Appendix 1: The Proof of Propositions 1 and 2
To solve problem (13), we conjecture the corresponding value function has the structure (16). Inserting this structure into the HJB equation (15), by the first-order conditions, the functions \(h_m^{*},\, h^{*},\, h_1 ^{*}\) and \(h_2 ^{*}\) which realize the minimum in (15) are given by
$$\begin{aligned}&h^{*}_m=-u \sigma _m \pi _f, \quad h^{*}=-(\pi _1+\pi _2)u \sigma , \\&h^{*}_1 =-b \pi _1 u-N x b u/(1-\gamma ), \quad h^* _2=Nx b u/(1-\gamma )-b \pi _2 u. \end{aligned}$$
Substituting the above expressions for \(h_m^{*},\, h^{*}, \,h_1 ^{*}\) and \(h_2 ^{*}\) into Eq. (15), by first-order conditions, we can achieve the expression for the optimal strategy \(\pi ^*\) shown in (19)–(21). Plugging \(\pi ^*\) into (15) implies
$$\begin{aligned}&\Bigg (\frac{1}{2(1-\gamma )}N_t\!+\! \frac{1\!-\!\gamma -u}{(\gamma +u)(1-\gamma )^2}b^2 N^2-\frac{\lambda _1\!+\!\lambda _2}{(1-\gamma )(\gamma \!+\!u)}N\\&\quad \!+\!\frac{(b^2\!+\!\sigma ^2)(\lambda _1^2+\lambda _2 ^2)\!+\!2 \sigma ^2\lambda _1 \lambda _2}{2(\gamma +u)b^2(2\sigma ^2 \!+\!b^2)}\Bigg )x^2\!+\!\left( \frac{1}{1-\gamma }G_t\!+\!\frac{b^2N}{1\!-\!\gamma } \!+\!\frac{\mu _m^2}{2(\gamma +u)\sigma _m^2}\!+\!r\right) \!=\!0 \end{aligned}$$
The above equation is ensured if the following equations are satisfied:
$$\begin{aligned}&\frac{1}{2(1-\gamma )}N_t+ \frac{1-\gamma -u}{(\gamma +u)(1-\gamma )^2}b^2 N^2-\frac{\lambda _1+\lambda _2}{(1-\gamma )(\gamma +u)}N \nonumber \\&\quad +\frac{(b^2+\sigma ^2)(\lambda _1^2+\lambda _2 ^2)+2 \sigma ^2\lambda _1 \lambda _2}{2(\gamma +u)b^2(2\sigma ^2 +b^2)}=0, \end{aligned}$$
(67)
$$\begin{aligned}&\frac{1}{1-\gamma }G_t+\frac{b^2N}{1-\gamma }+\frac{\mu _m^2}{2(\gamma +u) \sigma _m^2}+r=0. \end{aligned}$$
(68)
One can easily verify that expressions (17) and (18) are the solutions to the above two equations respectively. Moreover, inserting the specific expressions for \(k_1,\, k_2\) and \(k_3\) to (18), \(\gamma \) would be eliminated in expressions (20) and (21), which implies Proposition 2.
Appendix 2: The proof of Proposition 3
To solve the HJB equation (30), we conjecture the solution has the structure (31). Inserting (31) to (30), by the first-order condition, we acquire
$$\begin{aligned} \begin{aligned}&h^{*}=-(\pi _1+\pi _2)u \sigma ,\\&h_a^{*}=-\frac{\sigma _a u}{1-\gamma }(A_1 a+A_2+B_3 x),\\&h^{*}_1 =-b \pi _1 u-(B_1 x+B_3 a) b u/(1-\gamma ), \\&h_2^{*}=(B_1 x+B_3a) bu/(1-\gamma )-b \pi _2 u. \end{aligned} \end{aligned}$$
(69)
Parallel to the proof of Proposition 1, we substitute the above expressions in the HJB equation (30). By the first-order condition, calculation yields an optimal strategy \(\pi ^*=(\pi ^*_1, \pi _2 ^*)\) shown in (32)–(33). Plugging the optimal strategy into (30), the right side of Eq. (30) becomes an affine function of \(a^2,\, a,\, x^2\) and \(ax\). The equation has to be satisfied for all values of \(a\) and \(x\), which leads to the following system of ODEs:
$$\begin{aligned} {A_1}_t+&\frac{4(1-\gamma )}{\varGamma (2\sigma ^2+b^2)}-\varGamma (1-\gamma ) \Bigg [\frac{4 \sigma ^2}{\varGamma ^2(2\sigma ^2+b^2)^2}+b^2 \left( \frac{1}{\varGamma (2\sigma ^2+b^2)}-\frac{\varGamma _1 B_3}{\varGamma }\right) ^2 \nonumber \\&+b^2\left( \frac{1}{\varGamma (2\sigma ^2+b^2)}+\frac{\varGamma _1 B_3}{\varGamma }\right) ^2\Bigg ] +(1-\gamma )\frac{4 b^2 \varGamma _1 ^2 B_3 ^2}{\varGamma }\nonumber \\&+2 b^2 B_3 ^2 \varGamma _1+\sigma _a ^2 A_1 ^2 \varGamma _1-2\kappa A_1=0, \end{aligned}$$
(70)
$$\begin{aligned} {A_2}_t+&\sigma _a ^2 \varGamma _1 A_1 A_2+ \kappa (A_1 \theta -A_2)=0, \end{aligned}$$
(71)
$$\begin{aligned} {A_0}_t+&(1-\gamma ) \tilde{r} + b^2 B_1+\frac{1}{2} \sigma _a ^2 \varGamma _1 A_2 ^2+\frac{1}{2} \sigma _a ^2 A_1+\kappa A_2 \theta =0,\end{aligned}$$
(72)
$$\begin{aligned} {B_1}_t+&(1-\gamma )\Bigg [\frac{(\lambda _2-\lambda _1)^2}{\varGamma (2\sigma ^2+b^2)} +\frac{(\lambda _1+\lambda _2)^2}{\varGamma b^2}-\frac{2 \varGamma _1 B_1 (\lambda _1 +\lambda _2)}{\varGamma }- \frac{\sigma ^2(\lambda _2-\lambda _1)^2}{\varGamma (2\sigma ^2+b^2)^2} \nonumber \\&-\varGamma b^2 \left( \left( \frac{\lambda _2-\lambda _1}{2\varGamma (2\sigma ^2+b^2)}- \frac{\lambda _1+\lambda _2}{2 \varGamma b^2}+\frac{\varGamma _1 B_1}{\varGamma } \right) ^2\right. \nonumber \\&\left. +\left( \frac{\lambda _2-\lambda _1}{2\varGamma (2\sigma ^2+b^2)} +\frac{\lambda _1+\lambda _2}{2\varGamma b^2}-\frac{\varGamma _1 B_1}{\varGamma }\right) ^2 \right) \nonumber \\&+2 \varGamma _1 B_1 \frac{2 \varGamma _1 B_1-(\lambda _1+\lambda _2)}{\varGamma b^2}\Bigg ]+2b^2 \varGamma _1 B_1 ^2+ \sigma _a ^2 \varGamma _1 B_3 ^2-2 (\lambda _1+\lambda _2) B_1=0,\end{aligned}$$
(73)
$$\begin{aligned} {B_3}_t+&(1-\gamma ) \Bigg \{\frac{\lambda _2-\lambda _1}{\varGamma (2\sigma ^2+b^2)}-\frac{\varGamma _1 B_3(\lambda _1+\lambda _2)}{\varGamma }+\frac{\lambda _2-\lambda _1}{\varGamma (2\sigma ^2+b^2)} -\frac{2\sigma ^2 (\lambda _2-\lambda _1)}{\varGamma (2\sigma ^2+b^2)^2} \nonumber \\&\qquad \quad \qquad -b^2 \left[ \frac{2(\lambda _2-\lambda _1)}{\varGamma (2\sigma ^2+b^2)^2}+ 4 \varGamma _1 B_3 \left( \frac{\lambda _1+\lambda _2}{2\varGamma b^2}- \frac{\varGamma _1 B_1}{\varGamma }\right) \right] \nonumber \\&\qquad \quad \qquad +b^2 \varGamma _1 \left( \frac{4 \varGamma _1 B_1 B_3}{\varGamma }-\frac{(\lambda _1+\lambda _2) B_3}{\varGamma b^2}\right) \Bigg \} \nonumber \\&\qquad \qquad \quad + 2\varGamma _1 b^2 B_1 B_3 + \sigma _a ^2 \varGamma _1 A_1 B_3-(\lambda _1+\lambda _2)B_3-\kappa B_3=0, \end{aligned}$$
(74)
with terminal conditions \(A_1(T)=A_2(T)=A_0(T)=B_1(T)=B_3(T)=0\).
