Appendix 1: Proof of Proposition 1
We prove Proposition 1 by induction. For the induction basis at time t
N
= T we have
$$ \begin{array}{l}{J}_T^i\left({X}_T,{D}_T,{F}_T, T\right)=\left({F}_T+\frac{s}{2}\right){X}_T+\left[{\lambda}_0\right({X}_0-{X}_{t_m}\left)+{\lambda}_i\right({X}_{t_m}-{X}_T\left)+{D}_T+\frac{X_T}{2{q}_i}\right]{X}_T\\ {}\kern0.72em =\left({F}_T+\frac{s}{2}\right){X}_T+{\lambda}_i\left(\frac{\lambda_0}{\lambda_i}{X}_0-\frac{\lambda_0}{\lambda_i}{X}_{t_m}+{X}_{t_m}\right){X}_T\\ {}\kern3.3em +\left(\frac{1}{2{q}_i}-{\lambda}_i\right){X}_T^2+{X}_T{D}_T.\end{array} $$
We then obtain the optimization problem at the point t
n
with the following form
$$ \begin{array}{l}{J}_{t_n}^i\left({X}_{t_n},{D}_{t_n},{F}_{t_n},{t}_n\right)\\ {}\kern1.08em =\overset{ m in}{x_{t_n}}\left\{\left[,\right(,{F}_{t_n}+\frac{s}{2},\left),+,{\lambda}_0,\right(,{X}_0,-,{X}_{t_m},\left),+,{\lambda}_i,\right(,{X}_{t_m},-,{X}_{t_n},\left),+,{D}_{t_n},+,\frac{x_{t_n}}{2{q}_i},\right],{x}_{t_n}\right.\\ {}\left.+,{E}_{t_n},{J}_{t_{n+1}}^i,\left(,{X}_{t_n},-,{x}_{t_n},,,\left({D}_{t_n}+{\kappa}_i{x}_{t_n}\right),{e}^{-{\rho}_i\tau},,,{F}_{t_{n+1}},,,{t}_{n+1},\right)\right\}\\ {}=\overset{ m in}{x_{t_n}}\left\{\left[,\right(,{F}_{t_n}+\frac{s}{2},\left),+,{\lambda}_0,\right(,{X}_0,-,{X}_{t_m},\left),+,{\lambda}_i,\right(,{X}_{t_m},-,{X}_{t_n},\left),+,{D}_{t_n},+,\frac{x_{t_n}}{2{q}_i},\right],{x}_{t_n}\right.\\ {}\left({F}_{t_n}+\frac{s}{2}\right)\left({X}_{t_n}-{x}_{t_n}\right)+{\lambda}_i\left(\frac{\lambda_0}{\lambda_i}{X}_0-\frac{\lambda_0}{\lambda_i}{X}_{t_m}+{X}_{t_m}\right)\left({X}_{t_n}-{x}_{t_n}\right)+{\alpha}_{n+1}^i\left({X}_{t_n}-{x}_{t_n}\right){}^2\\ {}\left.+,{\beta}_{n+1}^i,\left({X}_{t_n}-{x}_{t_n}\right),\left(,{D}_{t_n},+,{\kappa}_i,{x}_{t_n},\right),{e}^{-{\rho}_i\tau},+,{\gamma}_{n+1}^i,\left(,{D}_{t_n},+,{\kappa}_i,{x}_{t_n},\right),{}^2,{e}^{-2{\rho}_i\tau}\right\}.\end{array} $$
(18)
We differentiate Eq. 18 with respect to \( {x}_{t_n} \) and set it equal to zero to find the optimal solution
$$ {x}_{t_n}=-\frac{1}{2}{\delta}_{n+1}^i\left({X}_{t_n}\left(-{\lambda}_i-2{\alpha}_{n+1}^i+{\beta}_{n+1}^i{e}^{-{\rho}_i\tau}{\kappa}_i\right)+{D}_{t_n}\left(1-{\beta}_{n+1}^i{e}^{-{\rho}_i\tau}+2{\gamma}_{n+1}^i{e}^{-2{\rho}_i\tau}{\kappa}_i\right)\right), $$
