By iterated expectations, it follows that

$$\begin{aligned} \begin{aligned}&E\left[ \frac{I(A=a^{I})}{\pi ^*(a^{I}\mid W)} \int _0^\tau \frac{h^*_{12}(s, a^D, a^I, W)dM^{F*}_{12}(s, a^I, W)}{S^*_1(s \mid a^I, W)}\right] \\&\quad =E\left[ \frac{\pi (a^{I} \mid W)}{\pi ^*(a^{I}\mid W)} \right. \\&\quad \left. \int _0^\tau \frac{h^*_{12}(s, a^D, a^I, W)S_1(s \mid a^I, W)}{S^*_1(s \mid a^I, W)} \left\{ \mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^I, W)-\mathop {}\!\textrm{d}\Lambda _{12}^*(s \mid a^I, W) \right\} \right] , \end{aligned} \end{aligned}$$

(23)

$$\begin{aligned} \begin{aligned}&E\left[ \frac{I(A=a^D)}{\pi ^*(a^D \mid W)} \int _0^\tau \frac{ h^*_{13}(s, a^D, a^I, W) dM^{F*}_{13}(s, a^D, W)}{S^*_1(s \mid a^D, W)} \right] \\&\quad =E\left[ \frac{\pi (a^{D} \mid W)}{\pi ^*(a^{D}\mid W)}\right. \\&\quad \left. \int _0^\tau \frac{h_{13,}^*(s, a^D, a^I, W)S_1(s \mid a^D, W)}{S^*_1(s \mid a^D, W)} \left\{ \mathop {}\!\textrm{d}\Lambda _{13}(s \mid a^D, W)-\mathop {}\!\textrm{d}\Lambda _{13}^*(s \mid a^D, W) \right\} \right] , \end{aligned} \end{aligned}$$

(24)

and

$$\begin{aligned} \begin{aligned}&E\left[ \frac{I(A=a^D)}{\pi ^*(a^D \mid W)} \frac{\eta h^*_{23}(T_1, a^D, a^I, W)}{S^*_1(T_1 \mid a^D, W)} \int _{T_1}^\tau \frac{ dM^{F*}_{23}(s, a^D, T_1, W)}{S^*_2(s \mid T_1, a^D, W)} \right] \\&\quad =E\left[ \frac{\pi (a^D \mid W)}{\pi ^*(a^{D}\mid W)} E \left\{ \frac{h^*_{23}(T_1, a^D, a^I, W)}{S^*_1(T_1 \mid a^D, W)} \int _{T_1}^\tau \frac{S_2(s \mid T_1, a^D, W)}{S^*_2(s \mid T_1, a^D, W)} \left\{ \mathop {}\!\textrm{d}\Lambda _{23}(s \mid T_1,a^D, W)\right. \right. \right. \\&\quad \left. \left. \left. -\mathop {}\!\textrm{d}\Lambda _{23}^*(s \mid T_1, a^D, W) \right\} \Big \vert a^D, W \right\} \right] \\&\quad =E\left[ \frac{\pi (a^D \mid W)}{\pi ^*(a^{D}\mid W)} \int _0^\tau \frac{h^*_{23}(s, a^D, a^I, W)}{S^*_1(s \mid a^D, W)} \left\{ 1-\frac{S_2(\tau \mid s, a^D, W)}{S_2^*(\tau \mid s, a^D, W)} \right\} \right. \\&\quad \left. S_1(s \mid a^D, W) \mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^D, W) \right] \end{aligned} \end{aligned}$$

(25)

