Appendix A: Proof of Theorem 3.2
For convenience, we use the short-hand notations:
$$\begin{aligned} \begin{aligned} \bigtriangledown _{x}\hat{f}(t)&=\bigtriangledown _{x}f \left( t,\hat{X}(t),\hat{u}^{(1)}(t),\hat{u}^{(2)}(t)\right) \\ \text {and}\ \ \bigtriangledown _{u^{(1)}}\hat{f}(t)&=\bigtriangledown _{u^{(1)}}f\left( t,\hat{X}(t),\hat{u}^{(1)}(t),\hat{u}^{(2)}(t)\right) \end{aligned} \end{aligned}$$
and similarly for other gradients.
Suppose that \(\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}\times \mathcal {A}^{(2)}_{\mathcal {G}}\) is a saddle point of the games (8). Then,
$$\begin{aligned} \lim _{y\rightarrow 0^{+}}\frac{1}{y}\left[ \mathcal {J}\left( \hat{u}^{(1)}+y\beta ^{(1)},\hat{\xi }^{(1)}+y\varsigma ^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) -\mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \right] \le 0 \end{aligned}$$
(46)
holds for all bounded \(\left( \beta ^{(1)},\varsigma ^{(1)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}\). According to the definition of \(\mathcal {J}\) in (7), it follows from (46) that
$$\begin{aligned}&E\left[ \int _{0}^{T}\left\{ \bigtriangledown _{x}\hat{f}\left( t\right) ^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t) +\bigtriangledown _{u^{(1)}}\hat{f}\left( t\right) ^{T}\beta ^{(1)}(t) \right\} \mathrm{d}t\right. \nonumber \\&\quad +\bigtriangledown g(\hat{X}(T))^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(T) +\sum _{l=1}^{n}\int _{0}^{T}\bigtriangledown _{x}h_{l}(t,\hat{X}(t)) \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\mathrm{d}\hat{\xi }^{(1)}_{l}(t) \nonumber \\&\quad \left. +\int _{0}^{T}h(t,\hat{X}(t))\mathrm{d}\varsigma ^{(1)}(t)+\sum _{l=1}^{n}\int _{0}^{T}\bigtriangledown _{x}k_{l}(t,\hat{X}(t)) \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\mathrm{d}\hat{\xi }^{(2)}_{l}(t) \right] \le 0. \end{aligned}$$
(47)
Now, by applying Itô formula to \(\bigtriangledown g(X(T))^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(T)\), we obtain
$$\begin{aligned}&E\left[ \bigtriangledown g(\hat{X}(T))^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(T)\right] = E\left[ \hat{p}(T)^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(T)\right] \nonumber \\&\quad = E\left[ \sum _{j=1}^{n}\left\{ \int _{0}^{T}\left[ \hat{p}_{j}(t)\left( \triangledown _{x}\hat{b}_{j}(t)^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t) +\triangledown _{u^{(1)}}\hat{b}_{j}(t)^{T}\beta ^{(1)}(t)\right) \right. \right. \right. \nonumber \\&\qquad +\sum _{l=1}^{n}\hat{q}_{jl}(t)\left( \triangledown _{x}\hat{\sigma }_{jl}(t)^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t) +\triangledown _{u^{(1)}}\hat{\sigma }_{jl}(t)^{T}\beta ^{(1)}(t)\right) \nonumber \\&\qquad +\sum _{l=1}^{n}\hat{\tilde{q}}_{jl}(t)\left( \triangledown _{x}\hat{\varpi }_{jl}(t)^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t)+\triangledown _{u^{(1)}}\hat{\varpi }_{jl}(t)^{T}\beta ^{(1)}(t)\right) \nonumber \\&\qquad \left. +\sum _{l=1}^{n}\int _{\mathbb {R}}\hat{r}_{jl}(t,z)\left( \triangledown _{x}\hat{\gamma }_{jl}(t,z)^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t) {+}\triangledown _{u^{(1)}}\hat{\gamma }_{jl}(t,z)^{T}\beta ^{(1)}(t)\right) \nu _{l}(\mathrm{d}z)\right] \mathrm{d}t \nonumber \\&\qquad -\left( \triangledown _{x} \hat{H}\left( t\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right) _{j} \breve{X}_{j}^{\left( u^{(1)},\xi ^{(1)}\right) }(t) \nonumber \\&\qquad +\,\sum _{l=1}^{n}\hat{p}_{j}(t)\triangledown _{x}\alpha _{jl}(t,\hat{X}(t))^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\mathrm{d}\hat{\xi }^{(1)}_{l}(t) \nonumber \\&\qquad +\sum _{l=1}^{n}\hat{p}_{j}(t)\alpha _{jl}(t,\hat{X}(t))\mathrm{d}\varsigma ^{(1)}_{l}(t) \nonumber \\&\qquad +\sum _{l=1}^{n}\hat{p}_{j}(t)\triangledown _{x}\lambda _{jl}(t,\hat{X}(t))^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\mathrm{d}\hat{\xi }^{(2)}_{l}(t) \nonumber \\&\qquad +\sum _{0\le t\le T}\sum _{l,m=1}^{n}\left( \triangledown _{x}\alpha _{jl}(\{t\},\hat{X}(t))^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\triangle \hat{\xi }^{(1)}_{l}(t) \right. \nonumber \\&\qquad +\,\triangledown _{x}\lambda _{jl}(\{t\},\hat{X}(t))^{T} \breve{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\triangle \hat{\xi }^{(2)}_{l}(t) \nonumber \\&\qquad \left. \left. \left. +\,\alpha _{jl}(\{t\},\hat{X}(t))\triangle \varsigma ^{(1)}_{l}(t)\right) \int _{\mathbb {R}_{0}}\hat{r}_{jm}(\{t\},z)N_{m}(\{t\},\mathrm{d}z)\right\} \!\!\right] , \end{aligned}$$
(48)
where \(\triangle \varsigma _{l}^{(i)}(t)=\varsigma _{l}^{(i)}(t)-\varsigma _{l}^{(i)}(t-)\) is the pure discontinuous part of \(\varsigma _{l}^{(i)}(t)\) and \(\varsigma _{l}^{(i),C}(t)\) is the continuous part of \(\varsigma _{l}^{(i)}(t)\), \(i=1,2\).
