Financial Transaction Tax: Policy Analytics Based on Optimal Trading

Abstract

Introducing a financial transaction tax (FTT) has recently attracted tremendous global attention, with both proponents and opponents disputing its dampening effects on financial markets. In this paper we present a model to show under some circumstances that there exists a win-win situation via optimal trading when the tax burden can be dispersed. The way to absorb FTT in our model is to adjust the bid–ask spread. In our optimal trading model that considers FTT, the representative traders depend on the liquidity (market depth) they supply to weight their associated transaction cost by adjusting the spread ex post. We illustrate the analytical properties and computational solutions of our model when finding the optimal trading strategy under different market situations in order to offset FTT. We also conduct a simulation study to show the superior performance of our proposed optimal trading strategy in comparison to the alternative strategies that do not consider absorbing FTT. The results demonstrate that there is indeed a win-win situation, because financial institutions will not be worse off if such an optimal trading strategy is applied to offset FTT and reduce their transaction cost.

Appendices

Appendix 1: Proof of Proposition 1

We use induction to prove Proposition 1. For the optimal value function of the last time period we obtain

$$\begin{aligned} J_T\left( X_T,D_T,F_T,T,x_{t_{N-1}}\right)&= \left( F_T+\frac{S(x_{t_{N-1}})}{2}\right) X_T\\&\quad +\left[ \lambda (X_0-X_T)+D_T+\frac{X_T}{2q}\right] X_T\\&= \left( F_T+\frac{\alpha x_{t_{N-1}}+\beta }{2}\right) X_T\\&\quad +\,\left[ \lambda (X_0-X_T)+D_T+\frac{X_T}{2q}\right] X_T. \end{aligned}$$

This is Eq. 2 with Eq. 4. We now verify that for every $t_n\in \{ t_{0},\ldots ,t_{N} \}$ the optimal value function has the form given by Eq. 2. We assume that Proposition 1 holds true for some $t_{n+1}\in \{ t_{1},\ldots ,t_{N} \}$ and then find for $t_n$:

$$\begin{aligned}&J_{t_n}\Big (X_{t_n},D_{t_n},F_{t_n},{t_n},x_{t_{n-1}}\Big ) \nonumber \\&\quad =\, \mathop {x_{t_n}}\limits ^{min}\left\{ \left[ \left( F_{t_n}+\frac{S(x_{t_{n-1}})}{2}\right) +\lambda \left( X_0-X_{t_n}\right) +D_{t_n}+\frac{x_{t_n}}{2q}\right] x_{t_n} \right. \nonumber \\&\qquad \left. +\,E_{t_n}J_{t_{n+1}}\left( X_{t_n}-x_{t_n},\Big (D_{t_n}+\kappa x_{t_n}\Big )e^{-\rho \tau }, F_{t_{n+1}},{t_{n+1}},x_{t_{n}}\right) \right\} \nonumber \\&\quad =\, \mathop {x_{t_n}}\limits ^{min}\Bigg \{\left[ \Big (F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}\Big )+\lambda \Big (X_0-X_{t_n}\Big ) +D_{t_n}+\frac{x_{t_n}}{2q}\right] x_{t_n} \nonumber \\&\qquad +\,\Big (F_{t_n}+\frac{\beta }{2}\Big )\Big (X_{t_n}-x_{t_n}\Big )+\lambda X_0\Big (X_{t_n}-x_{t_n}\Big ) +d_{n+1}\Big (X_{t_n}-x_{t_n}\Big )^2 \nonumber \\&\qquad +\,l_{n+1}\Big (X_{t_n}-x_{t_n}\Big )\Big (D_{t_n}+\kappa x_{t_n}\Big )e^{-\rho \tau }+f_{n+1}\Big (D_{t_n}+\kappa x_{t_n}\Big )^2e^{-2\rho \tau } \nonumber \\&\qquad +\,h_{n+1}\Big (X_{t_n}-x_{t_n}\Big )x_{t_n}+s_{n+1}\Big (D_{t_n}+\kappa x_{t_n}\Big )e^{-\rho \tau }x_{t_n} +u_{n+1}x_{t_n}^2 \Bigg \}. \end{aligned}$$

(9)

To obtain the minimum, we differentiate Eq. 9 with respect to $x_{t_n}$:

$$\begin{aligned} \frac{\partial J}{\partial x_{t_n}}&= F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}+\lambda \left( X_0-X_{t_n}\right) +D_{t_n}+\frac{x_{t_n}}{q}\\&-\,\left( F_{t_n}+\frac{\beta }{2}\right) -\lambda X_0-2d_{n+1}\left( X_{t_n}-x_{t_n}\right) \\&+\,l_{n+1}\left( \kappa X_{t_n}-\kappa x_{t_n}-D_{t_n}-\kappa x_{t_n}\right) e^{-\rho \tau }+f_{n+1}2\kappa \left( D_{t_n}+\kappa x_{t_n}\right) e^{-2\rho \tau } \\&+\,h_{n+1}\left( X_{t_n}-2x_{t_n}\right) +s_{n+1}\left( D_{t_n}+2\kappa x_{t_n}\right) e^{-\rho \tau }+2u_{n+1}x_{t_n} \\&= 2x_{t_n}\left[ \frac{1}{2q}+d_{n+1}-\kappa l_{n+1}e^{-\rho \tau }+{\kappa }^2f_{n+1}e^{-2\rho \tau }\right. \\&\left. -\,h_{n+1}+\kappa s_{n+1}e^{-\rho \tau }+u_{n+1}\right] +\,\frac{\alpha x_{t_{n-1}}}{2} \\&+\,X_{t_n}\left[ -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right] \\&+\,D_{t_n}\left[ 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right] . \end{aligned}$$