Appendix 3: The Proof of Proposition 4
To simplify the calculation, we define a new Brownian motion \(\hat{Z}_3\) under \(P\) as
$$\begin{aligned} d\hat{Z}_3=\frac{\sigma }{\sqrt{\sigma ^2+b^2}} d\hat{Z}(t)+ \frac{b}{\sqrt{\sigma ^2+b^2}} d\hat{Z_1}(t). \end{aligned}$$
(75)
Using standard Gaussian linear regression, the filtered model can be rewritten as
$$\begin{aligned} \frac{dP_1 (t)}{P_1 (t)}&=(\tilde{r}+\hat{a}(t)- \lambda _1 X(t))dt+\hat{\sigma } d \hat{Z}_3 (t), \end{aligned}$$
(76)
$$\begin{aligned} \frac{dP_2 (t)}{P_2 (t)}&=(\tilde{r}+\hat{a}(t)+ \lambda _2 X(t))dt +\rho \hat{\sigma } d\hat{Z_3}(t)+ \sqrt{1-\rho ^2} \hat{\sigma } d\hat{Z_4} (t), \end{aligned}$$
(77)
$$\begin{aligned} d\hat{a}(t)&=\kappa (\theta -\hat{a}(t))dt+\eta \left( \frac{1}{\hat{\sigma }}d\hat{Z_3}(t)+ \frac{1-\rho }{\rho _1 \hat{\sigma }} d\hat{Z_4}(t) \right) , \end{aligned}$$
(78)
where \(\hat{Z_4}\) is a Brownian motion under \(P\) independent of \(\hat{Z}_3\) and \(\hat{\sigma }= \sqrt{\sigma ^2+b^2},\, \eta \) is given by (38) and \((\rho , \rho _1)=\left( \frac{\sigma ^2}{b^2+\sigma ^2}, \sqrt{\frac{b^2}{b^2+\sigma ^2}}\right) \). Based on Girsanov’s theorem, we define a new Brownian motions under \(Q\) as
$$\begin{aligned} d\hat{Z}_{3}^Q(t)= d\hat{Z}_{3}(t)-h_1 (t)dt, \quad d \hat{Z}_4 ^Q (t)= d\hat{Z}_4 (t)-h_2(t)dt. \end{aligned}$$
Applying the dynamic programming principle, the robust HJB equation for the filtered model can be derived as:
$$\begin{aligned}&\sup \limits _{\pi \in \varPi } \inf \limits _{\varphi \in \mathcal {H}} \Bigg \{J_t \!+\! w J_w \left[ \tilde{r}\!+\! (\lambda _2 \pi _2\!-\!\lambda _1 \pi _1) x \!+\!\hat{a} (\pi _1\!+\!\pi _2)\!+\! h_1 \hat{\sigma } (\pi _1\!+\!\rho \pi _2)\!+\!h_2 \pi _2 \rho _1 \hat{\sigma } \right] \nonumber \\&\quad \!+\, J_x \left( -(\lambda _1\!+\!\lambda _2)x(t)\!+ \!h_1 \hat{\sigma }(1\!-\!\rho )\!-\!h_2 \rho _1 \hat{\sigma } \right) \!+\!J_{\hat{a}} \left( \kappa (\theta -\hat{a})\!+\!h_1\frac{\eta }{\hat{\sigma }}\!+\! h_2 \frac{\eta (1-\rho )}{\rho _1 \hat{\sigma }}\right) \nonumber \\&\quad +\,\frac{1}{2} w^2 J_{ww} \hat{\sigma }^2(\pi _1^2+\pi _2^2+2 \rho \pi _1 \pi _2)+ w J_{wx} \hat{\sigma }(1-\rho )(\pi _1-\pi _2) + \eta (\pi _1+\pi _2)J_{w \hat{a}}\nonumber \\&\quad \!+\, J_{xx}\hat{\sigma }^2(1-\rho )\!+\! \frac{\eta ^2(1-\rho )}{\rho _1^2 \hat{\sigma }^2}J_{\hat{a} \hat{a}} \!+\!\frac{(1\!-\!\gamma )J}{2u}\left( h_1 ^2+h_2 ^2\right) \Bigg \}\!=\!0. \end{aligned}$$
(79)
Substituting the conjecture of the value function (43) into (79), the first-order condition yields
$$\begin{aligned} h^* _1=&-\frac{\eta u}{(1-\gamma ) \hat{\sigma }}(\hat{A}_1 \hat{a} +\hat{A}_2+\hat{B}_3 x)- \hat{\sigma }(\pi _1+\rho \pi _2)u \nonumber \\&-\frac{\hat{\sigma }(1-\rho ) u}{1-\gamma } (\hat{B}_1 x+\hat{B}_2+\hat{B}_3 \hat{a}), \end{aligned}$$
(80)
$$\begin{aligned} h^* _2=&-\frac{\eta u (1-\rho )}{(1-\gamma ) \hat{\sigma } \rho _1}(\hat{A}_1 \hat{a} +\hat{A}_2+\hat{B}_3 x)- \hat{\sigma } \rho _1 \pi _2 u \nonumber \\&+\frac{\rho _1 \hat{\sigma } u}{1-\gamma } (\hat{B}_1 x+\hat{B}_2+\hat{B}_3 \hat{a}), \end{aligned}$$
(81)
and an optimal strategy shown in (44)–(45). Plugging these into (79), it turns out that our conjecture (43) indeed solves the HJB equation if the functions satisfy the following ODE system:
$$\begin{aligned} \hat{{A}_1}_t+\,&\frac{2(1+\eta \varGamma _1 \hat{A}_1)^2}{\hat{\sigma }^2(1+\rho )\varGamma }(1-\gamma )\!-\! 2 \kappa \hat{A}_1\!+\! 2 \hat{\sigma } ^2 (1\!-\!\rho ) (1\!-\!\gamma )\frac{\varGamma _1 ^2}{\varGamma }\hat{B}_3 ^2\!+\!2 \hat{\sigma }^2 \varGamma _1 (1\!-\!\rho ) \hat{B}_3 ^2 \nonumber \\ \!+\,&\,\frac{2(1-\rho )\eta ^2 \varGamma _1}{\rho _1 ^2 \hat{\sigma }^2} \hat{A}_1 ^2=0, \end{aligned}$$
(82)
$$\begin{aligned} \hat{ A_2}_t+&\kappa \theta \hat{A}_1 -\kappa \hat{A}_2+2(1-\rho )(1-\gamma )\hat{\sigma }^2\frac{\varGamma _1 ^2}{\varGamma } \hat{B}_2 \hat{B}_3 + 2 \eta \varGamma _1 (1-\gamma ) \frac{\hat{A}_2+\eta \varGamma _1 \hat{A}_1 \hat{A}_2}{\hat{\sigma }^2 (1+\rho ) \varGamma } \nonumber \\ +&\,2 \hat{\sigma }^2 (1-\rho ) \varGamma _1 \hat{B}_2 \hat{B}_3 + \frac{2(1-\rho ) \eta ^2 \varGamma _1}{\rho _1^2 \hat{\sigma }^2} \hat{A}_1 \hat{A}_2=0,\end{aligned}$$
(83)
$$\begin{aligned} \hat{ A_0}_t +\,&\tilde{r}(1-\gamma )+ \kappa \theta \hat{A}_2 \!+\!\hat{\sigma }^2 (1-\rho )\frac{\varGamma _1^2}{\varGamma } \hat{B}_2 ^2\!+\!(1-\gamma )\frac{(\eta \varGamma _1 \hat{A}_2)^2}{\hat{\sigma }^2 (1\!+\!\rho ) \varGamma } \nonumber \\ +&\hat{B}_2 ^2 \hat{\sigma } ^2 \varGamma _1 (1-\rho )\!+ \hat{\sigma } ^2 (1-\rho ) \hat{B}_1+\frac{(1-\rho )\eta ^2 \varGamma _1 }{\rho _1 ^2 \hat{\sigma }^2}\hat{A}_2 ^2+\frac{\eta ^2 (1-\rho )}{\rho _1 ^2 \hat{\sigma } ^2} \hat{A}_1=0, \end{aligned}$$
(84)
