t} \nonumber
\\
&\ge \int_{s}^{\rho_{1}}f^{u_{t}}_{t}{d}t + h^{i,j}\mathbf{1}_{\{\rho_{1}
= T\}} - [\hat{g}^{i,\gamma^{(1)}_{1}}_{\rho_{1}} - \check{g}^{ j,
\gamma^{(2)}_{1}}_{\rho_{1}} ] + Y^{\gamma_{1}}_{\rho_{1}}
\mathbf{1}_{\{\rho_{1} \lt T\}} - \int_{s}^{\rho_{1}}
Z^{u_{t}}_{t}{\,\textrm{d}}B_{t},
\end{align}
where, in contrast to the proof of Lemma 1, here 
$\alpha$ is arbitrary and so 
${\gamma^{(1)}_{1}}$ is not necessarily optimal at time 
$\rho_{1}$. This means that the inequality 
${L^{i,j}_{\rho_{1}}({\boldsymbol{Y}}) \le Y^{i,j}_{\rho_{1}}}$ must be enforced and the nonnegative term 
${\int_{s}^{\rho_{1}}{\,\textrm{d}}K^{i,j,+}_{t}}$ cannot be neglected. Proceeding iteratively, for 
$n = 1,\ldots,N,$ it follows that
\begin{align}
Y^{i,j}_{s} &\ge \int_{s}^{\rho_{N}}f^{u_{t}}_{t}{\,\textrm{d}}t + \sum_{n =
1}^{N}h^{\gamma_{n-1}}\mathbf{1}_{\{\rho_{n} = T,\; \rho_{n-1} \lt T\}} -
\sum_{n = 1}^{N}[\hat{g}^{\gamma^{(1)}_{n-1},\gamma^{(1)}_{n}}_{\rho_{n}}
- \check{g}^{\gamma^{(2)}_{n-1},\gamma^{(2)}_{n}}_{\rho_{n}} ] \nonumber
\\
&\quad\, + Y^{\gamma_{N}}_{\rho_{N}}\mathbf{1}_{\{\rho_{N} \lt T\}} -
\int_{s}^{\rho_{N}}Z^{u_{t}}_{t}{\,\textrm{d}}B_{t},
\label{eqn24}\end{align}(20)
from which we obtain
\begin{align}
\sum_{n = 1}^{N}\check{g}^{\gamma^{(2)}_{n-1},
\gamma^{(2)}_{n}}_{\rho_{n}} &\le - \int_{s}^{\rho_{N}}
f^{u_{t}}_{t}{\,\textrm{d}}t - \sum_{n = 1}^{N}h^{\gamma_{n-1}}
\mathbf{1}_{\{\rho_{n} = T,\; \rho_{n-1} \lt T\}} + \sum_{n =
1}^{N}\hat{g}^{\gamma^{(1)}_{n-1},\gamma^{(1)}_{n}}_{\rho_{n}} \nonumber
\\
&\quad\, + Y^{i,j}_{s} - Y^{\gamma_{N}}_{\rho_{N}}\mathbf{1}_{\{\rho_{N} \lt
T\}} + \int_{s}^{\rho_{N}}Z^{u_{t}}_{t}{\,\textrm{d}}B_{t}.
\label{eqn25}\end{align}(21)
We note that 
$\mathbb{P}(\{\rho_{N} \lt T \text{ for all }N \ge 1\}) = 0$ and the limits as 
$N \to \infty$ on both sides of (21) are well defined. As the switching costs are nonnegative, we have
\begin{equation}
0 \le \sum_{n \ge 1}\check{g}^{\zeta_{n-1},\zeta_{n}}_{\tau_{n}} \le -
\int_{s}^{T}f^{u_{t}}_{t}{\,\textrm{d}}t - h^{u_{T}} + \sum_{n \ge
1}\hat{g}^{\xi_{n-1},\xi_{n}}_{\sigma_{n}} + Y^{i,j}_{s} +
\int_{s}^{T}Z^{u_{t}}_{t}{\,\textrm{d}}B_{t}.
\label{eqn26}
\end{equation}(22)
Since 
$\alpha \in {\mathcal{A}^{i}_{s}}$, 
${Y^{i,j} \in
\mathcal{S}^{2}_{c}}$, 
$h^{i,j} \in L^{2}(\mathcal{F}_{T})$, and 
$f^{i,j}$, 
$Z^{i,j}$ belong to 
$\mathcal{H}^{2}$ for all 
$(i,j) \in
\Gamma$, the random variable on the right-hand side of (22) belongs to 
$L^{2}$ and we conclude that the control 
$\beta$ is square integrable.