with
$$ {\delta}_{n+1}^i={\left(\frac{1}{2{q}_i}+{\alpha}_{n+1}^i-{\beta}_{n+1}^i{e}^{-{\rho}_i\tau}{\kappa}_i+{\gamma}_{n+1}^i{e}^{-2{\rho}_i\tau}{\kappa}_i^2\right)}^{-1}. $$
We find that the optimal value function has the form given by Eq. 2 and the coefficients given by Eqs. 4–7. This completes the induction for t
n
∈ {t
m + 1, …, t
N
}. Since the reaction to the released news is uncertain, we change parameters to describe the market reaction. At t
m
we face the following problem
$$ \begin{array}{l}{J}_{t_n}^0\left({X}_{t_n},{D}_{t_n},{F}_{t_n},{t}_n\right)=\overset{min}{x_{t_n}}{E}_{t_n}\left\{\left[,\right(,{F}_{t_n},+,\frac{s}{2},\left),+,{\lambda}_0,\right(,{X}_0,-,{X}_{t_n},\left),+,{D}_{t_n},+,\frac{x_{t_n}}{2{q}_u},\right],{x}_{t_n}\right.\\ {}\left.+,{J}_{t_{n+1}},\left(,{X}_{t_n},-,{x}_{t_n},,,\left({D}_{t_n}+{\kappa}_u{x}_{t_n}\right),{e}^{-{\rho}_u\tau},,,{F}_{t_{n+1}},,,{t}_{n+1},\right),,\right\}\end{array} $$
(19)
where q
u
, κ
u
, and ρ
u
should indicate that the current value of q and the future value of κ and ρ are unknown for us.
Because the released news is modeled as a discrete random variable, we get
$$ \begin{array}{l}{E}_{t_n}{J}_{t_{n+1}}\left({X}_{t_n}-{x}_{t_n},\left({D}_{t_n}+{\kappa}_u{x}_{t_n}\right){e}^{-{\rho}_u\tau},{F}_{t_{n+1}},{t}_{n+1}\right)\\ {}\kern0.96em ={\displaystyle \sum_{i=1}^r{p}_i}\times {J}_{t_{n+1}}^i\left({X}_{t_n}-{x}_{t_n},\left({D}_{t_n}+{\kappa}_i{x}_{t_n}\right){e}^{-{\rho}_i\tau},{F}_{t_n},{t}_{n+1}\right).\end{array} $$
(20)
We define
$$ \frac{1}{\tilde{q}}:={\displaystyle \sum_{i=1}^r{p}_i}\times \frac{1}{q_i}, $$
and combining Eq. 19 with Eq. 20, we then find
$$ \begin{array}{l}{J}_{t_n}^0\left({X}_{t_n},{D}_{t_n},{F}_{t_n},{t}_n\right)=\overset{min}{x_{t_n}}\left\{\left[,\right(,{F}_{t_n},+,\frac{s}{2},\left),+,{\lambda}_0,\right(,{X}_0,-,{X}_{t_n},\Big)\right.+{D}_{t_n}+\frac{x_{t_n}}{2\tilde{q}}\Big]{x}_{t_n}+{\alpha}_{n+1}^i{\left({X}_{t_n}-{x}_{t_n}\right)}^2\\ {}+{\displaystyle \sum_{i=1}^r{p}_i}\times \left(\left({F}_{t_n}+\frac{s}{2}\right)\right({X}_{t_n}-{x}_{t_n}\Big)\\ {}+{\lambda}_i\left(\frac{\lambda_0}{\lambda_i}{X}_0-\frac{\lambda_0}{\lambda_i}{X}_{t_n}+{X}_{t_n}\right)\left({X}_{t_n}-{x}_{t_n}\right)+{\beta}_{n+1}^i\left({X}_{t_n}-{x}_{t_n}\right)\left({D}_{t_n}+{\kappa}_i{x}_{t_n}\right){e}^{-{\rho}_i\tau}\left.+,{\gamma}_{n+1}^i,\left(,{D}_{t_n},+,{\kappa}_i,{x}_{t_n},\right),{}^2,{e}^{-2{\rho}_i\tau},\Big)\right\}.\end{array} $$