$$\begin{aligned} \hat{B_1}_t +&\frac{(\lambda _2 -\lambda _1)^2}{2 \hat{\sigma }^2(1+\rho )\varGamma }(1-\gamma )+\frac{(\lambda _1+\lambda _2)^2}{2\hat{\sigma } ^2 (1-\rho )\varGamma } (1-\gamma )+\frac{2 \hat{\sigma } ^2 (1-\rho )\varGamma _1 ^2 \hat{B}_1 ^2}{\varGamma }(1-\gamma ) \nonumber \\ \!-\,&\frac{2 \varGamma _1 \hat{B}_1 (\lambda _1\!+\!\lambda _2)}{\varGamma }(1-\gamma )+\frac{2\eta ^2 \varGamma _1 ^2 \hat{B}_3 ^2}{\hat{\sigma }^2 (1+\rho ) \varGamma } (1-\gamma )+2 \hat{\sigma }^2 \varGamma _1 (1-\rho )\hat{B}_1 ^2 \nonumber \\ +\,&2\varGamma _1 \frac{(1-\rho )\eta ^2}{\rho _1 ^2 \hat{\sigma } ^2}\hat{B}_3^2=0, \end{aligned}$$
(85)
$$\begin{aligned} \hat{B_2}_t-&(\lambda _1+\lambda _2)+\kappa \theta \hat{B}_3+ 2 \hat{\sigma } ^2 (1-\gamma )(1-\rho )\frac{\varGamma _1^2}{\varGamma } \hat{B}_1 \hat{B}_2- (\lambda _1+\lambda _2)(1-\gamma )\frac{\varGamma _1}{\varGamma }\hat{B}_2 \nonumber \\ +\,&(1-\gamma )\frac{2\eta ^2 \varGamma _1 ^2 \hat{A}_2 \hat{B}_3}{\hat{\sigma } ^2(1+\rho )\varGamma }+2 \hat{\sigma } ^2 (1-\rho ) \varGamma _1 \hat{B}_1 \hat{B}_2+ \frac{2(1-\rho )\eta ^2 \varGamma _1}{\rho _1 ^2 \hat{\sigma } ^2}\hat{A}_2 \hat{B}_3 \nonumber \\ +\,&(1-\gamma )\frac{\eta \varGamma _1 (\lambda _2-\lambda _1)}{\hat{\sigma } ^2 (1+\rho ) \varGamma }\hat{A}_2 =0,\end{aligned}$$
(86)
$$\begin{aligned} \hat{B_3}_t-&\kappa \hat{B}_3+2(1-\gamma )(1-\rho )\hat{\sigma }^2 \frac{\varGamma _1 ^2}{\varGamma } \hat{B}_1 \hat{B}_3 -(1-\gamma )(\lambda _1+\lambda _2)\frac{\varGamma _1}{\varGamma } \hat{B}_3 \nonumber \\ +&\,(1-\gamma )(1+\eta \varGamma _1 \hat{A}_1)\frac{\lambda _2-\lambda _1}{\hat{\sigma }^2 (1+\rho )\varGamma } +(1-\gamma )\frac{2\eta ^2\varGamma _1 ^2 \hat{A}_1 \hat{B}_3+2\eta \varGamma _1 \hat{B}_3}{\hat{\sigma }^2 (1+\rho ) \varGamma }\nonumber \\ +&2\hat{\sigma }^2(1-\rho )\varGamma _1 \hat{B}_1 \hat{B}_3+ \frac{2(1-\rho )\eta ^2 \varGamma _1}{\rho _1 ^2 \hat{\sigma }^2 } \hat{A}_1 \hat{B}_3=0, \end{aligned}$$
(87)
with terminal conditions \(\hat{A}_1(T)=\hat{A}_2(T)=\hat{A}_0(T)=\hat{B}_1(T)=\hat{B}_2(T)=\hat{B}_3(T)\)
\(=0\).
Appendix 4
This section details the calculation of the four value functions under the strategies \(\pi ^{IM},\, \pi ^{IA},\, \pi ^{UIM}, \,\pi ^{UIA}\) in Section 5.3. We conjecture that the structures of value functions \(J^{IM}\) and \(J^{IA}\) are given by (57) and (60) for the observed case and (63) and (66) for unobserved case. Set \(\pi ^{IM}\) as an example, we hope to solve for the value function \(J^{IM}\) under the given strategy for ignoring mispricing \(\pi ^{IM}\). Inserting (55), (69) and (57) into HJB equation (30), we repeat the variables separation method similarly to “Appendix 2”. The following system of ODEs can be derived for (57):
$$\begin{aligned} \left( A_1 ^{IM}\right) _t&\!+\!\frac{4(1\!-!\gamma )}{\varGamma (2\sigma ^2\!+\!b^2)}\!-\!\varGamma (1\!-\!\gamma ) \Bigg [\frac{4 \sigma ^2}{\varGamma ^2(2\sigma ^2\!+\!b^2)^2}\!+\!b^2 \left( \frac{1}{\varGamma (2\sigma ^2\!+\!b^2)}-\frac{\varGamma _1 B_3}{\varGamma }\right) ^2 \nonumber \\&+b^2\left( \frac{1}{\varGamma (2\sigma ^2+b^2)}+\frac{\varGamma _1 B_3}{\varGamma }\right) ^2\Bigg ] +(1-\gamma )\frac{2 b^2 \varGamma _1 ^2 B_3 \left( B_3 ^{IM}\right) }{\varGamma } \nonumber \\&+2 b^2 \left( B_3 ^{IM}\right) ^2 \varGamma _1+\sigma _a ^2 \left( A_1 ^{IM}\right) ^2 \varGamma _1-2\kappa \left( A_1 ^{IM}\right) =0, \end{aligned}$$
(88)
$$\begin{aligned} \left( A_2^{IM} \right) _t&+ \sigma _a ^2 \varGamma _1 \left( A_1^{IM} \right) \left( A_2^{IM} \right) + \kappa \left[ \left( A_1^{IM} \right) \theta -\left( A_2^{IM} \right) \right] =0, \end{aligned}$$
(89)
$$\begin{aligned} \left( A_0^{IM} \right) _t&\!+\!(1\!-\!\gamma ) \tilde{r}\!+\!b^2 \left( B_1^{IM}\right) \!+\!\frac{1}{2} \sigma _a ^2 \varGamma _1 \left( A_2^{IM}\right) ^2\!+\!\frac{1}{2} \sigma _a ^2 \left( A_1^{IM}\right) +\kappa \theta \left( A_2^{IM}\right) =0,\end{aligned}$$
(90)
$$\begin{aligned} \left( B_1 ^{IM}\right) _t&+ 2b^2 \varGamma _1 \left( B_1 ^{IM}\right) ^2+ \sigma _a ^2 \varGamma _1 \left( B_3 ^{IM}\right) ^2-2 (\lambda _1+\lambda _2) \left( B_1 ^{IM}\right) =0,\end{aligned}$$
(91)
$$\begin{aligned} \left( B_3 ^{IM}\right) _t&+(1-\gamma ) \Bigg \{\frac{\lambda _2-\lambda _1}{\varGamma (2\sigma ^2+b^2)}-\frac{\varGamma _1 B_3 (\lambda _1\!+\!\lambda _2)}{\varGamma }\!+\! \frac{2 b^2 \varGamma _1 \left( B_1 ^{IM}\right) B_3}{\varGamma }\Bigg \} \nonumber \\&\!+\! 2\varGamma _1 b^2 \left( B_1 ^{IM}\right) \left( B_3 ^{IM}\right) \!+\! \sigma _a ^2 \varGamma _1 \left( A_1 ^{IM}\right) \left( B_3 ^{IM}\right) \nonumber \\&\!-\!(\lambda _1\!+\!\lambda _2) \left( B_3 ^{IM}\right) \!-\!\kappa \left( B_3 ^{IM}\right) \!=\!0, \end{aligned}$$
(92)
with terminal conditions \(A_1 ^{IM}(T)=A_2 ^{IM} (T)=A_0 ^{IM} (T)=B_1 ^{IM} (T)=B_3 ^{IM}(T)=0\) and \(A_1,\, A_2,\, A_3,\, B_1\) and \(B_3\) satisfy (70)–(74). Similarly, given the strategy \(\pi ^{IA}\), we obtain the following system of ODEs for the functions used in \(J^{IA}\) in (60).