Seting $\frac{\partial J}{\partial x_{t_n}}\mathop {=}\limits ^{!}0$ we obtain the optimal choice:

$$\begin{aligned} x_{t_n}&= wX_{t_n}+vD_{t_n}+m, \end{aligned}$$

(10)

where

$$\begin{aligned} w&= -\frac{1}{2}\delta _{n+1}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) , \nonumber \\ v&= -\frac{1}{2}\delta _{n+1}\left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) , \nonumber \\ m&= -\frac{1}{2}\delta _{n+1}\left( \frac{\alpha x_{t_{n-1}}}{2}\right) , \nonumber \\ \delta _{n+1}&= \left( \frac{1}{2q}+d_{n+1}-\kappa l_{n+1}e^{-\rho \tau }+{\kappa }^2f_{n+1}e^{-2\rho \tau }-h_{n+1}\right. \nonumber \\&\quad \left. +\,\kappa s_{n+1}e^{-\rho \tau }+u_{n+1}\right) ^{-1}. \end{aligned}$$

(11)

This proves Eq. 1 of Proposition 1. Putting Eq. 10 into Eq. 9 we find for the optimal value function:

$$\begin{aligned}&J_{t_n}\left( X_{t_n},D_{t_n},F_{t_n},{t_n},x_{t_{n-1}}\right) \\&\quad =\left[ \left( F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}\right) +\lambda \left( X_0-X_{t_n}\right) +D_{t_n}+\frac{wX_{t_n}+vD_{t_n}+m}{2q}\right] \\&\qquad \times \left( wX_{t_n}+vD_{t_n}+m\right) \\&\qquad +\,\left( F_{t_n}+\frac{\beta }{2}\right) \left( X_{t_n}-wX_{t_n}-vD_{t_n}-m\right) +\lambda X_0\left( X_{t_n}-wX_{t_n}-vD_{t_n}-m\right) \\&\qquad +\,d_{n+1}\left( X_{t_n}-wX_{t_n}-vD_{t_n}-m\right) ^2 +l_{n+1}\left( X_{t_n}-wX_{t_n}-vD_{t_n}-m\right) \\&\qquad \times \, \left( D_{t_n}+\kappa \left( wX_{t_n}+vD_{t_n}+m\right) \right) e^{-\rho \tau } \\&\qquad +\,f_{n+1}\left( D_{t_n}+\kappa \left( wX_{t_n}+vD_{t_n}+m\right) \right) ^2e^{-2\rho \tau }+h_{n+1}\left( X_{t_n}-wX_{t_n}-vD_{t_n}-m\right) \\&\qquad \times \, \left( wX_{t_n}+vD_{t_n}+m\right) \\&\qquad +\,s_{n+1}\left( D_{t_n}+\kappa \left( wX_{t_n}+vD_{t_n}+m\right) \right) e^{-\rho \tau }\left( wX_{t_n}+vD_{t_n}+m\right) \\&\qquad +\,u_{n+1}\left( wX_{t_n}+vD_{t_n}+m\right) ^2\\&\quad =X_{t_n}^2 \left( -\lambda w+\frac{w^2}{2q}+d_{n+1}\left( 1-w\right) ^2+\kappa l_{n+1}e^{-\rho \tau }w\left( 1-w\right) +{\kappa }^2f_{n+1}e^{-2\rho \tau }w^2 \right. \\&\qquad \left. +\,h_{n+1}\left( 1-w\right) +\kappa s_{n+1}e^{-\rho \tau }w^2+u_{n+1}w^2 \right) \\&\qquad +\,D_{t_n}^2 \left( v+\frac{v^2}{2q}+d_{n+1}v^2-l_{n+1}e^{-\rho \tau }v\left( 1+\kappa v\right) +f_{n+1}e^{-2\rho \tau }\left( 1+\kappa v\right) ^2-h_{n+1}v^2 \right. \\&\qquad \left. +\, s_{n+1}e^{-\rho \tau }v\left( 1+\kappa v\right) +u_{n+1}v^2\right) \\&\qquad +\,X_{t_n}D_{t_n} \Bigg (-\lambda v+w+\frac{wv}{2q}-2d_{n+1}\left( 1-w\right) v- l_{n+1}e^{-\rho \tau }\big (\left( 1-w\right) \left( 1+\kappa v\right) -\kappa vw\big )\\&\qquad +\,2{\kappa }f_{n+1}e^{-2\rho \tau }w\left( 1+\kappa v\right) +h_{n+1}v\left( 1-w\right) -h_{n+1}vw+\kappa s_{n+1}e^{-\rho \tau }vw \\&\qquad +\,s_{n+1}e^{-\rho \tau }w\left( 1+\kappa v\right) +2u_{n+1}wv\Bigg ) \\&\qquad +\,X_{t_n} \left( w\left( F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}+\lambda X_0\right) -\lambda m+\frac{mw}{q}+\left( 1-w\right) \left( F_{t_n}+\frac{\beta }{2}\right) \right. \\&\qquad +\,\lambda X_0\left( 1-w\right) -2d_{n+1}m\left( 1-w\right) - l_{n+1}e^{-\rho \tau }\left( \kappa m\left( 1-w\right) -\kappa wm\right) \\&\qquad \left. +\,2{\kappa }^2f_{n+1}e^{-2\rho \tau }wm+h_{n+1}m\left( 1-w\right) -h_{n+1}mw+2\kappa s_{n+1}e^{-\rho \tau }mw+2u_{n+1}mw\right) \\&\qquad +\,D_{t_n} \Bigg (v\left( F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}+\lambda X_0\right) +m+\frac{mw}{q}-v\left( F_{t_n}+\frac{\beta }{2}\right) -\lambda X_0v \\&\qquad +\,2d_{n+1}mv+l_{n+1}e^{-\rho \tau }\left( -\left( 1+\kappa v\right) m-\kappa vm\right) +2{\kappa }f_{n+1}e^{-2\rho \tau }\left( 1+\kappa v\right) m \\&\qquad -\,2h_{n+1}mv-h_{n+1}mw+ s_{n+1}e^{-\rho \tau }m\left( 1+\kappa v\right) +s_{n+1}e^{-\rho \tau }m\kappa v+2u_{n+1}mv\Bigg ) \\&\qquad +\, \Bigg (m\left( F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}+\lambda X_0\right) +\frac{m^2}{2q}-m\left( F_{t_n}+\frac{\beta }{2}\right) -\lambda X_0m+d_{n+1}m^2 \\&\qquad -\,l_{n+1}\kappa e^{-\rho \tau }m^2+{\kappa }^2m^2f_{n+1}e^{-2\rho \tau }-h_{n+1}m^2+s_{n+1}e^{-\rho \tau }m^2\kappa +u_{n+1}m^2\Bigg ). \end{aligned}$$