Next, by the definition of Hamiltonian (9), we have
$$\begin{aligned}&\bigtriangledown _{x}H\left( t,x,u^{(1)},u^{(2)},p,q,\tilde{q},r(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\xi ^{(1)},\mathrm{d}\xi ^{(2)}\right) \nonumber \\&\quad = \left[ \bigtriangledown _{x}f\left( t\right) +\sum _{j=1}^{n}\bigtriangledown _{x}b_{j}\left( t\right) p_{j} + \sum _{j,l=1}^{n}\bigtriangledown _{x}\sigma _{jl}\left( t\right) q_{jl}\right. \nonumber \\&\qquad \left. +\,\sum _{j,l=1}^{n}\int _{\mathbb {R}}\bigtriangledown _{x}\gamma _{jl}\left( t,z\right) r_{jl}(t,z)\nu _{l}(\mathrm{d}z) +\sum _{j,l=1}^{n}\bigtriangledown _{x}\varpi _{jl}\left( t\right) \tilde{q}_{jl} \right] \mathrm{d}t \nonumber \\&\qquad +\,\sum _{l=1}^{n}\left[ \left( \sum _{j=1}^{n} p_{j}\bigtriangledown _{x}\alpha _{jl}(t,x)+\bigtriangledown _{x}h_{l}(t,x)\right) \mathrm{d}\xi _{l}^{(1)}(t) \right. \nonumber \\&\qquad \left. +\,\left( \sum _{j=1}^{n}p_{j}\bigtriangledown _{x}\lambda _{jl}(t,x)+\bigtriangledown _{x}k_{l}(t,x)\right) \mathrm{d}\xi _{l}^{(2)}(t)\right] \nonumber \\&\qquad +\,\sum _{j,l,m=1}^{n}\left[ \bigtriangledown _{x}\alpha _{jl}\left( \{t\},x\right) \triangle \xi _{l}^{(1)}(t)\right. \nonumber \\&\qquad \left. +\,\bigtriangledown _{x}\lambda _{jl}\left( \{t\},x\right) \triangle \xi _{l}^{(2)}(t)\right] \int _{\mathbb {R}_{0}}r_{jm}(\{t\},z)N_{m}(\{t\},\mathrm{d}z) \end{aligned}$$
(49)
and
$$\begin{aligned}&\bigtriangledown _{u^{(1)}}H\left( t,x,u^{(1)},u^{(2)},p,q,r(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\xi ^{(1)},\mathrm{d}\xi ^{(2)}\right) \nonumber \\&\quad =\left[ \bigtriangledown _{u^{(1)}}f\left( t\right) +\sum _{j=1}^{n}\bigtriangledown _{u^{(1)}}b_{j}\left( t\right) p_{j}+\sum _{j,l=1}^{n}\bigtriangledown _{u^{(1)}}\sigma _{jl}\left( t\right) q_{jl} \right. \nonumber \\&\qquad \left. +\sum _{j,l=1}^{n}\bigtriangledown _{u^{(1)}}\varpi _{jl}\left( t\right) \tilde{q}_{jl} +\sum _{j,l=1}^{n}\int _{\mathbb {R}}\bigtriangledown _{u^{(1)}}\gamma _{jl}\left( t,z\right) r_{jl}(t,z)\nu _{l}(\mathrm{d}z)\right] \mathrm{d}t. \end{aligned}$$
(50)
Then, substituting (48), (49), (50) into (47), we get
$$\begin{aligned}&E\left[ \int _{0}^{T}\bigtriangledown _{u^{(1)}}H\left( t,\hat{X}(t),\hat{u}^{(1)},\hat{u}^{(2)},\hat{p},\hat{q},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) ^{T}\beta ^{(1)}(t) \right. \nonumber \\&\quad \left. +\int _{0}^{T}\hat{U}^{T}(t)\mathrm{d}\varsigma ^{(1),C}(t) +\sum _{0\le t\le T}\hat{V}^{T}(t)\triangle \varsigma ^{(1)}(t)\right] \le 0, \end{aligned}$$
(51)
where \(U_{l}(t)\) and \(V_{l}(t)\) are defined by (13) and (14), respectively.