Suppose

\(\pi \),

\(\Lambda _{13}\) and

\(\Lambda _{23}\) are correctly specified, but

\(\Lambda _{12}\) is not. Then the terms (

24) and (

25) are zero, and we have

$$\begin{aligned}&E \left[ {\tilde{\psi }}(Q^*)(Z, a^D, a^I, W)\right] \\&=E\Bigg [ e^{-\Lambda _{12}^*( \tau \mid a^I, W)- \Lambda _{13}(\tau \mid a^D, W)}\int _0^\tau e^{-\Lambda _{12}(s \mid a^I, W)+ \Lambda _{12}^*(s \mid a^I, W)} \left\{ \mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^I, W)\right. \\&\quad \left. -\mathop {}\!\textrm{d}\Lambda _{12}^*(s \mid a^I, W) \right\} \Bigg ]\\&-E\Bigg [\int _0^\tau S_2(\tau \mid s, a^D, W)\Omega _s(a^D, a^I, W) \left\{ \mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^I, W)-\mathop {}\!\textrm{d}\Lambda _{12}^*(s \mid a^I, W) \right\} \Bigg ] \\&+E\Bigg [\int _0^\tau \left\{ \int _0^s e^{-\Lambda _{12}(s \mid a^I, W)+ \Lambda _{12}^*(s \mid a^I, W)}\left\{ \mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^I, W)-\mathop {}\!\textrm{d}\Lambda _{12}^*(s \mid a^I, W) \right\} \right\} \\&\quad S_2(\tau \mid s a^D, W)\\&\quad \times e^{-\Lambda _{12}^*(s \mid a^I, W)- \Lambda _{13}(s \mid a^D, W)} \mathop {}\!\textrm{d}\Lambda _{12}^*(s \mid a^I, W) \Bigg ] \\&\quad + E\Bigg [1- e^{-\Lambda _{12}^*(\tau \mid a^I, W)- \Lambda _{13}(\tau \mid a^D, W)} \\&\quad - \int _0^\tau S_2(\tau \mid s, a^D, W)e^{-\Lambda _{12}^*(s \mid a^I, W)- \Lambda _{13}(s \mid a^D, W)} \mathop {}\!\textrm{d}\Lambda _{12}^*(s \mid a^I, W) \Bigg ]\\&\quad - \psi (Q^*; \tau , a^D, a^I) \\&\quad =E\Bigg [ e^{-\Lambda _{12}^*(\tau \mid a^I, W)- \Lambda _{13}(\tau \mid a^D, W)}\left\{ 1-e^{-\Lambda _{12}(\tau \mid a^I, W)+ \Lambda _{12}^*(\tau \mid a^I, W)} \right\} \Bigg ]\\&\quad -E\Bigg [\int _0^\tau S_2(\tau \mid s, a^D, W)\Omega _s(a^D, a^I, W) \left\{ \mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^I, W)-\mathop {}\!\textrm{d}\Lambda _{12}^*(s \mid a^I, W) \right\} \Bigg ] \\&\quad +E\Bigg [\int _0^\tau \left\{ 1-e^{-\Lambda _{12}(s \mid a^I, W)+ \Lambda _{12}^*(s \mid a^I, W)} \right\} \\&\quad S_2(\tau \mid s, a^D, W) e^{-\Lambda _{12}^*(s \mid a^I, W)- \Lambda _{13}(s \mid a^D, W)} \mathop {}\!\textrm{d}\Lambda _{12}^*(s \mid a^I, W) \Bigg ] \\&\quad + E\Bigg [1- e^{-\Lambda _{12}^*(\tau \mid a^I, W)- \Lambda _{13}(\tau \mid a^D, W)}\\&\quad - \int _0^\tau S_2(\tau \mid s, a^D, W)e^{-\Lambda _{12}^*(s \mid a^I, W)- \Lambda _{13}(s \mid a^D, W)} \mathop {}\!\textrm{d}\Lambda _{12}^*(s \mid a^I, W) \Bigg ]\\&\quad - \psi (Q^*; \tau , a^D, a^I) \\&\quad = E\Bigg [1- \Omega _t(a^D, a^I, W) + \int _0^\tau S_2(\tau \mid s, a^D, W)\Omega _s(a^D, a^I, W)\mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^I, W)\Bigg ]\\&\quad - \psi (Q^*; \tau , a^D, a^I) \\&\quad = \psi (Q; \tau , a^D, a^I)- \psi (Q^*; \tau , a^D, a^I) \end{aligned}$$

Similarly, suppose

\(\pi \),

\(\Lambda _{12}\) and

\(\Lambda _{23}\) are correctly specified, but

\(\Lambda _{13}\) is not. Then (

23) and (

25) are 0, and

$$\begin{aligned}&E \left[ {\tilde{\psi }}(Q^*)(Z, a^D, a^I, W)\right] \\&\quad =E\Bigg [ e^{-\Lambda _{12}(\tau \mid a^I, W)- \Lambda _{13}^*(\tau \mid a^D, W)}\left\{ 1-e^{-\Lambda _{13}(\tau \mid a^I, W)+ \Lambda _{13}^*(\tau \mid a^I, W)} \right\} \Bigg ]\\&\quad +E\Bigg [\int _0^\tau \left\{ 1-e^{-\Lambda _{13}(s \mid a^I, W)+ \Lambda _{13}^*(s \mid a^I, W)} \right\} \\&\quad S_2(\tau \mid s, a^D, W) e^{-\Lambda _{12}(s \mid a^I, W)- \Lambda _{13}^*(s \mid a^D, W)} \mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^I, W) \Bigg ] \\&\quad + E\Bigg [1- e^{-\Lambda _{12}(\tau \mid a^I, W)- \Lambda _{13}^*(\tau \mid a^D, W)} \\&\quad - \int _0^\tau S_2(\tau \mid s, a^D, W)e^{-\Lambda _{12}(s \mid a^I, W)- \Lambda _{13}^*(s \mid a^D, W)} \mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^I, W) \Bigg ]\\&\quad - \psi (Q^*; \tau , a^D, a^I) \\&\quad = E\Bigg [1- \Omega _t(a^D, a^I, W) + \int _0^\tau S_2(\tau \mid s, a^D, W)\Omega _s(a^D, a^I, W)\mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^I, W)\Bigg ]- \psi (Q^*; \tau , a^D, a^I) \\&\quad =\psi (Q; \tau , a^D, a^I)-\psi (Q^*; \tau , a^D, a^I) \end{aligned}$$

Finally, suppose

\(\pi \),

\(\Lambda _{12}\) and

\(\Lambda _{13}\) are correctly specified, but

\(\Lambda _{23}\) is not. Then (

23) and (

24) are 0, and

$$\begin{aligned}&E \left[ {\tilde{\psi }}(Q^*)(Z, a^D, a^I, W)\right] \\&\quad =E \left[ \int _0^\tau S_2^*(\tau \mid s, a^D, W)\left\{ 1- \frac{S_2(\tau \mid s, a^D, W)}{S_2^*(\tau \mid s, a^D, W)} \right\} \Omega _s(a^D, a^I, W) \mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^I, W) \right] \\&\quad + E\Bigg [1- \Omega _t(a^D, a^I, W) - \int _0^\tau S_2^*(\tau \mid s, a^D, W)\Omega _s(a^D, a^I, W) \mathop {}\!\textrm{d}\Lambda _{12}(s \mid a^I, W) \Bigg ]\\&\quad - \psi (Q^*; \tau , a^D, a^I) \\&\quad =\psi (Q; \tau , a^D, a^I)-\psi (Q^*; \tau , a^D, a^I) \end{aligned}$$