We then simplify the coefficients.

For $X_{t_n}^2$, we obtain:

$$\begin{aligned}&-\,\lambda w+\frac{w^2}{2q}+d_{n+1}\left( 1-w\right) ^2+\kappa l_{n+1}e^{-\rho \tau }w\left( 1-w\right) +{\kappa }^2f_{n+1}e^{-2\rho \tau }w^2 \\&\qquad +\,h_{n+1}\left( 1-w\right) +\kappa s_{n+1}e^{-\rho \tau }w^2+u_{n+1}w^2 \\&\quad = d_{n+1}+w\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) \\&\qquad +\,w^2\left( \frac{1}{2q}+d_{n+1}-\kappa l_{n+1}e^{-\rho \tau }+{\kappa }^2f_{n+1}e^{-2\rho \tau }-h_{n+1}+\kappa s_{n+1}e^{-\rho \tau }+u_{n+1}\!\!\right) \\&\quad = d_{n+1}-\frac{1}{2}\delta _{n+1}^{-1}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) ^2 \\&\qquad +\,\frac{1}{4}\delta _{n+1}^{-2}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) ^2\delta _{n+1} \\&\quad =d_{n+1}-\frac{1}{4}\delta _{n+1}^{-1}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) ^2. \end{aligned}$$

For $D_{t_n}^2$, we obtain:

$$\begin{aligned}&v+\frac{v^2}{2q}+d_{n+1}v^2-l_{n+1}e^{-\rho \tau }v\left( 1+\kappa v\right) +f_{n+1}e^{-2\rho \tau }\left( 1+\kappa v\right) ^2-h_{n+1}v^2 \\&\qquad +\,s_{n+1}e^{-\rho \tau }v\left( 1+\kappa v\right) +u_{n+1}v^2 \\&\quad = f_{n+1}e^{-2\rho \tau }+v\left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \\&\qquad +\,v^2\left( \frac{1}{2q}+d_{n+1}-\kappa l_{n+1}e^{-\rho \tau }+{\kappa }^2f_{n+1}e^{-2\rho \tau }-h_{n+1}+\kappa s_{n+1}e^{-\rho \tau }+u_{n+1}\right) \\&\quad = f_{n+1}e^{-2\rho \tau }-\frac{1}{2}\delta _{n+1}^{-1}\left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) ^2\\&\qquad +\,\frac{1}{4}\delta _{n+1}^{-2}\left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) ^2\delta _{n+1}\\&\quad = f_{n+1}e^{-2\rho \tau }-\frac{1}{4}\delta _{n+1}^{-1}\left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) ^2. \end{aligned}$$