Since the inequality (51) holds for all bounded \((\beta ^{(1)},\varsigma ^{(1)})\in \mathcal {A}^{(1)}_{\mathcal {G}}\), one can choose \(\varsigma ^{(1)}\equiv 0\) and hence
$$\begin{aligned} E\left[ \int _{0}^{T}\bigtriangledown _{u^{(1)}}H\left( t,\hat{X}(t),\hat{u}^{(1)},\hat{u}^{(2)},\hat{p},\hat{q},\hat{\tilde{q}},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) ^{T}\beta ^{(1)}(t)\right] \le 0 . \end{aligned}$$
(52)
In particular, given \(t\in [0,T]\), the inequality (52) holds for all bounded \(\left( \beta ^{(1)}(s),0\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}\), where
$$\begin{aligned} \beta ^{(1)}(s):=\left( 0,\ldots ,\beta ^{(1)}_{j}(s),\ldots ,0\right) ,\ \ \ \ \ j=1,\ldots ,n \end{aligned}$$
with
$$\begin{aligned} \beta ^{(1)}_{j}(s)=\theta _{j}^{(1)}\chi _{[t,t+h)}(s),\ \ \ \ \ s\in [0,T]. \end{aligned}$$
Here, \(t+h\le T\) and \(\theta _{j}^{(1)}=\theta _{j}^{(1)}(\omega )\) is a bounded \(\mathcal {G}^{(1)}_{t}\)-measurable random variable. Then (52) can be written as
$$\begin{aligned} E\left[ \int _{t}^{t+h}\theta _{j}^{(1)}\frac{\partial H}{\partial u_{j}^{(1)}}\left( t,\hat{X}(t),\hat{u}^{(1)},\hat{u}^{(2)},\hat{p},\hat{q},\hat{\tilde{q}},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right] \le 0. \end{aligned}$$
(53)
Since (53) holds for both \(\theta _{j}^{(1)}\) and \(-\theta _{j}^{(1)}\), we have
$$\begin{aligned} E\left[ \int _{t}^{t+h}\theta _{j}^{(1)}\frac{\partial H}{\partial u_{j}^{(1)}}\left( t,\hat{X}(t),\hat{u}^{(1)},\hat{u}^{(2)},\hat{p},\hat{q},\hat{\tilde{q}},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right] =0. \end{aligned}$$
(54)
Differentiating (54) with respect to h at \(h=0\), it follows that
$$\begin{aligned} E\left[ \theta _{j}^{(1)}\frac{\partial H}{\partial u_{j}^{(1)}}\left( t,\hat{X}(t),\hat{u}^{(1)},\hat{u}^{(2)},\hat{p},\hat{q},\hat{\tilde{q}},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right] =0 \end{aligned}$$
holds for all bounded \(\mathcal {G}^{(1)}_{t}\)-measurable random variable \(\theta _{j}^{(1)}\). As a result, we conclude that
$$\begin{aligned}&E\left[ \left. \frac{\partial H}{\partial u_{j}^{(1)}}\left( t,\hat{X}(t),\hat{u}^{(1)},\hat{u}^{(2)},\hat{p},\hat{q},\hat{\tilde{q}},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm{d}\hat{\xi }^{(2)}\right) \right| \mathcal {G}^{(1)}_{t}\right] =0,\\&\quad j=1,\ldots ,n, \end{aligned}$$
which leads to
$$\begin{aligned} E\left[ \left. \bigtriangledown _{u^{(1)}}H\left( t,\hat{X}(t),\hat{u}^{(1)},\hat{u}^{(2)},\hat{p},\hat{q},\hat{\tilde{q}},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right| \mathcal {G}^{(1)}_{t}\right] =0. \end{aligned}$$
Next, we prove (18) and (20). Note that the inequality (51) holds for all bounded \((\beta ^{(1)},\varsigma ^{(1)})\in \mathcal {A}^{(1)}_{\mathcal {G}}\). Thus, by letting \(\beta ^{(1)}=0\), we obtain
$$\begin{aligned} E\left[ \int _{0}^{T}\hat{U}^{T}(t)\mathrm{d}\varsigma ^{(1),C}(t) +\sum _{0\le t\le T}\hat{V}^{T}(t)\triangle \varsigma ^{(1)}(t)\right] \le 0. \end{aligned}$$