$$\begin{aligned} \left( A_1^{IA} \right) _t&\!+\!\frac{4(1-\gamma )}{\gamma (2\sigma ^2\!+\!b^2)}\!-\!\varGamma (1-\gamma ) \Bigg [\frac{4 \sigma ^2}{\gamma ^2(2\sigma ^2+b^2)^2}+b^2 \left( \frac{1}{\gamma (2\sigma ^2+b^2)}-\frac{\gamma _1 B_3}{\gamma }\right) ^2 \nonumber \\&+b^2\left( \frac{1}{\gamma (2\sigma ^2+b^2)}+\frac{B_3}{\gamma }\right) ^2\Bigg ]+(1-\gamma )\frac{4 b^2 \varGamma _1 ^2 B_3 \left( B_3^{IA}\right) }{\varGamma } \nonumber \\&+2 b^2 \left( B_3^{IA}\right) ^2 \varGamma _1+\sigma _a ^2 \left( A_1^{IA} \right) ^2 \varGamma _1-2\kappa \left( A_1^{IA} \right) =0, \end{aligned}$$
(93)
$$\begin{aligned} \left( A_2^{IA} \right) _t&+ \sigma _a ^2 \varGamma _1 \left( A_1^{IA} \right) \left( A_2^{IA} \right) + \kappa \left[ \left( A_1^{IA} \right) \theta -\left( A_2^{IA} \right) \right] =0, \end{aligned}$$
(94)
$$\begin{aligned} \left( A_0^{IA} \right) _t&\!+\!(1-\gamma ) \tilde{r} \!+\! b^2 \left( B_1^{IA}\right) \!+\!\frac{1}{2} \sigma _a ^2 \varGamma _1 \left( A_2^{IA}\right) ^2\!+\!\frac{1}{2} \sigma _a ^2 \left( A_1^{IA}\right) +\kappa \theta \left( A_2^{IA}\right) =0,\end{aligned}$$
(95)
$$\begin{aligned} \left( B_1^{IA}\right) _t&+(1-\gamma )\Bigg \{\frac{(\lambda _2-\lambda _1)^2}{\gamma (2\sigma ^2+b^2)}+\frac{(\lambda _1+\lambda _2)^2}{\gamma b^2}-\frac{2 B_1 (\lambda _1+\lambda _2)}{\gamma }- \frac{\sigma ^2(\lambda _2-\lambda _1)^2}{\gamma (2\sigma ^2+b^2)^2} \nonumber \\&-\varGamma b^2 \left[ \left( \frac{\lambda _2-\lambda _1}{2\gamma (2\sigma ^2+b^2)}- \frac{\lambda _1+\lambda _2}{2 \gamma b^2}+\frac{\varGamma _1 B_1}{\gamma } \right) ^2 \right. \nonumber \\&\left. +\left( \frac{\lambda _2-\lambda _1}{2\gamma (2\sigma ^2+b^2)} +\frac{\lambda _1+\lambda _2}{2\gamma b^2}-\frac{\varGamma _1 B_1}{\gamma }\right) ^2 \right] \nonumber \\&+2 \varGamma _1 \left( B_1^{IA}\right) \frac{2 \varGamma _1 B_1-(\lambda _1+\lambda _2)}{\gamma b^2}\Bigg \}+2b^2 \varGamma _1 \left( B_1^{IA}\right) ^2\nonumber \\&+\sigma _a ^2 \varGamma _1 \left( B_3^{IA}\right) ^2-2 (\lambda _1+\lambda _2) \left( B_1^{IA}\right) =0,\end{aligned}$$
(96)
$$\begin{aligned} \left( B_3^{IA}\right) _t&+(1-\gamma ) \left\{ \frac{\lambda _2-\lambda _1}{\gamma (2\sigma ^2+b^2)}-\frac{B_3 (\lambda _1+\lambda _2)}{\gamma }+\frac{\lambda _2-\lambda _1}{\gamma (2\sigma ^2+b^2)} -\frac{2\sigma ^2 (\lambda _2-\lambda _1)}{\gamma (2\sigma ^2+b^2)^2} \right. \nonumber \\&-b^2 \varGamma \left[ \frac{2(\lambda _2-\lambda _1)}{\gamma (2\sigma ^2+b^2)^2}+ \frac{4 B_3}{\gamma } \left( \frac{\lambda _1+\lambda _2}{2\gamma b^2}- \frac{\varGamma _1 B_1}{\gamma }\right) \right] \nonumber \\&\left. +\,b^2 \varGamma _1 \left( \frac{2 \varGamma _1 B_1 \left( B_3^{IA}\right) +2 \varGamma _1 B_3 \left( B_1^{IA}\right) }{\gamma }-\frac{(\lambda _1+\lambda _2) B_3}{\gamma b^2}\right) \right\} \nonumber \\&\!\!+\! 2\varGamma _1 b^2 \left( B_1^{IA}\right) \left( B_3^{IA}\right) \! +\! \sigma _a ^2 \varGamma _1 \left( A_1^{IA}\right) \left( B_3^{IA}\right) \!\!-\!\!(\lambda _1\!+\!\lambda _2)\left( B_3^{IA}\right) \nonumber \\&-\kappa \left( B_3^{IA}\right) \!\!=\!\!0, \end{aligned}$$
(97)
with terminal conditions \(A_1 ^{IA}(T)=A_2 ^{IA} (T)=A_0 ^{IA} (T)=B_1 ^{IA} (T)=B_3 ^{IA}(T)=0\) and \(A_1,\, A_2,\, A_3,\, B_1\) and \(B_3\) satisfy (70)–(74). In the unobserved case, given the strategy \(\pi ^{UIM}\), the system of ODEs needed to express the function \(J^{UIM}\) in (63) is
$$\begin{aligned}&\left( \hat{A}_1^{UIM}\right) _t+ \frac{4(1+\eta \varGamma _1 \hat{A}_1)}{\hat{\sigma }^2(1+\rho )\varGamma }(1-\gamma ) +4 \eta \varGamma _1 \left( \hat{A}_1 ^{UIM}\right) \frac{(1+\eta \varGamma _1 \hat{A}_1)}{\hat{\sigma }^2(1+\rho )\varGamma }(1-\gamma )\nonumber \\&\quad +\, 2 \hat{\sigma } ^2 (1-\rho ) (1-\gamma )\frac{\varGamma _1 ^2}{\varGamma }\hat{B}_3 \left( \hat{B}_3 ^{UIM}\right) +2 \hat{\sigma }^2 \varGamma _1 (1-\rho ) \left( \hat{B}_3 ^{UIM}\right) ^2 \nonumber \\&\quad +\frac{2(1-\rho )\eta ^2 \varGamma _1}{\rho _1 ^2 \hat{\sigma }^2} \left( \hat{A}_1 ^{UIM}\right) ^2-\hat{\sigma }^2 \varGamma \left[ \left( \frac{(1+\eta \varGamma _1 \hat{A}_1)}{\hat{\sigma }^2(1+\rho )\varGamma }\right) ^2(1+\rho )\right. \nonumber \\&\quad \left. +\left( \frac{\hat{B}_3 \varGamma _1}{\varGamma }\right) ^2(1-\rho ) - 2 \kappa \left( \hat{A}_1 ^{UIM}\right) \right] =0,\end{aligned}$$
(98)
$$\begin{aligned}&\left( \hat{A}_2 ^{UIM}\right) _t+ (1-\gamma ) \frac{2\eta \hat{A}_2 \varGamma _1}{\hat{\sigma }^2 (1+\rho ) \varGamma }+ \kappa \theta \left( \hat{A}_1 ^{UIM}\right) -\kappa \left( \hat{A}_2^{UIM}\right) \nonumber \\&\quad +\,2(1-\rho )(1-\gamma )\hat{\sigma }^2 \frac{\varGamma _1 ^2}{\varGamma }\left[ \left( \hat{B}_2 ^{UIM}\right) \hat{B}_3 + \left( \hat{B}_3 ^{UIM}\right) \hat{B}_2 \right] \nonumber \\&\quad +\,2 \eta (1-\gamma )\frac{\varGamma _1}{\varGamma }\left[ \frac{1+\eta \hat{A}_1 \varGamma _1}{\hat{\sigma }^2 (1+\rho )} \left( \hat{A}_2^{UIM}\right) +\frac{2\eta \varGamma _1 \hat{A}_2}{\hat{\sigma }^2 (1+\rho )} \left( \hat{A}_1^{UIM}\right) \right] \nonumber \\&\quad +\, 2 \hat{\sigma }^2 (1-\rho )\varGamma _1 \left( \hat{B}_2 ^{UIM}\right) \left( \hat{B}_3 ^{UIM}\right) + \frac{2(1-\rho ) \eta ^2 \varGamma _1}{\rho _1^2 \hat{\sigma }^2} \left( \hat{A}_1 ^{UIM}\right) \left( \hat{A}_2 ^{UIM}\right) \nonumber \\&\quad -\frac{2(1+\eta \varGamma _1 \hat{A}_1)\eta \hat{A}_2 \varGamma _1}{\hat{\sigma } ^2 (1+\rho ) \varGamma }-2 (1-\rho )\hat{\sigma }^2 \frac{\hat{B}_2 \hat{B}_3 \varGamma _1 ^2}{\varGamma }=0,\end{aligned}$$