(21)
We differentiate Eq. 21 and set it equal to zero to find the optimal solution
$$ {x}_{t_n}=-\frac{1}{2}{\delta}_{n+1}^0\left({\displaystyle \sum_{i=1}^r{p}_i}\times \left[{X}_{t_n}\left(-{\lambda}_i-2{\alpha}_{n+1}^i+{\beta}_{n+1}^i{\kappa}_i{e}^{-{\rho}_i\tau}\right)+{D}_{t_n}\left(1-{\beta}_{n+1}^i{e}^{-{\rho}_i\tau}+2{\gamma}_{n+1}^i{\kappa}_i{e}^{-2{\rho}_i\tau}\right)\right]\right), $$
(22)
with
$$ {\delta}_{n+1}^0={\left[{\displaystyle \sum_{i=1}^r{p}_i}\times \left(\frac{1}{2{q}_i}+{\alpha}_{n+1}^i-{\beta}_{n+1}^i{\kappa}_i{e}^{-{\rho}_i\tau}+{\gamma}_{n+1}^i{\kappa}_i^2{e}^{-2{\rho}_i\tau}\right)\right]}^{-1}. $$
Inserting Eq. 22 into Eq. 23, we find that the optimal value has the form given by Eq. 3 with the coefficients given by Eqs. 4–7, which is the form
$$ {J}_{t_n}^0\left({X}_{t_n},{D}_{t_n},{F}_{t_n},{t}_n\right)={X}_{t_n}\times \left(\left({F}_{t_n}+\frac{s}{2}\right)+{\lambda}_0{X}_0\right)+{\alpha}_n^0{X}_{t_n}^2+{\beta}_n^0{D}_{t_n}{X}_{t_n}+{\gamma}_n^0{D}_{t_n}^2. $$
Since the optimal value function has the same form as the optimal value function obtained by Sun et al. (2013), we can use their result to obtain the proof for the remainder of the Proposition 1.
Appendix 2: Proof of Proposition 2
For the induction basis at time t
N
= T we have
$$ \begin{array}{l}{J}_T^i\left({X}_T,{D}_T,{F}_T, T\right)=\left({F}_T+\frac{s}{2}\right){X}_T+\left[{\lambda}_0\right({X}_0-{X}_{t_m}\left)+{\lambda}_i\right({X}_{t_m}-{X}_T\left)+{D}_T+\frac{X_T}{2{q}_i}\right]{X}_T\\ {}\kern1.08em =\left({F}_T+\frac{s}{2}\right){X}_T+{\lambda}_i\left(\frac{\lambda_0}{\lambda_i}{X}_0-\frac{\lambda_0}{\lambda_i}{X}_{t_m}+{X}_{t_m}\right){X}_T\\ {}+\left(\frac{1}{2{q}_i}-{\lambda}_i\right){X}_T^2+{X}_T{D}_T.\end{array} $$
For the induction step for some t
n
∈ {t
m + 1, …, t
N − 1} we get