(55)
In order to prove (18), we choose \(\varsigma ^{(1)}\) in the following way:
$$\begin{aligned} \varsigma ^{(1)}(t):=\left( 0,\ldots ,\varsigma ^{(1)}_{l}(t),\ldots ,0\right) ,\quad l=1,\ldots ,n. \end{aligned}$$
with
$$\begin{aligned} \mathrm{d}\varsigma _{l}^{(1)}(t)=a_{l}^{(1)}(t)\mathrm{d}t, \quad t\in [0,T], \end{aligned}$$
where \(a_{l}^{(1)}(t)\ge 0\) is continuous \(\mathcal {G}^{(1)}_{t}\)-adapted stochastic process. Then, it follows from (55) that
$$\begin{aligned} E\left[ \int _{0}^{T}\hat{U}_{l}(t)a_{l}^{(1)}(t)\mathrm{d}t\right] \le 0 \end{aligned}$$
(56)
holds for all \(\mathcal {G}^{(1)}_{t}\)-adapted \(a_{l}^{(1)}(t)\ge 0\), which implies that
$$\begin{aligned} E\left[ \left. \hat{U}_{l}(t)\right| \ \mathcal {G}^{(1)}_{t}\right] \le 0, \quad \text {for}\ \text {almost} \ \text {all}\quad t\in [0,T],\quad l=1,2,\ldots ,n. \end{aligned}$$
(57)
Moreover, by choosing
$$\begin{aligned} \varsigma ^{(1)}(t):=\left( 0,\ldots ,\varsigma ^{(1)}_{l}(t),\ldots ,0\right) ,\quad l=1,\ldots ,n, \end{aligned}$$
where
$$\begin{aligned} \varsigma _{l}^{(1)}(t)=\hat{\xi }_{l}^{(1),C}(t), \quad t\in [0,T], \end{aligned}$$
together with (55), we have
$$\begin{aligned} E\left[ \int _{0}^{T}\hat{U}_{l}(t)\mathrm{d}\hat{\xi }_{l}^{(1),C}(t)\right] \le 0. \end{aligned}$$
(58)
Similarly, by letting \(\varsigma _{l}^{(1)}(t)=-\hat{\xi }_{l}^{(1),C}(t)\), we get
$$\begin{aligned} E\left[ \int _{0}^{T}\hat{U}_{l}(t)\mathrm{d}\hat{\xi }_{l}^{(1),C}(t)\right] \ge 0. \end{aligned}$$
(59)
Combining (58) and (59), we obtain
$$\begin{aligned} E\left[ \int _{0}^{T}\hat{U}_{l}(t)\mathrm{d}\hat{\xi }_{l}^{(1),C}(t)\right] = E\left[ \int _{0}^{T}E\left[ \left. \hat{U}_{l}(t)\right| \ \mathcal {G}^{(1)}_{t}\right] \mathrm{d}\hat{\xi }_{l}^{(1),C}(t)\right] =0. \end{aligned}$$
(60)
Since \(\hat{\xi }_{l}^{(1),C}(\cdot )\) is a singular control, we have \(\mathrm{d}\hat{\xi }_{l}^{(1),C}(t)\ge 0\). Hence, it follows from (57) and (60) that
$$\begin{aligned} E\left[ \left. \hat{U}_{l}(t)\right| \mathcal {G}^{(1)}_{t}\right] \mathrm{d}\hat{\xi }_{l}^{(1),C}(t)=0,\quad t\in [0,T],\quad l=1,2,\ldots ,n. \end{aligned}$$
In order to prove (20), we fix \(t\in [0,T]\) and choose \(\varsigma ^{(1)}:=\left( 0,\ldots ,\varsigma ^{(1)}_{l},\ldots ,0\right) \), \(l=1,2,\ldots ,n\), such that
$$\begin{aligned} \mathrm{d}\varsigma ^{(1)}_{l}(s)=a^{(1)}_{l}(\omega )\delta _{t}(s),\quad s\in [0,T], \end{aligned}$$
where \(a^{(1)}_{l}(\omega )\ge 0\) is bounded \(\mathcal {G}^{(1)}_{t}\)-measurable and \(\delta _{t}(s)\) is the unit point mass at t. In this case, it follows from (55) that
$$\begin{aligned} E\left[ \hat{V}_{l}(t)a^{(1)}_{l}(\omega )\right] \le 0 \end{aligned}$$
holds for all bounded \(\mathcal {G}^{(1)}_{t}\)-measurable \(a^{(1)}_{l}(\omega )\ge 0\). This gives
$$\begin{aligned} E\left[ \left. \hat{V}_{l}(t)\right| \ \mathcal {G}^{(1)}_{t}\right] \le 0, \quad l=1,2,\ldots ,n. \end{aligned}$$