(ii) We only show the first equality in (19) as the second follows via similar arguments. We proceed by showing that, for every 
$\alpha \in \mathcal{A}^{i}_{s},$ we have
\begin{equation}
Y^{i,j}_{s} \ge J^{i,j}_{s}(\gamma(\alpha,\overline{\beta^{*}}(\alpha))).
\label{eqn27}
\end{equation}(23)
Taking conditional expectations in (20) above, we obtain
\begin{align}
Y^{i,j}_{s} &\ge \mathbb{E}\bigg[\int_{s}^{\rho_{N}}f^{u_{t}}_{t}{\,\textrm{d}}t +
\sum_{n = 1}^{N}h^{\gamma_{n-1}}\mathbf{1}_{\{\rho_{n} = T,\; \rho_{n-1}
\lt T\}} - \sum_{n = 1}^{N}[\hat{g}^{\gamma^{(1)}_{n-1},
\gamma^{(1)}_{n}}_{\rho_{n}} -
\check{g}^{\gamma^{(2)}_{n-1},\gamma^{(2)}_{n}}_{\rho_{n}} ] \biggm|
j_{s}\bigg] \nonumber
\\
&\quad\, + \mathbb{E}[Y^{\gamma_{N}}_{\rho_{N}}\mathbf{1}_{\{\rho_{N} \lt
T\}} \ \,\,|\, \ \, j_{s}].
\label{eqn28}\end{align}(24)
Using (i) above, we have 
${\gamma(\alpha,\overline{\beta^{*}}(\alpha)) \in
\mathcal{G}^{i,j}_{s}}$, so taking the limit 
$N \to \infty$ in (24) proves inequality (23).
Next, for each integer 
$k \ge 0,$ let 
$\alpha^{*}_{k}$ denote the truncation of the control 
$\alpha^{*}$ from Lemma 1 to the first k switches: 
$\alpha^{*}_{k} = (\sigma^{*}_{n},\xi^{*}_{n})_{0 \le n \le k}$ with 
$(T,\xi^{*}_{k})$ appended. Then 
$\alpha^{*}_{k} \in
{\mathcal{A}^{i}_{s}}$ for each k and 
${J^{i,j}_{s}}(\gamma(\alpha^{*}_{k},\overline{\beta^{*}}
(\alpha^{*}_{k}))) \to
{J^{i,j}_{s}}(\gamma(\alpha^{*},\overline{\beta^{*}}(\alpha^{*})))$ by the nonanticipative properties of 
$\overline{\beta^{*}}$ and as 
$\gamma(\alpha^{*},\overline{\beta^{*}}(\alpha^{*})) \in
{\mathcal{G}^{i,j}_{s}}$. The claim
\[
\mathop{\textrm{ess sup}}_{\alpha \in \mathcal{A}^{i}_{s}}
J^{i,j}_{s}(\gamma(\alpha,\overline{\beta^{*}}(\alpha))) = Y^{i,j}_{s}
\]
is then proved by passing to the limit 
$k \to \infty$ in
\[
J^{i,j}_{s}(\gamma(\alpha^{*}_{k},\overline{\beta^{*}} (\alpha^{*}_{k})))
\le \mathop{\textrm{ess sup}}_{\alpha \in \mathcal{A}^{i}_{s}}
J^{i,j}_{s}(\gamma(\alpha,\overline{\beta^{*}}(\alpha))) \le Y^{i,j}_{s},
\]
and using Lemma 1.
Proof of Theorem 3. (i) By construction, we have 
$\alpha^{*} =
\overline{\alpha^{*}}(\beta^{*})$ and 
$\beta^{*} =
\overline{\beta^{*}}(\alpha^{*})$ so that, by Lemma 1,
\begin{equation}
Y^{i,j}_{s} = J^{i,j}_{s}(\gamma(\alpha^{*},\beta^{*})) =
J^{i,j}_{s}(\gamma(\overline{\alpha^{*}}(\beta^{*}),\beta^{*})) =
J^{i,j}_{s}(\gamma(\alpha^{*},\overline{\beta^{*}} (\alpha^{*}))),
\end{equation}
and, by Lemma 2,
\[
\mathop{\textrm{ess sup}}_{\alpha \in \mathcal{A}^{i}_{s}} J^{i,j}_{s}
(\gamma(\alpha,\overline{\beta^{*}}(\alpha))) = Y^{i,j}_{s} =
\mathop{\textrm{ess inf}}_{\beta \in \mathcal{B}^{ j}_{s}}
J^{i,j}_{s}(\gamma(\overline{\alpha^{*}}(\beta),\beta)).