For $X_{t_n}D_{t_n}$, we obtain:

$$\begin{aligned}&-\,\lambda v+w+\frac{wv}{2q}-2d_{n+1}(1-w)v-l_{n+1}e^{-\rho \tau }((1-w)(1+\kappa v)-\kappa vw) \\&\qquad +\,2{\kappa }f_{n+1}e^{-2\rho \tau }w(1+\kappa v)+h_{n+1}v(1-w)-h_{n+1}vw+\kappa s_{n+1}e^{-\rho \tau }vw \\&\qquad +\,s_{n+1}e^{-\rho \tau }w(1+\kappa v)+2u_{n+1}wv \\&\quad = l_{n+1}e^{-\rho \tau }+w\left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \\&\qquad +\,v\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) \\&\qquad +\,2vw\left( \frac{1}{2q}+d_{n{+}1}{-}\kappa l_{n+1}e^{-\rho \tau }+{\kappa }^2f_{n+1}e^{-2\rho \tau }-h_{n{+}1}{+}\kappa s_{n+1}e^{-\rho \tau }+u_{n+1}\right) \\&\quad = l_{n+1}e^{-\rho \tau } -\,\frac{1}{2}\delta _{n+1}^{-1}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) \\&\qquad \times \left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \\&\qquad -\,\frac{1}{2}\delta _{n+1}^{-1}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) \\&\qquad \times \left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \\&\qquad +\,2\frac{1}{4}\delta _{n+1}^{-1}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) \\&\qquad \times \left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \\&\quad = l_{n+1}e^{-\rho \tau }-\frac{1}{2}\delta _{n+1}^{-1}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) \\&\qquad \times \left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) . \end{aligned}$$

For $X_{t_n}$, we obtain:

$$\begin{aligned}&w\left( F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}+\lambda X_0\right) -\lambda m+\frac{mw}{q}+\left( 1-w\right) \left( F_{t_n}+\frac{\beta }{2}\right) +\lambda X_0\left( 1-w\right) \\&\qquad -\,2d_{n+1}m\left( 1-w\right) - l_{n+1}e^{-\rho \tau }\left( \kappa m\left( 1-w\right) -\kappa wm\right) +2{\kappa }^2f_{n+1}e^{-2\rho \tau }wm \\&\qquad +\,h_{n+1}m\left( 1-w\right) -h_{n+1}mw+2\kappa s_{n+1}e^{-\rho \tau }mw+2u_{n+1}mw \\&\quad = \left( F_{t_n}+\frac{\beta }{2}+\lambda X_0\right) +w\left( \frac{\alpha x_{t_{n-1}}}{2}\right) +m\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) \\&\qquad +\,2wm\left( \frac{1}{2q}+d_{n+1}-\kappa l_{n+1}e^{-\rho \tau }+{\kappa }^2f_{n+1}e^{-2\rho \tau }-h_{n+1}+\kappa s_{n+1}e^{-\rho \tau }+u_{n+1}\right) \\&\quad = \left( F_{t_n}+\frac{\beta }{2}+\lambda X_0\right) -\frac{1}{2}\delta _{n+1}^{-1}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) \left( \frac{\alpha x_{t_{n-1}}}{2}\right) \\&\qquad -\,\frac{1}{2}\delta _{n+1}^{-1}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) \left( \frac{\alpha x_{t_{n-1}}}{2}\right) \\&\qquad +\,2\frac{1}{4}\delta _{n+1}^{-2}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) \left( \frac{\alpha x_{t_{n-1}}}{2}\right) \delta _{n+1} \\&\quad = \left( F_{t_n}+\frac{\beta }{2}+\lambda X_0\right) -\frac{1}{2}\delta _{n+1}^{-1}\left( -\lambda -2d_{n+1}+\kappa l_{n+1}e^{-\rho \tau }+h_{n+1}\right) \left( \frac{\alpha x_{t_{n-1}}}{2}\right) . \end{aligned}$$

For $D_{t_n}$, we obtain:

$$\begin{aligned}&v\left( F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}+\lambda X_0\right) +m+\frac{mw}{q}-v\left( F_{t_n}+\frac{\beta }{2}\right) -\lambda X_0v+2d_{n+1}mv \\&\quad +\,l_{n+1}e^{-\rho \tau }\left( -\left( 1+\kappa v\right) m-\kappa vm\right) +2{\kappa }f_{n+1}e^{-2\rho \tau }\left( 1+\kappa v\right) m-2h_{n+1}mv-h_{n+1}mw\\&\quad +\,s_{n+1}e^{-\rho \tau }m\left( 1+\kappa v\right) +s_{n+1}e^{-\rho \tau }m\kappa v+2u_{n+1}mv \\&\quad = v\left( \frac{\alpha x_{t_{n-1}}}{2}\right) +m\left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \\&\qquad +\,2vm\left( \frac{1}{2q}+d_{n+1}-\kappa l_{n+1}e^{-\rho \tau }+{\kappa }^2f_{n+1}e^{-2\rho \tau }-h_{n+1}+\kappa s_{n+1}e^{-\rho \tau }+u_{n+1}\right) \\&\quad = -\frac{1}{2}\delta _{n+1}^{-1}\left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \left( \frac{\alpha x_{t_{n-1}}}{2}\right) \\&\qquad -\,\frac{1}{2}\delta _{n+1}^{-1}\left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \left( \frac{\alpha x_{t_{n-1}}}{2}\right) \\&\quad +\,2\frac{1}{4}\delta _{n+1}^{-2}\left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \left( \frac{\alpha x_{t_{n-1}}}{2}\right) \delta _{n+1} \\&\quad = -\frac{1}{2}\delta _{n+1}^{-1}\left( 1-l_{n+1}e^{-\rho \tau }+2{\kappa }f_{n+1}e^{-2\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \left( \frac{\alpha x_{t_{n-1}}}{2}\right) . \end{aligned}$$