(61)
Let \(\xi _{l}^{(1),d}(t)\) denote the pure discontinuous part of \(\xi _{l}^{(1)}\), \(l=1,2,\ldots ,n\). Choosing \(\varsigma _{l}^{(1)}(t)=\xi _{l}^{(1),d}(t)\) and \(\varsigma _{l}^{(1)}(t)=-\xi _{l}^{(1),d}(t)\) respectively, it is clear from (55) that
$$\begin{aligned} E\left[ \sum _{0\le t\le T}\hat{V}_{l}(t)\triangle \xi _{l}^{(1)}(t)\right] =E\left[ \sum _{0<t\le T}E\left[ \left. \hat{V}_{l}(t)\right| \ \mathcal {G}^{(1)}_{t}\right] \triangle \xi _{l}^{(1)}(t)\right] =0. \end{aligned}$$
(62)
It is obvious that \(\triangle \xi _{l}^{(1)}(t)\ge 0\), for \(\xi _{l}^{(1)}(\cdot )\) is a singular control. Thus, we conclude from (61) and (62) that
$$\begin{aligned} E\left[ \left. \hat{V}_{l}(t)\right| \ \mathcal {G}^{(1)}_{t}\right] \triangle \xi _{l}^{(1)}(t)=0 \quad \text {for}\ \text {all} \quad t\in [0,T], \quad l=1,2,\ldots ,n. \end{aligned}$$
Next, suppose that
$$\begin{aligned} \lim _{y\rightarrow 0^{+}}\frac{1}{y}\left[ \mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)}+y\beta ^{(2)},\hat{\xi }^{(2)}+y\varsigma ^{(2)}\right) -\mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \right] \ge 0 \end{aligned}$$
(63)
holds for all bounded \(\left( \beta ^{(2)},\varsigma ^{(2)}\right) \in \mathcal {A}^{(2)}_{\mathcal {G}}\). Then, by following a similar approach, we have (19), (21) and
$$\begin{aligned} E\left[ \left. \bigtriangledown _{u^{(2)}}H\left( t,\hat{X}(t),\hat{u}^{(1)},\hat{u}^{(2)},\hat{p},\hat{q},\hat{\tilde{q}},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right| \mathcal {G}^{(2)}_{t}\right] =0. \end{aligned}$$
This completes the proof. \(\square \)
Appendix B: Proof of Theorem 3.3
In the following, we use the notation
$$\begin{aligned}&\hat{H}(t)\left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) =H\left( t,\hat{x},\hat{u}^{(1)},\hat{u}^{(2)},\hat{p},\hat{q},\hat{\tilde{q}},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) ,\\&H\left( t,u^{(1)}\right) \left( \mathrm{d}t,\mathrm{d}\xi ^{(1)},\mathrm{d}\hat{\xi }^{(2)}\right) \\&\quad =H\left( t,x^{\left( u^{(1)},\xi ^{(1)}\right) },u^{(1)},\hat{u}^{(2)},\hat{p},\hat{q},\hat{\tilde{q}},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\xi ^{(1)},\mathrm{d}\hat{\xi }^{(2)}\right) ,\\&H\left( t,u^{(2)}\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm{d}\xi ^{(2)}\right) \\&\quad =H\left( t,x^{\left( u^{(2)},\xi ^{(2)}\right) },\hat{u}^{(1)},u^{(2)}\hat{p},\hat{q},\hat{\tilde{q}},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm{d}\xi ^{(2)}\right) \end{aligned}$$
and similarly with \(\hat{f}(t)\), \(f\left( t,X^{\left( u^{(i)},\xi ^{(i)}\right) }(t),u^{(i)}\right) \), \(\hat{b}(t)\), \(b\left( t,X^{\left( u^{(i)},\xi ^{(i)}\right) }(t),u^{(i)}\right) \), \(\hat{\sigma }(t)\), \(\sigma \left( t,X^{\left( u^{(i)},\xi ^{(i)}\right) }(t),u^{(i)}\right) \), \(\hat{\gamma }(t,z)\), \(\gamma \left( t,X^{\left( u^{(i)},\xi ^{(i)}\right) }(t),u^{(i)},z\right) \), \(i=1,2\).