\]
Since 
$\overline{\beta^{*}}(\alpha) \in {\mathcal{B}^{ j}_{s}}$ for every 
$\alpha \in {\mathcal{A}^{i}_{s}}$ and 
${\overline{\alpha^{*}}(\beta) \in \mathcal{A}^{i}_{s}}$ for every 
$\beta \in {\mathcal{B}^{ j}_{s}}$, almost surely we have
\[
\hat{V}^{i,j}_{s} :\!= \mathop{\textrm{ess inf}}_{\beta \in \mathcal{B}^{ j}_{s}}
\mathop{\textrm{ess sup}}_{\alpha \in \mathcal{A}^{i}_{s}}
J^{i,j}_{s}(\gamma(\alpha,\beta)) \le Y^{i,j}_{s} \le \mathop{\textrm{ess sup}}_{\alpha \in
\mathcal{A}^{i}_{s}}\mathop{\textrm{ess inf}}_{\beta \in \mathcal{B}^{ j}_{s}}
J^{i,j}_{s}(\gamma(\alpha,\beta)) =\!: \check{V}^{i,j}_{s},
\]
which completes the proof since 
${\hat{V}^{i,j}_{s} \ge
\check{V}^{i,j}_{s}}$ a.s.
(ii) For all 
$\overline{\alpha} \in
{\mathcal{A}^{i}_{s}},$ we have a.s.,
\begin{align}
\mathop{\textrm{ess inf}}\limits_{\beta \in \mathcal{B}^{ j}_{s}} J^{i,j}_{s}
(\gamma(\overline{\alpha}(\beta),\beta)) & \le \mathop{\textrm{ess inf}}\limits_{\beta \in
\mathcal{B}^{ j}_{s}}\mathop{\textrm{ess sup}}_{\alpha \in
\mathcal{A}^{i}_{s}}J^{i,j}_{s}(\gamma(\alpha,\beta))
\\
& = Y^{i,j}_{s} = \mathop{\textrm{ess inf}}\limits_{\beta \in \mathcal{B}^{ j}_{s}}
J^{i,j}_{s}(\gamma(\overline{\alpha^{*}}(\beta),\beta)),
\end{align}
and the corresponding statement for 
$\overline{\beta^{*}}$ is proved analogously. Since 
${\overline{\alpha^{*}} \in \mathcal{A}^{i}_{s}}$ and 
${\overline{\beta^{*}} \in \mathcal{B}^{ j}_{s}},$ the proof is complete.
Remark 4. In proving Theorem 3 we established the following. For players 1 and 2 respectively, there exist nonanticipative strategies 
$\overline{\alpha^{*}}$ and 
$\overline{\beta^{*}}$ as well as controls 
$\alpha^{*}$ and 
$\beta^{*}$ which satisfy the following:
• the controls
$\alpha^{*}$ and $\beta^{*}$ and nonanticipative strategies $\overline{\alpha^{*}}$ and $\overline{\beta^{*}}$ are related by $\alpha^{*} = \overline{\alpha^{*}}(\beta^{*})$ and $\beta^{*} =
\overline{\beta^{*}}(\alpha^{*})$;•
$\alpha^{*}$ and $\beta^{*}$ are jointly admissible;• when player 2 (the minimiser) uses the nonanticipative strategy
$\overline{\beta^{*}}$, then the use of the control $\alpha^{*}$ by player 1 (the maximiser) gives the maximum possible value for the switching game over all controls $\alpha$ such that $(\alpha,\overline{\beta^{*}}(\alpha))$ is jointly admissible, including all square-integrable controls $\alpha$;• when player 1 uses the nonanticipative strategy
$\overline{\alpha^{*}}$, then the use of the control $\beta^{*}$ by player 2 gives the minimum possible value for the switching game over all controls $\beta$ such that $(\overline{\alpha^{*}}(\beta),\beta)$ is jointly admissible, including all square-integrable controls $\beta$;• the strategies
$\overline{\alpha^{*}}$ and $\overline{\beta^{*}}$ are best responses in the robust sense Reference Ball, Chudoung and Day[1], Reference Bayraktar, Cosso and Pham[2], Reference Krasovski and Subbotin[22].