$$ \begin{array}{l}{J}_{t_n}^i\left({X}_{t_n},{D}_{t_n},{F}_{t_n},{t}_n\right)=\overset{ m in}{x_{t_n}}\left\{\left[,\right(,{F}_{t_n},+,\frac{s}{2},\left),+,{\lambda}_0,\right(,{X}_0,-,{X}_{t_m},\left),+,{\lambda}_i,\right(,{X}_{t_m},-,{X}_{t_n},\left),+,{D}_{t_n},+,\frac{x_{t_n}}{2{q}_i},\right],{x}_{t_n}\right.\\ {}\left.+,{E}_{t_n},{J}_{t_{n+1}}^i,\left(,{X}_{t_n},-,{x}_{t_n},,,\left({D}_{t_n}+{\kappa}_i{x}_{t_n}\right),{e}^{-{\rho}_i\tau},,,{F}_{t_{n+1}},,,{t}_{n+1},\right)\right\}\\ {}=\overset{ m in}{x_{t_n}}\left\{\left[,\right(,{F}_{t_n}+\frac{s}{2},\left),+,{\lambda}_0,\right(,{X}_0,-,{X}_{t_m},\left),+,{\lambda}_i,\right(,{X}_{t_m},-,{X}_{t_n},\left),+,{D}_{t_n},+,\frac{x_{t_n}}{2{q}_i},\right],{x}_{t_n}\right.\\ {}+\left({a}_i\times {a}_i^{N-\left( n+1\right)}{F}_n+\frac{s}{2}\right)\left({X}_{t_n}-{x}_{t_n}\right)+{\lambda}_i\left(\frac{\lambda_0}{\lambda_i}{X}_0-\frac{\lambda_0}{\lambda_i}{X}_{t_m}+{X}_{t_m}\right)\left({X}_{t_n}-{x}_{t_n}\right)\\ {}+{b}_{n+1}^i{\left({X}_{t_n}-{x}_{t_n}\right)}^2\\ {}+{c}_{n+1}^i{\left({D}_{t_n}+{\kappa}_i{x}_{t_n}\right)}^2{e}^{-2{\rho}_i\tau}+{d}_{n+1}^i{v}_i{F}_{t_n}^2+{g}_{n+1}^i\left({X}_{t_n}-{x}_{t_n}\right)\left({D}_{t_n}+{\kappa}_i{x}_{t_n}\right){e}^{-{\rho}_i\tau}\\ {}\left.+,{h}_{n+1}^i,\left({X}_{t_n}-{x}_{t_n}\right),{a}_i,{F}_{t_n},+,{l}_{n+1}^i,\left(,{D}_{t_n},+,{\kappa}_i,{x}_{t_n},\right),{e}^{-{\rho}_i\tau},{a}_i,{F}_{t_n}\right\}.\end{array} $$
(23)
To obtain the minimum, we differentiate Eq. 23 with respect to \( {x}_{t_n} \)
$$ \begin{array}{l}\frac{\partial J}{\partial {x}_{t_n}}=\left({F}_{t_n}+\frac{s}{2}\right)+{\lambda}_0\left({X}_0-{X}_{t_m}\right)+{\lambda}_i\left({X}_{t_m}-{X}_{t_n}\right)+{D}_{t_n}+\frac{x_{t_n}}{q_i}\\ {}\kern0.36em -\left({a}_i^{N- n}{F}_n+\frac{s}{2}\right)-{\lambda}_i\left(\frac{\lambda_0}{\lambda_i}{X}_0-\frac{\lambda_0}{\lambda_i}{X}_{t_m}+{X}_{t_m}\right)-2{b}_{n+1}^i\left({X}_{t_n}-{x}_{t_n}\right)\\ {}\kern0.36em +2{\kappa}_i{c}_{n+1}^i\left({D}_{t_n}+{\kappa}_i{x}_{t_n}\right){e}^{-2{\rho}_i\tau}+{g}_{n+1}^i{e}^{-{\rho}_i\tau}\left[{\kappa}_i\left({X}_{t_n}-{x}_{t_n}\right)-\right({D}_{t_n}+{\kappa}_i{x}_{t_n}\left)\right]\\ {}\kern0.36em -{h}_{n+1}^i{a}_i{F}_{t_n}+{l}_{n+1}^i{\kappa}_i{e}^{-{\rho}_i\tau}{a}_i{F}_{t_n}\\ {}\kern1.44em ={x}_{t_n}\left(\frac{1}{q_i}+2{b}_{n+1}^i-2{g}_{n+1}^i{\kappa}_i{e}^{-{\rho}_i\tau}+{c}_{n+1}^i2{\kappa}_i^2{e}^{-2{\rho}_i\tau}\right)\\ {}\kern0.36em +{X}_{t_n}\left(-{\lambda}_i-2{b}_{n+1}^i+{g}_{n+1}^i{e}^{-{\rho}_i\tau}{\kappa}_i\right)+{D}_{t_n}\left(1+{c}_{n+1}^i2{\kappa}_i{e}^{-2{\rho}_i\tau}-{g}_{n+1}^i{e}^{-{\rho}_i\tau}\right)\\ {}\kern0.36em +{F}_{t_n}\left(1-{a}_i^{N- n}-{h}_{n+1}^i{a}_i+{l}_{n+1}^i{\kappa}_i{e}^{-{\rho}_i\tau}{a}_i\right).\end{array} $$
(24)
Setting \( \frac{\partial J}{\partial {x}_{t_n}}\overset{!}{=}0 \) for Eq. 24 to obtain the optimal choice
$$ {x}_{t_n}= o{D}_{t_n}+ w{X}_{t_n}+ u{F}_{t_n}, $$
(25)
where
$$ \begin{array}{l} o\kern0.96em =\kern0.48em -\frac{1}{2}{\delta}_{n+1}^i\left(1+{c}_{n+1}^i2{\kappa}_i{e}^{-2{\rho}_i\tau}-{g}_{n+1}^i{e}^{-{\rho}_i\tau}\right),\\ {} w\kern0.85em =\kern0.48em -\frac{1}{2}{\delta}_{n+1}^i\left(-{\lambda}_i-2{b}_{n+1}^i+{g}_{n+1}^i{e}^{-{\rho}_i\tau}{\kappa}_i\right),\\ {} u\kern0.85em =\kern0.48em -\frac{1}{2}{\delta}_{n+1}^i\left(1-{a}_i^{N- n}-{h}_{n+1}^i{a}_i+{l}_{n+1}^i{\kappa}_i{e}^{-{\rho}_i\tau}{a}_i\right),\\ {}{\delta}_{n+1}^i=\kern0.48em {\left(\frac{1}{2{q}_i}+{b}_{n+1}^i-{g}_{n+1}^i{\kappa}_i{e}^{-{\rho}_i\tau}+{c}_{n+1}^i{\kappa}_i^2{e}^{-2{\rho}_i\tau}\right)}^{-1}.\end{array} $$
Putting Eq. 25 into Eq. 23 we obtain the optimal value function given by Eq. 9 and find the coefficients given by Eqs. 11–17. This completes the induction for t
n
∈ {t
m + 1, …, t
N
}. We are unsure about market reaction to the released news and the following change of the parameters that describe the market. At t
m
we face the following problem
$$ \begin{array}{l}{J}_{t_n}^0\left({X}_{t_n},{D}_{t_n},{F}_{t_n},{t}_n\right)=\overset{min}{x_{t_n}}{E}_{t_n}\left\{\left[,\right(,{F}_{t_n},+,\frac{s}{2},\left),+,{\lambda}_0,\right(,{X}_0,-,{X}_{t_n},\left),+,{D}_{t_n},+,\frac{x_{t_n}}{2{q}_u},\right],{x}_{t_n}\right.\\ {}\left.+,{J}_{t_{n+1}},\left(,{X}_{t_n},-,{x}_{t_n},,,\left({D}_{t_n}+{\kappa}_u{x}_{t_n}\right),{e}^{-{\rho}_u\tau},,,{F}_{t_{n+1}},,,{t}_{n+1},\right)\right\}\end{array} $$
(26)
where q
u
, κ
u
, a
u
and ρ
u
should indicate that the current value of q and the future value of κ, ρ, and a
u
are unknown.