[I] By the definition of \(\mathcal {J}\left( u^{(1)},\xi ^{(1)};u^{(2)},\xi ^{(2)}\right) \), we have
$$\begin{aligned} \mathcal {J}\left( u^{(1)},\xi ^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) -\mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) =I_{1}+I_{2}+I_{3}+I_{4}, \end{aligned}$$
(64)
where
$$\begin{aligned} I_{1}&= E\left[ \int _{0}^{T}\left( f\left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t),u^{(1)}\right) -\hat{f}(t)\right) \mathrm{d}t\right] ,\\ I_{2}&= E\left[ g(X^{\left( u^{(1)},\xi ^{(1)}\right) }(T))-g(\hat{X}(T))\right] ,\\ I_{3}&= E\left[ \int _{0}^{T} h\left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) \mathrm{d}\xi ^{(1)}(t)-\int _{0}^{T} h\left( t,\hat{X}(t)\right) \mathrm{d}\hat{\xi }^{(1)}(t)\right] ,\\ I_{4}&= E\left[ \int _{0}^{T}\left( k(t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t))-k(t,\hat{X}(t))\right) \mathrm{d}\hat{\xi }^{(2)}(t)\right] . \end{aligned}$$
It follows from the definition of Hamiltonian (9) that
$$\begin{aligned} I_{1}&= E\left[ \int _{0}^{T} H\left( t,u^{(1)}\right) \left( \mathrm{d}t,\mathrm{d}\xi ^{(1)},\mathrm{d}\hat{\xi }^{(2)}\right) - \hat{H}(t)\left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right. \nonumber \\&\quad \ -\,\int _{0}^{T}\left\{ \left( b\left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t), u^{(1)}\right) -\hat{b}(t)\right) ^{T}\hat{p}(t) \right. \nonumber \\&\quad \ -\,tr\left[ \left( \sigma \left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t),u^{(1)}\right) -\hat{\sigma }(t)\right) ^{T} \hat{q}(t)\right] \nonumber \\&\quad \ -\,\sum _{j,l=1}^{n}\int _{\mathbb {R}}\left( \gamma _{j,l}\left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t),u^{(1)},z\right) -\hat{\gamma }_{j,l}(t,z)\right) r_{j,l}(t,z)\nu _{l}(\mathrm{d}z) \nonumber \\&\quad \ \left. -\,\,tr\left[ \left( \varpi \left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t),u^{(1)}\right) -\hat{\varpi }(t)\right) ^{T} \hat{\tilde{q}}(t)\right] \right\} \mathrm{d}t \nonumber \\&\quad \ +\,\int _{0}^{T}\left\{ \hat{p}^{T}(t)\alpha (t,\hat{X}(t))+h(t,\hat{X}(t)) \right\} \mathrm{d}\hat{\xi }^{(1)}(t) \nonumber \\&\quad \ -\,\int _{0}^{T}\left\{ \hat{p}^{T}(t)\alpha \left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) +h\left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) \right\} \mathrm{d}\xi ^{(1)}(t) \nonumber \\&\quad \ -\,\int _{0}^{T}\left\{ \hat{p}^{T}(t)\left[ \lambda \left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) -\lambda (t,\hat{X}(t))\right] \right. \nonumber \\&\quad \ \left. +\,\left[ k\left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) -k(t,\hat{X}(t))\right] \right\} \mathrm{d}\hat{\xi }^{(2)}(t) \nonumber \\&\quad \ -\,\sum _{0<t\le T}\sum _{j,l,m=1}^{n}\left( \alpha _{jl}\left( \{t\},X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) \triangle \xi _{l}^{(1)}(t)\right. \nonumber \\&\quad \ \left. -\,\alpha _{jl}(\{t\},\hat{X}(t))\triangle \hat{\xi }_{l}^{(1)}(t) \right) \int _{\mathbb {R}_{0}}\hat{r}_{jm}(\{t\},z)N_{m}(\{t\},\mathrm{d}z) \nonumber \\&\quad \ -\,\sum _{0<t\le T}\sum _{j,l,m=1}^{n}\left( \lambda _{jl}\left( \{t\},X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) \right. \nonumber \\&\quad \ \left. \left. -\,\lambda _{jl}(\{t\},\hat{X}(t)) \right) \triangle \hat{\xi }_{l}^{(2)}(t) \int _{\mathbb {R}_{0}}\hat{r}_{jm}(\{t\},z)N_{m}(\{t\},\mathrm{d}z) \right] . \end{aligned}$$
(65)
Let \( \tilde{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t):=X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)-\hat{X}(t)\), \(0\le t\le T\). Since g(x) is concave in x, we have
$$\begin{aligned} I_{2} \leqslant&E\left[ \left( X^{\left( u^{(1)},\xi ^{(1)}\right) }(T)-\hat{X}(T)\right) ^{T}\triangledown g(\hat{X}(T)) \right] =E\left[ \left( \tilde{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(T)\right) ^{T}\hat{p}(T)\right] . \end{aligned}$$
By applying Itô formula to \(\left( \tilde{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(T)\right) ^{T}\hat{p}(T)\), we obtain