Without $X_{t_n}$ or $D_{t_n}$, we obtain:

$$\begin{aligned}&m\left( F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}+\lambda X_0\right) +\frac{m^2}{2q}-m\left( F_{t_n}+\frac{\beta }{2}\right) -\lambda X_0m+d_{n+1}m^2 \\&\qquad -\,l_{n+1}\kappa e^{-\rho \tau }m^2+{\kappa }^2m^2f_{n+1}e^{-2\rho \tau }-h_{n+1}m^2+s_{n+1}e^{-\rho \tau }m^2\kappa +u_{n+1}m^2\\&\quad = m\left( \frac{\alpha x_{t_{n-1}}}{2}\right) +m^2\Bigg (\frac{1}{2q}+d_{n+1}-\kappa l_{n+1}e^{-\rho \tau }+{\kappa }^2f_{n+1}e^{-2\rho \tau }\\&\qquad -\,h_{n+1}+\kappa s_{n+1}e^{-\rho \tau }+u_{n+1}\Bigg ) \\&\quad = -\frac{1}{2}\delta _{n+1}^{-1}\left( \frac{\alpha x_{t_{n-1}}}{2}\right) ^2+\frac{1}{4}\delta _{n+1}^{-2}\left( \frac{\alpha x_{t_{n-1}}}{2}\right) ^2\delta _{n+1} \\&\quad = -\frac{1}{4}\delta _{n+1}^{-1}\left( \frac{\alpha x_{t_{n-1}}}{2}\right) ^2. \end{aligned}$$

From this we obtain Eq. 2 through Eq. 3. This concludes the proof.

Appendix 2: Proof of Proposition 2

We use induction to prove Proposition 2. For the last period we have:

$$\begin{aligned} J_T\left( X_T,D_T,F_T,T,x_{t_{N-1}}\right)&= \left( F_T+\frac{\alpha x_{t_{N-1}}+\beta }{2}\right) X_T\\&+\,\left[ \lambda (X_0-X_T)+D_T+\frac{X_T}{2q}\right] X_T. \end{aligned}$$

This satisfies Eq. 6 with Eq. 8. Using that Eq. 6 holds true for some $t_{n+1}$ we find for $t_n$:

$$\begin{aligned}&J_{t_n}\left( X_{t_n},D_{t_n},F_{t_n},{t_n},x_{t_{n-1}}\right) \nonumber \\&\quad =\; \mathop {x_{t_n}}\limits ^{min}\Bigg \{ \left[ \left( F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}\right) +\lambda \left( X_0-X_{t_n}\right) +D_{t_n}+\frac{x_{t_n}}{2q}\right] x_{t_n}\nonumber \\&\qquad +\,E_{t_n}J_{t_{n+1}}\left( X_{t_n}-x_{t_n},\left( D_{t_n}+\kappa x_{t_n}\right) e^{-\rho \tau },F_{t_{n+1}},{t_{n+1}},x_{t_{n}}\right) \Bigg \} \nonumber \\&\quad =\; \mathop {x_{t_n}}\limits ^{min}\Bigg \{ \left[ \left( F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}\right) +\lambda \left( X_0-X_{t_n}\right) +D_{t_n}+\frac{x_{t_n}}{2q}\right] x_{t_n}\nonumber \\&\qquad \!+\!\,\left( a\times a^{N-\left( n+1\right) }F_{t_n}\!+\!\frac{\beta }{2}\right) \left( X_{t_n}\!-\!x_{t_n}\right) \!+\!\lambda X_0\left( X_{t_n}\!-\!x_{t_n}\right) \!+\!b_{n+1}\left( X_{t_n}-x_{t_n}\right) ^2 \nonumber \\&\qquad +\,c_{n+1}\left( D_{t_n}+\kappa x_{t_n}\right) ^2e^{-2\rho \tau }{+}d_{n+1}mF_{t_n}^2+g_{n+1}\left( X_{t_n}-x_{t_n}\right) \left( D_{t_n}{+}\kappa x_{t_n}\right) e^{{-}\rho \tau } \nonumber \\&\qquad +\,h_{n+1}\left( X_{t_n}-x_{t_n}\right) aF_{t_n}+l_{n+1}\left( D_{t_n}+\kappa x_{t_n}\right) e^{-\rho \tau }aF_{t_n} \nonumber \\&\qquad +\,v_{n+1}\left( X_{t_n}-x_{t_n}\right) x_{t_n}\!+\!w_{n+1}aF_{t_n}x_{t_n}+s_{n+1}\left( D_{t_n}\!+\!\kappa x_{t_n}\right) e^{-\rho \tau }x_{t_n}+u_{n+1}x_{t_n}^2\Bigg \}.\nonumber \\ \end{aligned}$$