(99)
$$\begin{aligned}&\left( \hat{A}_0 ^{UIM}\right) _t + \tilde{r} (1-\gamma )+ \kappa \theta \left( \hat{A}_2^{UIM} \right) + \hat{\sigma }^2 (1-\rho )\frac{\varGamma _1^2}{\varGamma } \hat{B}_2 \hat{B}_2^{UIM} \nonumber \\&\quad +\, \hat{\sigma } ^2 (1-\rho ) \left( \hat{B}_1 ^{UIM}\right) +\frac{(1-\rho )\eta ^2 \varGamma _1 }{\rho _1 ^2 \hat{\sigma }^2} \left( \hat{A}_2 ^{UIM}\right) ^2+\frac{\eta ^2 (1-\rho )}{\rho _1 ^2 \hat{\sigma } ^2} \left( \hat{A}_1 ^{UIM}\right) \nonumber \\&\quad +\,\hat{\sigma } ^2 \varGamma _1 (1-\rho )\left( \hat{B}_2 ^{UIM}\right) ^2+(1-\gamma )\frac{2\eta ^2 \varGamma _1^2 \hat{A}_2}{\hat{\sigma }^2 (1+\rho ) \varGamma } \hat{A}_2 ^{UIM}\nonumber \\&\quad -\,\hat{\sigma } ^2 \varGamma \left[ \left( \frac{\hat{A}_2 \eta \varGamma _1}{\hat{\sigma } ^2 (1+\rho ) \varGamma }\right) ^2+\left( \frac{\hat{B}_2 \varGamma _1}{\varGamma }\right) ^2 (1-\rho ) \right] =0, \end{aligned}$$
(100)
$$\begin{aligned}&\left( \hat{B}_1 ^{UIM}\right) _t +2 \hat{\sigma }^2 \varGamma _1 (1-\rho )\left( \hat{B}_1 ^{UIM}\right) ^2 + 2\varGamma _1 \frac{(1-\rho )\eta ^2}{\rho _1 ^2 \hat{\sigma } ^2}\left( \hat{B}_3 ^{UIM}\right) ^2=0, \end{aligned}$$
(101)
$$\begin{aligned}&\left( \hat{B}_2 ^{UIM} \right) _t-(\lambda _1+\lambda _2)+\kappa \theta \left( \hat{B}_3 ^{UIM}\right) + 2 \hat{\sigma } ^2 (1-\gamma )(1-\rho )\frac{\varGamma _1^2}{\varGamma } \left( \hat{B}_1 ^{UIM} \right) \hat{B}_2 \nonumber \\&\quad -\,(\lambda _1+\lambda _2)(1-\gamma )\frac{\varGamma _1}{\varGamma }\hat{B}_2+ (1-\gamma )\frac{2 \eta ^2 \varGamma _1 ^2 \left( \hat{A}_2 ^{UIM} \right) \left( \hat{B}_3 ^{UIM} \right) }{\hat{\sigma } ^2(1+\rho )\varGamma } \nonumber \\&\quad +\,2 \hat{\sigma } ^2 (1-\rho ) \varGamma _1 \left( \hat{B}_1 ^{UIM} \right) \left( \hat{B}_2 ^{UIM} \right) + \frac{2(1-\rho )\eta ^2 \varGamma _1}{\rho _1 ^2 \hat{\sigma } ^2}\left( \hat{A}_2 ^{UIM} \right) \left( \hat{B}_3 ^{UIM} \right) \nonumber \\&\quad +\,(1-\gamma )\frac{\eta \varGamma _1 (\lambda _2-\lambda _1)}{\hat{\sigma } ^2 (1+\rho ) \varGamma }\hat{A}_2 =0,\end{aligned}$$
(102)
$$\begin{aligned}&\left( \hat{B}_3 ^{UIM} \right) _t-\kappa \left( \hat{B}_3 ^{UIM} \right) +2(1-\gamma )(1-\rho )\hat{\sigma }^2 \frac{\varGamma _1 ^2}{\varGamma } \left( \hat{B}_1 ^{UIM} \right) \hat{B}_3 -(1-\gamma )(\lambda _1+\lambda _2)\frac{\varGamma _1}{\varGamma } \hat{B}_3 \nonumber \\&\quad \!+\,(1\!-\!\gamma )(1\!+\!\eta \varGamma _1 \hat{A}_1)\frac{\lambda _2-\lambda _1}{\hat{\sigma }^2 (1+\rho )\varGamma } \!+\!(1-\gamma )\frac{2\eta ^2\varGamma _1 ^2 \hat{A}_1\!+\!2\eta \varGamma _1 }{\hat{\sigma }^2 (1+\rho ) \varGamma }\left( \hat{B}_3 ^{UIM} \right) \nonumber \\&\quad \!+\,2\hat{\sigma }^2(1\!-\!\rho )\varGamma _1 \left( \hat{B}_1 ^{UIM} \right) \left( \hat{B}_3 ^{UIM} \right) \!+\! \frac{2(1\!-\!\rho )\eta ^2 \varGamma _1}{\rho _1 ^2 \hat{\sigma }^2 } \left( \hat{A}_1 ^{UIM} \right) \left( \hat{B}_3 ^{UIM} \right) =0, \end{aligned}$$
(103)
with terminal conditions \(\hat{A}_1 ^{UIM}(T) =\hat{A}_2 ^{UIM}(T)=\hat{A}_0 ^{UIM}(T)=\hat{B}_1 ^{UIM}(T)=\hat{B}_2 ^{UIM}(T)=\hat{B}_3 ^{UIM}(T)=0\) and \(\hat{A}_1,\, \hat{A}_2\), \(\hat{A}_3\), \(\hat{B}_1\), \(\hat{B}_2\) and \(\hat{B}_3\) satisfy (82)–(87). Finally, again in the unobserved case, given the strategy \(\pi ^{UIA}\), the functions needed to express \(J^{UIA}\) in (66) solve the following system of ODEs:
$$\begin{aligned}&\left( \hat{A}_1^{UIA}\right) _t+ \frac{4(1+\eta \hat{A}_1)}{\hat{\sigma }^2(1+\rho )\gamma }(1-\gamma ) +4 \eta \varGamma _1 \left( \hat{A}_1 ^{UIA}\right) \frac{(1+\eta \hat{A}_1)}{\hat{\sigma }^2(1+\rho )\gamma }(1-\gamma )- 2 \kappa \left( \hat{A}_1 ^{UIA}\right) \nonumber \\&\quad +\, 2 \hat{\sigma } ^2 (1-\rho ) (1-\gamma )\frac{\varGamma _1}{\gamma }\hat{B}_3 \left( \hat{B}_3 ^{UIA}\right) +2 \hat{\sigma }^2 \varGamma _1 (1-\rho ) \left( \hat{B}_3 ^{UIA}\right) ^2 \nonumber \\&\quad +\,\frac{2(1-\rho )\eta ^2 \varGamma _1}{\rho _1 ^2 \hat{\sigma }^2} \left( \hat{A}_1 ^{UIA}\right) ^2-\hat{\sigma }^2 \varGamma \left[ \left( \frac{(1+\eta \hat{A}_1)}{\hat{\sigma }^2(1+\rho )\gamma }\right) ^2(1+\rho )+\left( \frac{\hat{B}_3 }{\gamma }\right) ^2(1-\rho )\right] =0,\nonumber \\ \end{aligned}$$
(104)