$$\begin{aligned} I_{2}&\le E\left[ \int _{0}^{T}\left\{ \hat{p}^{T}(t)\left( b\left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t),u^{(1)}\right) -\hat{b}(t)\right) \right. \right. \nonumber \\&\quad +\,tr\left[ \left( \sigma \left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t),u^{(1)}\right) -\hat{\sigma }(t)\right) ^{T}\hat{q}(t) \right] \nonumber \\&\quad +\,\sum _{j,l=1}^{n}\int _{\mathbb {R}_{0}}\left( \gamma _{jl}\left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t),z\right) -\hat{\gamma }_{jl}(t,z)\right) \hat{r}_{jl}(t,z)\nu _{l}(\mathrm{d}z) \nonumber \\&\quad \left. +\,tr\left[ \left( \varpi \left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t),u^{(1)}\right) -\hat{\varpi }(t)\right) ^{T}\hat{\tilde{q}}(t) \right] \right\} \mathrm{d}t \nonumber \\&\quad -\,\int _{0}^{T}\left( \tilde{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) ^{T} \triangledown _{x}\hat{H}(t)\left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \nonumber \\&\quad +\,\int _{0}^{T}\hat{p}^{T}(t)\alpha \left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) \mathrm{d}\xi ^{(1)}(t) -\int _{0}^{T}\hat{p}^{T}(t)\alpha (t,\hat{X}(t))\mathrm{d}\hat{\xi }^{(1)}(t) \nonumber \\&\quad +\,\int _{0}^{T}\hat{p}^{T}(t)\left\{ \lambda \left( t,X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) -\lambda (t,\hat{X}(t))\right\} \mathrm{d}\hat{\xi }^{(2)}(t) \nonumber \\&\quad +\, \sum _{0<t\le T}\sum _{j,l,m=1}^{n}\left( \alpha _{jl}\left( \{t\},X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) \triangle \xi _{l}^{(1)}(t)\right. \nonumber \\&\quad \left. -\,\quad \alpha _{jl}(\{t\},\hat{X}(t))\triangle \hat{\xi }_{l}^{(1)}(t) \right) \int _{\mathbb {R}_{0}}\hat{r}_{jm}(\{t\},z)N_{m}(\{t\},\mathrm{d}z) \nonumber \\&\quad +\, \sum _{0<t\le T}\sum _{j,l,m=1}^{n}\left( \lambda _{jl}\left( \{t\},X^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) \right. \nonumber \\&\quad \left. \left. -\,\lambda _{jl}(\{t\},\hat{X}(t)) \right) \triangle \hat{\xi }_{l}^{(2)}(t) \int _{\mathbb {R}_{0}}\hat{r}_{jm}(\{t\},z)N_{m}(\{t\},\mathrm{d}z) \right] . \end{aligned}$$
(66)
Substituting (65) and (66) into (64), we obtain
$$\begin{aligned}&\mathcal {J}\left( u^{(1)},\xi ^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) -\mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \nonumber \\&\quad \leqslant E\left[ \int _{0}^{T} H\left( t,u^{(1)}\right) \left( \mathrm{d}t,\mathrm{d}\xi ^{(1)},\mathrm{d}\hat{\xi }^{(2)}\right) -\int _{0}^{T}\hat{H}(t)\left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right. \nonumber \\&\qquad \left. -\int _{0}^{T}\left( \tilde{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\right) ^{T} \triangledown _{x}\hat{H}(t)\left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right] . \end{aligned}$$
(67)
Since \((x,u^{(1)},\xi ^{(1)})\rightarrow H\left( t,x^{\left( u^{(1)},\xi ^{(1)}\right) },u^{(1)},\hat{u}^{(2)},\hat{p},\hat{q},\hat{r}(\cdot )\right) \left( \mathrm{d}t,\mathrm{d}\xi ^{(1)},\mathrm{d}\hat{\xi }^{(2)}\right) \) is concave, we have
$$\begin{aligned}&H\left( t,u^{(1)}\right) \left( \mathrm{d}t,\mathrm{d}\xi ^{(1)},\mathrm{d}\hat{\xi }^{(2)}\right) -\hat{H}(t)\left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \\&\quad \le \left[ \triangledown _{x}\hat{H}(t)\left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right] ^{T} \tilde{X}^{\left( u^{(1)},\xi ^{(1)}\right) }(t)\\&\qquad +\left[ \triangledown _{u^{(1)}}\hat{H}(t)\left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)}, \mathrm {d}\hat{\xi }^{(2)} \right) \right] ^{T} \left( u^{(1)}(t)-\hat{u}^{(1)}(t)\right) \\&\qquad +\left[ \triangledown _{\xi ^{(1)}}\hat{H}(t)\left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right] ^{T} \left( \mathrm{d}\xi ^{(1)}(t)-\mathrm{d}\hat{\xi }^{(1)}(t)\right) . \end{aligned}$$
Then, (67) leads to
$$\begin{aligned}&\mathcal {J}\left( u^{(1)},\xi ^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) -\mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \nonumber \\&\quad =E\left[ \int _{0}^{T} E\left[ \left. \triangledown _{u^{(1)}}\hat{H}(t)\left( \mathrm{d}t,\mathrm{d}\hat{\xi }^{(1)},\mathrm {d}\hat{\xi }^{(2)}\right) \right| \ \mathcal {G}^{(1)}_{t}\right] ^{T} \left( u^{(1)}(t)-\hat{u}^{(1)}(t)\right) \right. \nonumber \\&\qquad +\int _{0}^{T}E\left[ \left. \hat{U}^{T}(t) \right| \mathcal {G}^{(1)}_{t} \right] \left( \mathrm{d}\xi ^{(1)}(t)-\mathrm{d}\hat{\xi }^{(1)}(t)\right) \nonumber \\&\qquad \left. +\sum _{0<t\le T}\sum _{l=1}^{n}E\left[ \left. \hat{V}_{l}(t)\right| \ \mathcal {G}^{(1)}_{t} \right] \left( \triangle \xi _{l}^{(1)}(t)-\triangle \hat{\xi }_{l}^{(1)}(t)\right) \right] . \end{aligned}$$
(68)
Since the control \(\left( \hat{u}^{(1)},\hat{\xi }^{(1)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}\) satisfies (17), (18) and (20), we conclude that
$$\begin{aligned}&\mathcal {J}\left( u^{(1)},\xi ^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) -\mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \\&\quad \le E\left[ \int _{0}^{T}E\left[ \left. \hat{U}^{T}(t) \right| \mathcal {G}^{(1)}_{t} \right] \mathrm{d}\xi ^{(1)}(t)+\sum _{0<t\le T}\sum _{l=1}^{n}E\left[ \left. \hat{V}_{l}(t)\right| \ \mathcal {G}^{(1)}_{t} \right] \triangle \xi ^{(1)}_{l}(t)\right] =0 \end{aligned}$$
holds for all \(\left( u^{(1)},\xi ^{(1)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}\). Therefore, we obtain (24).