Because the released news is modeled as a discrete random variable, we obtain
$$ \begin{array}{l}\kern0.6em {E}_{t_n}{J}_{t_{n+1}}\left({X}_{t_n}-{x}_{t_n},\left({D}_{t_n}+{\kappa}_u{x}_{t_n}\right){e}^{-{\rho}_u\tau},{a}_u{F}_{t_{n+1}},{t}_{n+1}\right)\\ {}={\displaystyle \sum_{i=0}^r{p}_i}\times {J}_{t_{n+1}}^i\left({X}_{t_n}-{x}_{t_n},\left({D}_{t_n}+{\kappa}_i{x}_{t_n}\right){e}^{-{\rho}_i\tau},{a}_i{F}_{t_n},{t}_{n+1}\right).\end{array} $$
(27)
We use \( {a}_i={e}^{\mu_i\tau} \) and \( {v}_i={e}^{\left(2{\mu}_i+{\sigma}_i^2\right)\times \tau} \) and define
$$ \frac{1}{\overline{q}}={\displaystyle \sum_{i=1}^r{p}_i}\times \frac{1}{q_i},\kern0.33em \overline{a}={\displaystyle \sum_{i=1}^r{p}_i}\times \left({a}_i^{N+1-\left( m+1\right)}\right),\kern0.33em \overline{\lambda}={\displaystyle \sum_{i=1}^r{p}_i}\times {\lambda}_i. $$
Combining Eq. 26 and Eq. 27 with this definitions, we find that
$$ \begin{array}{l}\kern0.72em {J}_{t_n}^0\left({X}_{t_n},{D}_{t_n},{F}_{t_n},{t}_n\right)\\ {}=\left[\left({F}_{t_n}+\frac{s}{2}\right)+{\lambda}_0\right({X}_0-{X}_{t_n}\left)+{D}_{t_n}+\frac{x_{t_n}}{2\overline{q}}\right]{x}_{t_n}\\ {}\kern-1em +\left({F}_n\overline{a}+\frac{s}{2}\right)\left({X}_{t_n}-{x}_{t_n}\right)+\overline{\lambda}\left(\frac{\lambda_0}{\overline{\lambda}}{X}_0-\frac{\lambda_0}{\overline{\lambda}}{X}_{t_n}+{X}_{t_n}\right)\left({X}_{t_n}-{x}_{t_n}\right)\kern-2.5em +\left({\displaystyle \sum_{i=1}^r{p}_i}\times {b}_{n+1}^i\right)\left({X}_{t_n}-{x}_{t_n}\right){}^2\\ {}\kern0.6em +{D}_{t_n}^2\left({\displaystyle \sum_{i=1}^r{p}_i}\times {c}_{n+1}^i{e}^{-2{\rho}_i\tau}\right)\kern0.5em +2{D}_{t_n}{x}_{t_n}\left({\displaystyle \sum_{i=1}^r{p}_i}\times {c}_{n+1}^i{e}^{-2{\rho}_i\tau}{\kappa}_i\right)+{x}_{t_n}^2\left({\displaystyle \sum_{i=1}^r{p}_i}\times {c}_{n+1}^i{e}^{-2{\rho}_i\tau}{\kappa}_i^2\right)\\ {}+\left({\displaystyle \sum_{i=1}^r{p}_i}\times {d}_{n+1}^i{v}_i\right){F}_{t_n}^2+\left({X}_{t_n}-{x}_{t_n}\right)\left({D}_{t_n}\right({\displaystyle \sum_{i=1}^r{p}_i}\times {g}_{n+1}^i{e}^{-{\rho}_i\tau}\left)\kern0.5em +\right({\displaystyle \sum_{i=1}^r{p}_i}\times {g}_{n+1}^i{e}^{-{\rho}_i\tau}{\kappa}_i\left){x}_{t_n}\right)\\ {}\kern0.6em +\left({\displaystyle \sum_{i=1}^r{p}_i}\times {h}_{n+1}^i{a}_i\right)\left({X}_{t_n}-{x}_{t_n}\right){F}_{t_n}\kern0.5em +{D}_{t_n}{F}_{t_n}\left({\displaystyle \sum_{i=1}^r{p}_i}\times {l}_{n+1}^i{a}_i{e}^{-{\rho}_i\tau}\right)+{x}_{t_n}{F}_{t_n}\left({\displaystyle \sum_{i=1}^r{p}_i}\times {l}_{n+1}^i{a}_i{e}^{-{\rho}_i\tau}{\kappa}_i\right).\end{array} $$