$$\begin{aligned}&\left( \hat{A}_2 ^{UIA}\right) _t+ (1-\gamma ) \frac{2\eta \hat{A}_2}{\hat{\sigma }^2 (1+\rho ) \gamma }+ \kappa \theta \left( \hat{A}_1 ^{UIA}\right) -\kappa \left( \hat{A}_2^{UIA}\right) \nonumber \\&\quad +\,2(1-\rho )(1-\gamma )\hat{\sigma }^2 \frac{\varGamma _1 }{\gamma }\left[ \left( \hat{B}_2 ^{UIA}\right) \hat{B}_3 + \left( \hat{B}_3 ^{UIA}\right) \hat{B}_2 \right] \nonumber \\&\quad +\,2 \eta (1-\gamma )\frac{\varGamma _1}{\gamma }\left[ \frac{1+\eta \hat{A}_1}{\hat{\sigma }^2 (1+\rho )} \left( \hat{A}_2^{UIA}\right) +\frac{2\eta \ \hat{A}_2}{\hat{\sigma }^2 (1+\rho )} \left( \hat{A}_1^{UIA}\right) \right] \nonumber \\&\quad +\, 2 \hat{\sigma }^2 (1-\rho )\varGamma _1 \left( \hat{B}_2 ^{UIA}\right) \left( \hat{B}_3 ^{UIA}\right) + \frac{2(1-\rho ) \eta ^2 \varGamma _1}{\rho _1^2 \hat{\sigma }^2} \left( \hat{A}_1 ^{UIA}\right) \left( \hat{A}_2 ^{UIA}\right) \nonumber \\&\quad -\,\frac{2(1+\eta \hat{A}_1)\eta \hat{A}_2 \varGamma }{\hat{\sigma } ^2 (1+\rho ) \gamma ^2}-2 (1-\rho )\hat{\sigma }^2 \frac{\hat{B}_2 \hat{B}_3 \varGamma }{\gamma ^2}=0,\end{aligned}$$
(105)
$$\begin{aligned}&\left( \hat{A}_0 ^{UIA}\right) _t + \tilde{r} (1-\gamma )+ \kappa \theta \left( \hat{A}_2^{UIA} \right) + \hat{\sigma }^2 (1-\rho )\frac{\varGamma _1}{\gamma } \hat{B}_2 \hat{B}_2^{UIA} +(1-\gamma )\frac{2\eta ^2 \varGamma _1 \hat{A}_2}{\hat{\sigma }^2 (1+\rho ) \gamma } \hat{A}_2 ^{UIA} \nonumber \\&\quad +\, \hat{\sigma } ^2 (1-\rho ) \left( \hat{B}_1 ^{UIA}\right) +\frac{(1-\rho )\eta ^2 \varGamma _1 }{\rho _1 ^2 \hat{\sigma }^2} \left( \hat{A}_2 ^{UIA}\right) ^2+\frac{\eta ^2 (1-\rho )}{\rho _1 ^2 \hat{\sigma } ^2} \left( \hat{A}_1 ^{UIA}\right) \nonumber \\&\quad +\,\hat{\sigma } ^2 \varGamma _1 (1-\rho )\left( \hat{B}_2 ^{UIA}\right) ^2-\hat{\sigma } ^2 \varGamma \left[ \left( \frac{\hat{A}_2 \eta }{\hat{\sigma } ^2 (1+\rho ) \gamma }\right) ^2+\left( \frac{\hat{B}_2}{\gamma }\right) ^2 (1-\rho ) \right] =0,\end{aligned}$$
(106)
$$\begin{aligned}&\left( \hat{B}_1 ^{UIA}\right) _t+2(1-\gamma )\left[ \frac{\eta (\lambda _2-\lambda _1)\hat{B}_3}{\hat{\sigma } ^2 (1+\rho )\gamma } -\frac{\lambda _1+\lambda _2}{\gamma }\hat{B}_1+\frac{(\lambda _2-\lambda _1)^2}{2\hat{\sigma }^2 (1+\rho ) \gamma } +\frac{(\lambda _1+\lambda _2)^2}{2 \hat{\sigma } ^2 \gamma (1-\rho )}\right. \nonumber \\&\quad +\,\hat{\sigma }^2 (1-\rho )\varGamma _1\left( \frac{2 \hat{B}_1}{\gamma }-\frac{\lambda _1+\lambda _2}{\hat{\sigma }^2 \gamma (1-\rho )}\right) \left( \hat{B}_1 ^{UIA} \right) +\frac{2 \eta ^2 \varGamma _1 \hat{B}_3 \left( \hat{B}_3^{UIA}\right) }{\hat{\sigma }^2 (1+\rho )\gamma }\nonumber \\&\quad +\,\left( \hat{B}_3^{UIA}\right) ^2\varGamma _1 \frac{\eta ^2 (1-\rho )}{\rho _1 ^2 (1-\gamma ) \hat{\sigma }^2} -\hat{\sigma } ^2\varGamma (1+\rho )\left( \frac{\frac{\lambda _2-\lambda _1}{2}+\eta \hat{B}_3}{\hat{\sigma }^2(1+\rho )\gamma }\right) ^2\nonumber \\&\quad -\left. \hat{\sigma } ^2 \varGamma (1-\rho )\left( \frac{\hat{B}_1}{\gamma }+ \hat{\sigma }^2 \varGamma _1 \left( \hat{B}_1^{UIA}\right) ^2\frac{1-\rho }{1-\gamma }- \frac{\lambda _1+\lambda _2}{2 \hat{\sigma }^2 (1-\rho )\gamma }\right) ^2\right] =0, \end{aligned}$$
(107)
$$\begin{aligned}&\left( \hat{B}_2 ^{UIA}\right) _t+(1-\gamma )\left[ \frac{\eta (\lambda _2-\lambda _1) \hat{A}_2}{\hat{\sigma }^2 (1+\rho )\gamma } -\frac{\lambda _1+\lambda _2}{\gamma } \hat{B}_2-\frac{\lambda _1+\lambda _2}{1-\gamma }+\frac{\kappa \theta \left( \hat{B}_3 ^{UIA}\right) }{1-\gamma } \right. \nonumber \\&\,\hat{\sigma }^2(1-\rho )\varGamma _1\left( \frac{2\hat{B}_1 \left( \hat{B}_2 ^{UIA}\right) }{\gamma }+\frac{2\hat{B}_1 \left( \hat{B}_2 ^{UIA}\right) }{\gamma }-\left( \hat{B}_2 ^{UIA}\right) \frac{\lambda _1+\lambda _2}{\hat{\sigma }^2 \gamma (1-\rho )}\right) \nonumber \\&\quad +\,\eta \varGamma _1\frac{2\eta \left( \hat{B}_3 ^{UIA}\right) \hat{A}_1+2\eta \left( \hat{B}_3 ^{UIA}\right) \hat{A}_2}{\hat{\sigma }^2(1+\rho )\gamma }+ 2 \left( \hat{B}_1 ^{UIA}\right) \left( \hat{B}_2 ^{UIA}\right) \hat{\sigma }^2 \varGamma _1 \frac{1-\rho }{1-\gamma }\nonumber \\&\quad +\, 2\left( \hat{A}_2 ^{UIA}\right) \left( \hat{B}_3 ^{UIA}\right) \frac{(1-\rho ) \eta ^2 \varGamma _1}{\rho _1 ^2(1-\gamma )\hat{\sigma }^2}-2\hat{\sigma }^2\varGamma (1-\rho )\left( \frac{\hat{B}_1 \hat{B}_2}{\gamma ^2}\!-\!\frac{\hat{B}_2(\lambda _1\!+\!\lambda _2)}{2\hat{\sigma }^2\gamma (1\!-\!\rho )}\right) \nonumber \\&\quad \!+\left. \frac{2\eta \hat{A}_2 \varGamma }{\hat{\sigma }^2\gamma }(\frac{\eta \hat{B}_3\!+\!\frac{\lambda _2-\lambda _1}{2}}{\hat{\sigma }^2(1+\rho )\gamma }) + \eta \varGamma _1 \frac{\lambda _2-\lambda _1}{\hat{\sigma }^2 (1+\rho )\gamma } \left( \hat{A}_2 ^{UIA} \right) \right] =0,\end{aligned}$$
(108)