[II] Following a similar argument as in [I], we conclude that
$$\begin{aligned} \mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) =\inf _{\left( u^{(2)},\xi ^{(2)}\right) \in \mathcal {A}^{(2)}_{\mathcal {G}}} \mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};u^{(2)},\xi ^{(2)}\right) \end{aligned}$$
holds, if the regular–singular control \(\left( \hat{u}^{(2)},\hat{\xi }^{(2)}\right) \in \mathcal {A}^{(2)}_{\mathcal {G}}\) satisfies (17), (18) and (20).
[III] If both (I) and (II) hold, we have
$$\begin{aligned} \mathcal {J}\left( u^{(1)},\xi ^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \le \mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \le \mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};u^{(2)},\xi ^{(2)}\right) \end{aligned}$$
for all \(\left( u^{(1)},\xi ^{(1)};u^{(2)},\xi ^{(2)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}\times \mathcal {A}^{(2)}_{\mathcal {G}}\). Then,
$$\begin{aligned} \mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \ge&\sup _{\left( u^{(1)},\xi ^{(1)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}}\mathcal {J}\left( u^{(1)},\xi ^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \\ \ge&\inf _{\left( u^{(2)},\xi ^{(2)}\right) \in \mathcal {A}^{(2)}_{\mathcal {G}}}\sup _{\left( u^{(1)},\xi ^{(1)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}} \mathcal {J}\left( u^{(1)},\xi ^{(1)};u^{(2)},\xi ^{(2)}\right) \end{aligned}$$
and
$$\begin{aligned} \mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \le&\inf _{\left( u^{(2)},\xi ^{(2)}\right) \in \mathcal {A}^{(2)}_{\mathcal {G}}}\mathcal {J}\left( \hat{u}^{(1)},\hat{\xi }^{(1)};u^{(2)},\xi ^{(2)}\right) \\ \le&\sup _{\left( u^{(1)},\xi ^{(1)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}}\inf _{\left( u^{(2)},\xi ^{(2)}\right) \in \mathcal {A}^{(2)}_{\mathcal {G}}}\mathcal {J}\left( u^{(1)},\xi ^{(1)};u^{(2)},\xi ^{(2)}\right) . \end{aligned}$$
Therefore, we conclude that
$$\begin{aligned}&\inf _{\left( u^{(2)},\xi ^{(2)}\right) \in \mathcal {A}^{(2)}_{\mathcal {G}}}\sup _{\left( u^{(1)},\xi ^{(1)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}} \mathcal {J}\left( u^{(1)},\xi ^{(1)};u^{(2)},\xi ^{(2)}\right) \\&\quad \le \sup _{\left( u^{(1)},\xi ^{(1)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}} \inf _{\left( u^{(2)},\xi ^{(2)}\right) \in \mathcal {A}^{(2)}_{\mathcal {G}}}\mathcal {J}\left( u^{(1)},\xi ^{(1)};u^{(2)},\xi ^{(2)}\right) . \end{aligned}$$
This in conjunction with the following inequality:
$$\begin{aligned}&\sup _{\left( u^{(1)},\xi ^{(1)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}}\inf _{\left( u^{(2)},\xi ^{(2)}\right) \in \mathcal {A}^{(2)}_{\mathcal {G}}}\mathcal {J}\left( u^{(1)},\xi ^{(1)};u^{(2)},\xi ^{(2)}\right) \\&\quad \le \inf _{\left( u^{(2)},\xi ^{(2)}\right) \in \mathcal {A}^{(2)}_{\mathcal {G}}}\sup _{\left( u^{(1)},\xi ^{(1)}\right) \in \mathcal {A}^{(1)}_{\mathcal {G}}} \mathcal {J}\left( u^{(1)},\xi ^{(1)};u^{(2)},\xi ^{(2)}\right) , \end{aligned}$$
gives (8). Then, \(\left( \hat{u}^{(1)},\hat{\xi }^{(1)};\hat{u}^{(2)},\hat{\xi }^{(2)}\right) \) is the saddle point of the zero-sum game (8). \(\square \)