(28)
We then obtain the solution that minimizes Eq. 28 is
$$ \begin{array}{c}{x}_m=-\frac{1}{2}{\delta}_{m+1}\Big(\left(1+2{\displaystyle \sum_{i=1}^r{p}_i}\times {c}_{n+1}^i{e}^{-2{\rho}_i\tau}{\kappa}_i-{\displaystyle \sum_{i=1}^r{p}_i}\times {g}_{n+1}^i{e}^{-{\rho}_i\tau}\right){D}_{t_m}\\ {}+\left(-\overline{\uplambda}-2{\displaystyle \sum_{i=1}^r{p}_i}\times {b}_{n+1}^i+{\displaystyle \sum_{i=1}^r{p}_i}\times {g}_{n+1}^i{e}^{-{\rho}_i\tau}{\kappa}_i\right){X}_{t_m}\\ {}+\left(1-\overline{a}-{\displaystyle \sum_{i=1}^r{p}_i}\times {h}_{n+1}^i{a}_i+{\displaystyle \sum_{i=1}^r{p}_i}\times {l}_{n+1}^i{a}_i{e}^{-{\rho}_i\tau}{\kappa}_i\right){F}_{t_m}\Big),\end{array} $$
(29)
With
$$ {\delta}_{m+1}={\left(\frac{1}{2\overline{q}}+{\displaystyle \sum_{i=1}^r{p}_i}\times {b}_{n+1}^i-{\displaystyle \sum_{i=1}^r{p}_i}\times {g}_{n+1}^i{e}^{-{\rho}_i\tau}{\kappa}_i+{\displaystyle \sum_{i=1}^r{p}_i}\times {c}_{n+1}^i{e}^{-2{\rho}_i\tau}{\kappa}_i^2\right)}^{-1}. $$
Inserting Eq. 29 into Eq. 28, we find the optimal value function given by Eq. 9 and the coefficients given by Eqs. 11–17. For the induction step for some t
n
∈ {t
0, …, t
m − 1} we get
$$ \begin{array}{c}{J}_{t_n}^0\left({X}_{t_n},{D}_{t_n},{F}_{t_n},{t}_n\right)=\overset{ m in}{x_{t_n}}\left\{\right[\left({F}_{t_n}+\frac{s}{2}\right)+{\uplambda}_0\left({X}_0-{X}_{t_n}\right)+{D}_{t_n}+\frac{x_{t_n}}{2{q}_0}\Big]{x}_{t_n}\\ {}+{E}_{t_n}{J}_{t_{n+1}}^0\left({X}_{t_n}-{x}_{t_n},\left({D}_{t_n}+{\kappa}_0{x}_{t_n}\right){e}^{-{\rho}_0\tau},{F}_{t_{n+1}},{t}_{n+1}\right)\Big\}\\ {}=\overset{ m in}{x_{t_n}}\left\{\right[\left({F}_{t_n}+\frac{s}{2}\right)+{\uplambda}_0\left({X}_0-{X}_{t_n}\right)+{D}_{t_n}+\frac{x_{t_n}}{2{q}_0}\Big]{x}_{t_n}\\ {}+\left({a}_0\times {a}_0^{m-\left( n+1\right)}\times \overline{a}{F}_n+\frac{s}{2}\right)\left({X}_{t_n}-{x}_{t_n}\right)+{\uplambda}_0{X}_0\left({X}_{t_n}-{x}_{t_n}\right)+{b}_{n+1}^0\left({X}_{t_n}-{x}_{t_n}\right){}^2\\ {}+{c}_{n+1}^0{\left({D}_{t_n}+{\kappa}_0{x}_{t_n}\right)}^2{e}^{-2{\rho}_0\tau}+{d}_{n+1}^0{v}_0{F}_{t_n}^2+{g}_{n+1}^0\left({X}_{t_n}-{x}_{t_n}\right)\left({D}_{t_n}+{\kappa}_0{x}_{t_n}\right){e}^{-{\rho}_0\tau}\\ {}+{h}_{n+1}^0\left({X}_{t_n}-{x}_{t_n}\right){a}_0{F}_{t_n}+{l}_{n+1}^0\left({D}_{t_n}+{\kappa}_0{x}_{t_n}\right){e}^{-{\rho}_0\tau}{a}_0{F}_{t_n}\Big\}.\end{array} $$