$$\begin{aligned}&\left( \hat{B}_3 ^{UIA}\right) _t+(1-\gamma )\left[ (\lambda _2-\lambda _1)(\hat{\sigma }^2 (1+\rho )\gamma )+\frac{2\eta \hat{B}_3}{\hat{\sigma }^2 (1+\rho )\gamma }+\frac{(1+\eta \hat{A}_1)(\lambda _2-\lambda _1)}{\hat{\sigma }^2 (1+\rho )\gamma } \right. \nonumber \\&\quad +\, \hat{\sigma }^2(1-\rho )\varGamma _1\left( \frac{2\hat{B}_1 \left( \hat{B}_3 ^{UIA}\right) }{\gamma }+\frac{2 \hat{B}_3 \left( \hat{B}_1 ^{UIA}\right) }{\gamma }-\frac{\left( \hat{B}_3 ^{UIA}\right) (\lambda _1+\lambda _2)}{\hat{\sigma }^2 \gamma (1-\rho )}\right) \nonumber \\&\quad +\,\eta \varGamma _1 \left( \frac{2\eta \hat{A}_1 \left( \hat{B}_3 ^{UIA}\right) +2\eta \hat{B}_3 \left( \hat{A}_1 ^{UIA}\right) }{\hat{\sigma }^2 (1+\rho )\gamma }+\frac{2 \left( \hat{B}_3 ^{UIA}\right) }{\hat{\sigma }^2 (1+\rho )\gamma }+ \frac{\hat{A}_1 (\lambda _2-\lambda _1)}{\hat{\sigma }^2 (1+\rho )\gamma }\right) \nonumber \\&\quad +\,2\left( \hat{B}_1 ^{UIA}\right) \left( \hat{B}_3 ^{UIA}\right) \hat{\sigma }^2 \varGamma _1 \frac{1-\rho }{1-\gamma }+ 2\left( \hat{A}_1 ^{UIA}\right) \left( \hat{B}_3 ^{UIA}\right) \frac{(1-\rho )\eta ^2 \varGamma _1}{\rho _1^2 (1-\gamma )\hat{\sigma }^2} \nonumber \\&\quad -\,\frac{2\varGamma (1+\eta \hat{A}_1)}{\gamma }\left( \frac{\lambda _2-\lambda _1}{2\hat{\sigma }^2 (1+\rho )\gamma } + \frac{\eta \hat{B}_6}{\hat{\sigma }^2(1+\rho )\gamma }\right) -\frac{\hat{B}_3(\lambda _ 1+\lambda _2)}{\gamma }\nonumber \\&\quad -\left. \frac{\kappa \left( \hat{B}_3 ^{UIA}\right) }{1-\gamma }-2(1-\rho )\hat{\sigma } ^2 \varGamma \frac{\hat{B}_3}{\gamma }\left( \frac{\hat{B}_1}{\gamma } -\frac{\lambda _1+\lambda _2}{2\hat{\sigma }^2 (1-\rho )\gamma }\right) \right] =0, \end{aligned}$$
(109)
with terminal conditions \(\hat{A}_1 ^{UIA}(T) =\hat{A}_2 ^{UIA}(T)=\hat{A}_0 ^{UIA}(T)=\hat{B}_1 ^{UIA}(T)=\hat{B}_2 ^{UIA}(T)=\hat{B}_3 ^{UIA}(T)=0\) and \(\hat{A}_1,\, \hat{A}_2, \,\hat{A}_3,\, \hat{B}_1,\, \hat{B}_2\) and \(\hat{B}_3\) satisfy (82)–(87).
Appendix 5
The parameter estimation of the mispricing model (23)–(25) is quite straightforward since the likelihood function of the observed prices \(X_1(t),X_2(t)\) can be computed explicitly. In fact, \(a(t)\) and \(X(t)\) are Ornstein–Uhlenbeck processes given by
$$\begin{aligned} a(t)&=\theta +(a_0-\theta )e^{-\kappa t}+\sigma _a e^{-\kappa t} \int \limits _0^t e^{\kappa u}dZ_a(u), \end{aligned}$$
(110)
$$\begin{aligned} X(t)&=X_0e^{-(\lambda _1+\lambda _2)t} + be^{-(\lambda _1+\lambda _2)t} \left( \int \limits _0^t e^{(\lambda _1+\lambda _2)u} dZ_1(u) - \int \limits _0^t e^{(\lambda _1+\lambda _2)u} dZ_2(u) \right) . \end{aligned}$$
(111)
Substituting back to (23)–(24), we have
$$\begin{aligned} \ln \left( \frac{P_1(t)}{P_1(0)}\right)&= (2\tilde{r}-\sigma ^2-b^2)t/2 + \int \limits _0^t a(s)ds - \lambda _1 \int \limits _0^t X(s) ds + \sigma Z(t) + b Z_1(t) \\&=\! (2\tilde{r}\!-\!\sigma ^2-b^2)t/2 \!+\! \int \limits _0^t \left( \theta \!+\!(a_0\!-\!\theta )e^{-\kappa s}\!+\!\sigma _a e^{-\kappa s} \int \limits _0^s e^{\kappa u}dZ_a(u) \right) ds \\&\quad - \lambda _1 \int \limits _0^t \left( X_0e^{-(\lambda _1+\lambda _2)s} + be^{-(\lambda _1+\lambda _2)s} \left( \int \limits _0^s e^{(\lambda _1+\lambda _2)u} dZ_1(u) \right. \right. \nonumber \\&\quad \left. \left. - \int \limits _0^s e^{(\lambda _1+\lambda _2)u} dZ_2(u) \right) \right) ds+ \sigma Z(t) + b Z_1(t) \\&= (2\tilde{r}+2\theta -\sigma ^2-b^2)t/2 + (a_0-\theta )(1-e^{-\kappa t})/\kappa \\&\quad -X_0\lambda _1(1-e^{-(\lambda _1+\lambda _2)t})/(\lambda _1+\lambda _2) \\&\quad +\int \limits _0^t \int \limits _u^t \sigma _a e^{\kappa (u-s)} ds \ dZ_a(u)\\&\quad + \int \limits _0^t b \left( 1-\int \limits _u^t \lambda _1 e^{(\lambda _1+\lambda _2)(u-s)}ds \right) dZ_1(u) \\&\quad + \int \limits _0^t \int \limits _u^t \lambda _1 b e^{(\lambda _1+\lambda _2)(u-s)}ds \ dZ_2(u) + \sigma Z(t),\\ \ln \left( \frac{P_2(t)}{P_2(0)}\right)&= (2\tilde{r}-\sigma ^2-b^2)t/2 + \int \limits _0^t a(s)ds + \int \limits _0^t \lambda _2 X(s) ds + \sigma Z(t) + b Z_2(t) \\&=(2\tilde{r}+2\theta -\sigma ^2-b^2)t/2 + (a_0-\theta )(1-e^{-\kappa t})/\kappa \\&\quad + X_0\lambda _2(1-e^{-(\lambda _1+\lambda _2)t})/(\lambda _1+\lambda _2)\\&\quad +\int \limits _0^t \int \limits _u^t \sigma _a e^{\kappa (u-s)} ds \ dZ_a(u)\\&\quad + \int \limits _0^t b \left( 1-\int \limits _u^t \lambda _2 b e^{(\lambda _1+\lambda _2)(u-s)}ds \right) dZ_2(u) \\&\quad + \int \limits _0^t \int \limits _u^t \lambda _2 b e^{(\lambda _1+\lambda _2)(u-s)}ds \ dZ_1(u) + \sigma Z(t). \end{aligned}$$
Let \(Q_1(t)=\ln (P_1(t)/P_1(0))\) and \(Q_2(t)=\ln (P_2(t)/P_2(0))\). The joint distribution of \((Q_1(s),Q_2(s),Q_1(t),Q_2(t))\) with \(s<t\) is bivariate Normal with mean and covariance given by
$$\begin{aligned} E(Q_1(t))&= (2\tilde{r}+2\theta -\sigma ^2-b^2)t/2 + (a_0-\theta )(1-e^{-\kappa t})/ \kappa \nonumber \\&\quad - X_0\lambda _1(1-e^{-(\lambda _1+\lambda _2)t})/(\lambda _1+\lambda _2), \end{aligned}$$
(112)
$$\begin{aligned} E(Q_2(t))&= (2\tilde{r}+2\theta -\sigma ^2-b^2)t/2 + (a_0-\theta )(1-e^{-\kappa t})/ \kappa \nonumber \\&\quad + X_0\lambda _2(1-e^{-(\lambda _1+\lambda _2)t})/(\lambda _1+\lambda _2), \end{aligned}$$
(113)