Let us emphasise that 
$\overline{\alpha^{*}}$ is not necessarily a best response strategy in the sense that
\[
J^{i,j}_{s}(\gamma(\overline{\alpha^{*}}(\beta),\beta)) = \mathop{\textrm{ess sup}}_{\alpha
\in \mathcal{A}^{i}_{s}}J^{i,j}_{s}(\gamma(\alpha,\beta)) \quad \text{for
all }\beta \in \mathcal{B}^{ j}_{s},
\]
and correspondingly for 
$\overline{\beta^{*}}$. In the game with initial data (s, i, j), for player 1, we can define a mapping 
$\overline{\alpha}
\colon {\mathcal{B}^{ j}_{s} \to \mathcal{A}^{i}_{s}}$ such that, for each 
$\beta \in {\mathcal{B}^{ j}_{s}}$ a.s.,
$$
J^{i,j}_{s}(\gamma(\overline{\alpha}(\beta),\beta)) \ge
J^{i,j}_{s}(\gamma(\alpha,\beta))\quad\text{a.s.\ for all } \alpha \in
\mathcal{A}^{i}_{s},
$$
but this mapping is generally not nonanticipative since its output 
$\overline{\alpha}(\beta)$ can depend on the entire trajectory corresponding to the input 
$\beta$. For example, let 
$\beta \in
{\mathcal{B}^{ j}_{s}}$ be a given switching control for player 2. Let 
$u^{\beta}$ denote the stochastic process that indicates player 2’s current mode according to 
$\beta$,
\[
u^{\beta}_{t} = \zeta_{0}\mathbf{1}_{[\tau_{0},\tau_{1}]}(t) +
\sum\limits_{n \ge 1}\zeta_{n}\mathbf{1}_{(\tau_{n},\tau_{n+1}]}(t),
\qquad t \in [s,T].
\]
For 
$i \in \Gamma^{1}$ and 
$t \in [s,T]$, let 
${\tilde{f}^{i}_{t}(\beta) = f^{i,u^{\beta}_{t}}_{t}}$ denote the controllable part of player 1’s instantaneous reward when player 2’s switching control 
$\beta$ is given and fixed. Similarly, let 
${\tilde{h}^{i}_{s,T}(\beta)} :\!= {h^{i,u^{\beta}_{T}}
+ \sum_{n=1}^{\infty}\check{g}^{\zeta_{n-1},
\zeta_{n}}_{\tau_{n}}\mathbf{1}_{\{\tau_{n} \ge s\}}}$ denote the controllable part of player 1’s terminal reward given player 2’s switching control 
$\beta$, and inclusive of the payments received from player 2’s switching costs 
${\check{g}^{\zeta_{n-1},\zeta_{n}}_{\tau_{n}}}$ from time s onwards. With these definitions, we can evaluate a control 
$\alpha \in
\mathcal{A}^{i}_{s}$ for player 1 according to
\begin{align}
\tilde{J}_{s}^{i}(\alpha;\,\beta) & = \mathbb{E}\bigg[\int_{s}^{T}
\tilde{f}^{u^{\alpha}_{t}}_{t}(\beta){\,\textrm{d}}t -
\sum_{n=1}^{\infty}\hat{g}^{\xi_{n-1},\xi_{n}}_{\sigma_{n}} +
\tilde{h}^{u^{\alpha}_{T}}_{s,T}(\beta) \biggm| j_{s}\bigg],
\qquad \alpha \in \mathcal{A}^{i}_{s},
\label{eqn29}
\end{align}(25)
and define the corresponding optimal value 
${\tilde{V}_{s}^{i}(\beta) = \mathop{\textrm{ess sup}}_{\alpha \in
\mathcal{A}_{s}^{i}}\tilde{J}_{s}^{i}(\alpha;\,\beta)}$. Using the results in Reference Djehiche, Hamadène, Morlais and Zhao[12] and Reference Martyr[24], we can prove the existence of an optimal control 
$\alpha^{*} \in {\mathcal{A}^{i}_{s}}$ for this problem. The nonanticipativity issue arises from the dependence of (25) on the expected future rewards due to player 2’s control 
$\beta$.
Acknowledgements
Randall Martyr and John Moriarty would like to thank the UK Engineering and Physical Sciences Research Council (EPSRC) for its financial support via grants EP/N013492/1 and EP/P002625/1, respectively.