(30)
To obtain the minimum we differentiate Eq. 30 with respect to \( {x}_{t_n} \)
$$ \begin{array}{l}\frac{\partial J}{\partial {x}_{t_n}}=\left({F}_{t_n}+\frac{s}{2}\right)+{\uplambda}_0\left({X}_0-{X}_{t_n}\right)+{D}_{t_n}+\frac{x_{t_n}}{q_0}\\ {}-\left({a}_0^{m- n}\times \overline{a}{F}_n+\frac{s}{2}\right)-{\uplambda}_0{X}_0-2{b}_{n+1}^0\left({X}_{t_n}-{x}_{t_n}\right)\\ {}+2{\kappa}_0{c}_{n+1}^0\left({D}_{t_n}+{\kappa}_0{x}_{t_n}\right){e}^{-2{\rho}_0\tau}+{g}_{n+1}^0{e}^{-{\rho}_0\tau}\left[{\kappa}_0\left({X}_{t_n}-{x}_{t_n}\right)-\right({D}_{t_n}+{\kappa}_0{x}_{t_n}\left)\right]\\ {}-{h}_{n+1}^0{a}_0{F}_{t_n}+{l}_{n+1}^0{\kappa}_0{e}^{-{\rho}_0\tau}{a}_0{F}_{t_n}\ \\ {}\kern1.2em =\kern0.24em {x}_{t_n}\left(\frac{1}{q_0}+2{b}_{n+1}^0-2{g}_{n+1}^0{\kappa}_0{e}^{-{\rho}_0\tau}+{c}_{n+1}^02{\kappa}_0^2{e}^{-2{\rho}_0\tau}\right)\\ {}+{X}_{t_n}\left(-{\uplambda}_0-2{b}_{n+1}^0+{g}_{n+1}^0{e}^{-{\rho}_0\tau}{\kappa}_0\right)+{D}_{t_n}\left(1+{c}_{n+1}^02{\kappa}_0{e}^{-2{\rho}_0\tau}-{g}_{n+1}^0{e}^{-{\rho}_0\tau}\right)\\ {}+{F}_{t_n}\left(1-{a}_0^{m- n}\times \overline{a}-{h}_{n+1}^0{a}_0+{l}_{n+1}^0{\kappa}_0{e}^{-{\rho}_0\tau}{a}_0\right).\end{array} $$
(31)
Setting \( \frac{\partial J}{\partial {x}_{t_n}}\overset{!}{=}0 \) for Eq. 31 to obtain the optimal choice
$$ {x}_{t_n}= o{D}_{t_n}+ w{X}_{t_n}+ u{F}_{t_n}, $$
(32)
Where
$$ \begin{array}{l}\kern0.48em o\kern0.48em =-\frac{1}{2}{\delta}_{n+1}^0\left(1+{c}_{n+1}^02{\kappa}_0{e}^{-2{\rho}_0\tau}-{g}_{n+1}^0{e}^{-{\rho}_0\tau}\right),\\ {}\kern0.48em w\kern0.36em =-\frac{1}{2}{\delta}_{n+1}^0\left(-{\uplambda}_0-2{b}_{n+1}^0+{g}_{n+1}^0{e}^{-{\rho}_0\tau}{\kappa}_0\right),\\ {}\kern0.48em u\kern0.48em =-\frac{1}{2}{\delta}_{n+1}^0\left(1-{a}_0^{m- n}\times \overline{a}-{h}_{n+1}^0{a}_0+{l}_{n+1}^0{\kappa}_0{e}^{-{\rho}_0\tau}{a}_0\right),\\ {}{\delta}_{n+1}^0\kern0.36em =\kern0.36em {\left(\frac{1}{2{q}_0}+{b}_{n+1}^0-{g}_{n+1}^0{\kappa}_0{e}^{-{\rho}_0\tau}+{c}_{n+1}^0{\kappa}_0^2{e}^{-2{\rho}_0\tau}\right)}^{-1}.\end{array} $$
Putting Eq. 32 into Eq. 30 we obtain the optimal value function given by Eq. 9 and the coefficients given by Eqs. 11–17. This concludes the induction.