$$\begin{aligned} Cov(Q_1(s),Q_1(t))&= I_1(s,t)+I_2(s,t,\lambda _1,\lambda _1)+I_5(s,t,\lambda _1,\lambda _1) + \sigma ^2 s, \end{aligned}$$
(114)
$$\begin{aligned} Cov(Q_2(s),Q_2(t))&= I_1(s,t)+I_2(s,t,\lambda _2,\lambda _2)+I_5(s,t,\lambda _2,\lambda _2) + \sigma ^2 s, \end{aligned}$$
(115)
$$\begin{aligned} Cov(Q_1(s),Q_2(t))&= I_1(s,t)+I_3(s,t,\lambda _1,\lambda _2)+I_4(s,t,\lambda _1,\lambda _2) + \sigma ^2 s, \end{aligned}$$
(116)
$$\begin{aligned} Cov(Q_2(s),Q_1(t))&= I_1(s,t)+I_3(s,t,\lambda _2,\lambda _1)+I_4(s,t,\lambda _2,\lambda _1) + \sigma ^2 s, \end{aligned}$$
(117)
where
$$\begin{aligned} I_1(s,t)&\!=\! \int \limits _0^s \left( \int \limits _u^s \sigma _a \exp (\kappa (u-v)) \, dv\right) \left( \int \limits _u^t \sigma _a \exp (\kappa (u-v)) \, dv \right) du \nonumber \\&= -\frac{\sigma _a^2 e^{-\kappa (s+t)} \left( (2-2 \kappa s) e^{\kappa (s+t)}-2 e^{\kappa s}+e^{2 \kappa s}-2 e^{\kappa t}+1\right) }{2 k^3},\end{aligned}$$
(118)
$$\begin{aligned} I_2(s,t,c,d)&= \int \limits _0^s b^2 \left( 1-c \int \limits _u^s \exp \left( -\left( \lambda _1+\lambda _2\right) (u-v)\right) \, dv\right) \nonumber \\&\times \left( 1-d \int \limits _u^t \exp \left( -\left( \lambda _1+\lambda _2\right) (u-v)\right) \, dv\right) \, du \nonumber \\&= \frac{b^2}{2 \left( \lambda _1+\lambda _2\right) {}^3} \left( c d \left( -2 e^{\left( \lambda _1+\lambda _2\right) s}+e^{\left( \lambda _1+\lambda _2\right) (t-s)}+e^{\left( \lambda _1+\lambda _2\right) (s+t)} \right. \right. \nonumber \\&-\left. 2 e^{\left( \lambda _1+\lambda _2\right) t}+2\right) +2 \left( \lambda _1+\lambda _2\right) \left( \left( \lambda _1+\lambda _2\right) s \left( c+d+\lambda _1+\lambda _2\right) +c d s\right. \nonumber \\&\left. +\left. c \left( -e^{\left( \lambda _1+\lambda _2\right) s}\right) +c+d e^{\left( \lambda _1+\lambda _2\right) (t-s)}-d e^{\left( \lambda _1+\lambda _2\right) t}\right) \right) ,\end{aligned}$$
(119)
$$\begin{aligned} I_3(s,t,c,d)&= \int \limits _0^s b^2 \left( 1-c \int \limits _u^s \exp \left( -\left( \lambda _1+\lambda _2\right) (u-v)\right) \, dv\right) \nonumber \\&\times \left( d \int \limits _u^t \exp \left( -\left( \lambda _1+\lambda _2\right) (u-v)\right) \, dv\right) \, du \nonumber \\&= \frac{b^2 d}{2 \left( \lambda _1+\lambda _2\right) {}^3} \left( 2 c e^{\left( \lambda _1+\lambda _2\right) s}-c e^{\left( \lambda _1+\lambda _2\right) (s+t)}+ 2 \left( c+\lambda _1+\lambda _2\right) e^{\left( \lambda _1+\lambda _2\right) t} \right. \nonumber \\&+\, e^{\left( \lambda _1+\lambda _2\right) (-s)} \left( -2 c e^{\left( \lambda _1+\lambda _2\right) s}-2 \left( \lambda _1+\lambda _2\right) s \left( c+\lambda _1+\lambda _2\right) e^{\left( \lambda _1+\lambda _2\right) s}\right. \nonumber \\&\left. +\left. c e^{\left( \lambda _1+\lambda _2\right) t}-2 \left( c+\lambda _1+\lambda _2\right) e^{\left( \lambda _1+\lambda _2\right) t}\right) \right) ,\end{aligned}$$
(120)
$$\begin{aligned} I_4(s,t,c,d)&= \int \limits _0^s b^2 \left( c \int \limits _u^s \exp \left( -\left( \lambda _1+\lambda _2\right) (u-v)\right) \, dv\right) \nonumber \\&\times \left( 1-d \int \limits _u^t \exp \left( -\left( \lambda _1+\lambda _2\right) (u-v)\right) \, dv\right) \, du \nonumber \\&= \frac{b^2 c}{2 \left( \lambda _1+\lambda _2\right) {}^3} \left( 2 \left( d+\lambda _1+\lambda _2\right) e^{\left( \lambda _1+\lambda _2\right) s}-d e^{\left( \lambda _1+\lambda _2\right) (s+t)} +2 d e^{\left( \lambda _1+\lambda _2\right) t} \right. \nonumber \\&+\,e^{\left( \lambda _1+\lambda _2\right) (-s)} \left( -2 \left( d+\lambda _1+\lambda _2\right) e^{\left( \lambda _1+\lambda _2\right) s}\right. \nonumber \\&\left. -\left. 2 \left( \lambda _1+\lambda _2\right) s \left( d+\lambda _1+\lambda _2\right) e^{\left( \lambda _1+\lambda _2\right) s}-d e^{\left( \lambda _1+\lambda _2\right) t}\right) \right) , \end{aligned}$$
(121)
$$\begin{aligned} I_5(s,t,c,d)&= \int \limits _0^s b^2 \left( c \int \limits _u^s \exp \left( -\left( \lambda _1+\lambda _2\right) (u-v)\right) \, dv\right) \nonumber \\&\times \left( d \int \limits _u^t \exp \left( -\left( \lambda _1+\lambda _2\right) (u-v)\right) \, dv\right) \, du \nonumber \\&= \frac{b^2 c d}{2 \left( \lambda _1+\lambda _2\right) {}^3} \left( 2 \lambda _1 s+2 \lambda _2 s+e^{\left( \lambda _1+\lambda _2\right) (-(s-t))} \left( \left( e^{\left( \lambda _1+\lambda _2\right) s}-1\right) {}^2\right. \right. \nonumber \\&\left. \quad -\left. 2 e^{\left( \lambda _1+\lambda _2\right) (2 s-t)}\right) +2\right) . \end{aligned}$$
(122)
Given the historical stock prices \(\{P_1(t_i),P_2(t_i)\}_{i=1}^n\), we can obtain \(\{Q_1(t_i),Q_2(t_i)\}_{i=1}^n\) and they have multivariate Normal distribution with mean and covariance matrix by (112)–(122). The parameters can be then estimated by maximizing the likelihood function of \(\{Q_1(t_i),Q_2(t_i)\}\) subject to the constraints \(\lambda _1+\lambda _2>0 \text { and } b,\sigma ,\sigma _a,\kappa >0\).