(12)

To obtain the minimum we differentiate Eq. 12 with respect to $x_{t_n}$:

$$\begin{aligned} \frac{\partial J}{\partial x_{t_n}}&= \left( F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}\right) +\lambda \left( X_0-X_{t_n}\right) +D_{t_n}+\frac{x_{t_n}}{q}\\&-\,\left( a^{N-n}F_{t_n}+\frac{\beta }{2}\right) -\lambda X_0-2b_{n+1}\left( X_{t_n}-x_{t_n}\right) \\&+\,2\kappa c_{n+1}\left( D_{t_n}+\kappa x_{t_n}\right) e^{-2\rho \tau }+g_{n+1} e^{-\rho \tau }\left[ \kappa \left( X_{t_n}-x_{t_n}\right) -\left( D_{t_n}+\kappa x_{t_n}\right) \right] \\&-\,h_{n+1}aF_{t_n} +l_{n+1}\kappa e^{-\rho \tau }aF_{t_n}+v_{n+1}\left( X_{t_n}-2x_{t_n}\right) +w_{n+1}aF_{t_n}\\&+\,s_{n+1}\left( D_{t_n}+\kappa 2x_{t_n}\right) e^{-\rho \tau }+2u_{n+1}x_{t_n} \\&= x_{t_n}\Bigg (\frac{1}{q}+2b_{n+1}-2g_{n+1}\kappa e^{-\rho \tau }+c_{n+1}2\kappa ^2 e^{-2\rho \tau }-2v_{n+1}\\&+\,2s_{n+1}\kappa e^{-\rho \tau }+2u_{n+1}\Bigg )+X_{t_n}\left( -\lambda -2b_{n+1}+g_{n+1} e^{-\rho \tau }\kappa +v_{n+1}\right) \\&+\,D_{t_n}\left( 1+c_{n+1}2\kappa e^{-2\rho \tau }-g_{n+1} e^{-\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \\&+\,F_{t_n}\left( 1-a^{N-n}-h_{n+1}a+l_{n+1}\kappa e^{-\rho \tau }a+w_{n+1}a\right) +\frac{\alpha x_{t_{n-1}}}{2}. \end{aligned}$$

Setting $\frac{\partial J}{\partial x_{t_n}}\mathop {=}\limits ^{!}0$ we obtain the optimal choice:

$$\begin{aligned} x_{t_n}=\gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi , \end{aligned}$$

(13)

where

$$\begin{aligned} \gamma&= -\frac{1}{2}\delta _{n+1}\left( 1+c_{n+1}2\kappa e^{-2\rho \tau }-g_{n+1} e^{-\rho \tau }+s_{n+1}e^{-\rho \tau }\right) , \nonumber \\ \varphi&= -\frac{1}{2}\delta _{n+1}\left( -\lambda -2b_{n+1}+g_{n+1} e^{-\rho \tau }\kappa +v_{n+1}\right) , \nonumber \\ \eta&= -\frac{1}{2}\delta _{n+1}\left( 1-a^{N-n}-h_{n+1}a+l_{n+1}\kappa e^{-\rho \tau }a+w_{n+1}a\right) , \nonumber \\ \varpi&= -\frac{1}{2}\delta _{n+1}\left( \frac{\alpha x_{t_{n-1}}}{2}\right) , \nonumber \\ \delta _{n+1}&= \left( \frac{1}{2q}\!+\!b_{n+1}\!-\!g_{n+1}\kappa e^{-\rho \tau }\!+\!c_{n+1}\kappa ^2 e^{-2\rho \tau }-v_{n+1}+s_{n+1}\kappa e^{-\rho \tau }+u_{n+1}\right) ^{-1}.\nonumber \\ \end{aligned}$$

(14)

From this we find, that Eq. 5 holds true. We then obtain:

$$\begin{aligned}&J_{t_n}\left( X_{t_n},D_{t_n},F_{t_n},{t_n},x_{t_{n-1}}\right) \\&\quad =\, \Bigg [\left( F_{t_n}+\frac{\alpha x_{t_{n-1}}+\beta }{2}\right) +\lambda \left( X_0-X_{t_n}\right) +D_{t_n}\\&\qquad +\,\frac{\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) }{2q}\Bigg ]\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \\&\qquad +\,\left( a\times a^{N-\left( n+1\right) }F_{t_n}+\frac{\beta }{2}\right) \Big (X_{t_n}-\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \Big ) \\&\qquad +\,\lambda X_0\Big (X_{t_n}-\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \Big )\\&\qquad +\,b_{n+1}\Big (X_{t_n}-\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \Big )^2 \\&\qquad +\,c_{n+1}\Big (D_{t_n}+\kappa \left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \Big )^2e^{-2\rho \tau }+d_{n+1}mF_{t_n}^2 \\&\qquad +\,g_{n+1}\Big (X_{t_n}-\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \Big )\\&\qquad \times \,\Big (D_{t_n}+\,\kappa \left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \Big )e^{-\rho \tau } \\&\qquad +\,h_{n+1}\Big (X_{t_n}-\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \Big )aF_{t_n}\\&\qquad +\,l_{n+1}\Big (D_{t_n}+\kappa \left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \Big )e^{-\rho \tau }aF_{t_n} \\&\qquad +\,v_{n+1}\Big (X_{t_n}-\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \Big )\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \\&\qquad +\,s_{n+1}\Big (D_{t_n}+\kappa \left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \Big )e^{-\rho \tau }\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) \\&\qquad +\,u_{n+1}\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) ^2+w_{n+1}aF_{t_n}\left( \gamma D_{t_n}+\varphi X_{t_n}+\eta F_{t_n}+\varpi \right) . \end{aligned}$$

We now sort:

$$\begin{aligned}&J_{t_n}\left( X_{t_n},D_{t_n},F_{t_n},{t_n},x_{t_{n-1}}\right) \\&\quad =\, D_{t_n}\Big (\gamma \frac{\beta }{2}+\lambda X_0 \gamma +\frac{2\gamma \varpi }{2q}+\varpi -\gamma \frac{\beta }{2}-\lambda X_0 \gamma +2b_{n+1}\gamma \varpi \\&\qquad +\,2c_{n+1}e^{-2\rho \tau }\left( 1+\kappa \gamma \right) \kappa \varpi -g_{n+1}e^{-\rho \tau }\left( \kappa \varpi \gamma +\varpi \left( 1+\kappa \gamma \right) \right) \\&\qquad -\,v_{n+1}2\gamma \varpi +s_{n+1}e^{-\rho \tau }\left( 2\kappa \gamma \varpi +\varpi \right) +\,2u_{n+1}\gamma \varpi \Big )\\&\qquad +\,F_{t_n}\Big (\eta \frac{\beta }{2}+\lambda X_0 \eta +\frac{2\eta \varpi }{2q}+\varpi -a^{N-n}\varpi -\eta \frac{\beta }{2}-\lambda X_0 \eta +2b_{n+1}\eta \varpi \\&\qquad +\,2c_{n+1}e^{-2\rho \tau }\kappa ^2\eta \varpi -\,2g_{n+1}e^{-\rho \tau }\kappa \eta \varpi -h_{n+1}a\varpi +l_{n+1}e^{-\rho \tau }a\kappa \varpi \\&\qquad -\,v_{n+1}2\eta \varpi +\,2s_{n+1}e^{-\rho \tau }\kappa \eta \varpi +2u_{n+1}\eta \varpi +w_{n+1}a\varpi \Big ) \\&\qquad +\,X_{t_n}\Big (\varphi \frac{\beta }{2}+\lambda X_0\varphi +\frac{2\varphi \varpi }{2q}-\lambda \varpi +\frac{\beta }{2}-\varphi \frac{\beta }{2}+\lambda X_0-\lambda X_0\varphi \\&\qquad -\,2b_{n+1}\big (1-\varphi \big )\varpi +2c_{n+1}e^{-2\rho \tau }\kappa ^2\varphi \varpi +\,g_{n+1}e^{-\rho \tau }\Big (\big (1-\varphi \big )\kappa \varpi -\varpi \kappa \varphi \Big )\\&\qquad -v_{n+1}\big (2\varphi \varpi -\varpi \big ) +\,2s_{n+1}e^{-\rho \tau }\kappa \varphi \varpi +2u_{n+1}\varphi \varpi \Big ) \\&\qquad +\,D_{t_n}^2\Big (\gamma +\frac{\gamma ^2}{2q}+b_{n+1}\gamma ^2+c_{n+1}e^{-2\rho \tau }\big (1+\kappa \gamma \big )^2 -\,g_{n+1}e^{-\rho \tau }\gamma \big (1+\kappa \gamma \big )\\&\qquad -\,v_{n+1}\gamma ^2 +\,s_{n+1}e^{-\rho \tau }\big (\kappa \gamma ^2+\gamma \big )+u_{n+1}\gamma ^2\Big ) \\&\qquad +\,F_{t_n}^2\Big (\eta {+}\frac{\eta ^2}{2q}{-}a^{N-n}\eta +b_{n{+}1}\eta ^2{+}c_{n+1}e^{-2\rho \tau }\kappa ^2\eta ^2+\,d_{n+1}m-g_{n+1}e^{-\rho \tau }\kappa \eta ^2\\&\qquad -\,h_{n+1}a\eta +\,l_{n+1}e^{-\rho \tau }a\kappa \eta -v_{n+1}\eta ^2+s_{n+1}e^{-\rho \tau }\kappa \eta ^2+u_{n+1}\eta ^2+w_{n+1}a\eta \Big ) \\&\qquad +\,X_{t_n}^2\Big (-\lambda \varphi +\frac{\varphi ^2}{2q}+b_{n+1}\left( 1-\varphi \right) ^2+c_{n+1}e^{-2\rho \tau }\kappa ^2\varphi ^2\\&\qquad +\,g_{n+1}e^{-\rho \tau }\left( 1-\varphi \right) \kappa \varphi -v_{n+1}\left( \varphi ^2-\varphi \right) +\,s_{n+1}e^{-\rho \tau }\kappa \varphi ^2+u_{n+1}\varphi ^2\Big )\\&\qquad +\, D_{t_n}F_{t_n}\Big (\gamma +\frac{2\gamma \eta }{2q}+\eta -a^{N-n}\gamma +2b_{n+1}\gamma \eta +2c_{n+1}e^{-2\rho \tau }\left( 1+\kappa \gamma \right) y\kappa \eta \\&\qquad -\,g_{n+1}e^{-\rho \tau }\left( \kappa \eta \gamma +\eta \left( 1+\kappa \gamma \right) \right) -\,h_{n+1}a\gamma +l_{n+1}e^{-\rho \tau }a\left( 1+\kappa \gamma \right) -v_{n+1}2\gamma \eta \\&\qquad +s_{n+1}e^{-\rho \tau }\left( 2\kappa \gamma \eta +\eta \right) +\,2u_{n+1}\gamma \eta +w_{n+1}a\gamma \Big ) \\&\qquad +\,X_{t_n}D_{t_n}\Big (-\lambda \gamma +\frac{2\gamma \varphi }{2q}+\varphi -2b_{n+1}\left( 1-\varphi \right) \gamma +2c_{n+1}e^{-2\rho \tau }\left( 1+\kappa \gamma \right) \kappa \varphi \\&\qquad +\,g_{n+1}e^{-\rho \tau }\left( \left( 1-\varphi \right) \left( 1+\kappa \gamma \right) -\gamma \kappa \varphi \right) -v_{n+1}\left( 2\gamma \varphi -\gamma \right) \\&\qquad +\,s_{n+1}e^{-\rho \tau }\left( 2\kappa \gamma \varphi +\varphi \right) +2u_{n+1}\gamma \varphi \Big )\\&\qquad +\,X_{t_n}F_{t_n}\Big (\varphi +\frac{2\varphi \eta }{2q}-\lambda \eta +a^{N-n}-a^{N-n}\varphi -2b_{n+1}\left( 1-\varphi \right) \eta \\&\qquad +\,2c_{n+1}e^{-2\rho \tau }\kappa ^2\varphi \eta +\,g_{n+1}e^{-\rho \tau }\left( \left( 1-\varphi \right) \kappa \eta -\eta \kappa \varphi \right) +h_{n+1}a\left( 1-\varphi \right) \\&\qquad +\,l_{n+1}e^{-\rho \tau }a\kappa \varphi -v_{n+1}\left( 2\varphi \eta -\eta \right) +\,2s_{n+1}e^{-\rho \tau }\kappa \varphi \eta +2u_{n+1}\varphi \eta +w_{n+1}a\varphi \Big ) \\&\qquad +\,X_{t_n}x_{t_{n-1}}\left( \varphi \frac{\alpha }{2}\right) +D_{t_n}x_{t_{n-1}}\left( \gamma \frac{\alpha }{2}\right) +F_{t_n}x_{t_{n-1}} \left( \eta \frac{\alpha }{2}\right) +x_{t_{n-1}} \left( \varpi \frac{\alpha }{2}\right) \\&\qquad +\Big (\varpi \frac{\beta }{2}+\lambda X_0 \varpi +\frac{\varpi ^2}{2q}-\varpi \frac{\beta }{2}-\lambda X_0 \varpi +b_{n+1}\varpi ^2+c_{n+1}e^{-2\rho \tau }\kappa ^2\varpi ^2\\&\qquad -\,g_{n+1}e^{-\rho \tau }\kappa \varpi ^2 -\,v_{n+1}\varpi ^2+s_{n+1}e^{-\rho \tau }\kappa \varpi ^2+u_{n+1}\varpi ^2\Big ). \end{aligned}$$

For $D_{t_n}^2$, we obtain:

$$\begin{aligned}&\gamma +\frac{\gamma ^2}{2q}+b_{n+1}\gamma ^2+c_{n+1}e^{-2\rho \tau }\left( 1+\kappa \gamma \right) ^2-g_{n+1}e^{-\rho \tau }\gamma \left( 1+\kappa \gamma \right) -v_{n+1}\gamma ^2 \\&\qquad +\,s_{n+1}e^{-\rho \tau }\left( \kappa \gamma ^2+\gamma \right) +u_{n+1}\gamma ^2 \\&\quad =\, \gamma ^2\left( \frac{1}{2q}+b_{n+1}+\kappa ^2c_{n+1}e^{-2\rho \tau }-g_{n+1}\kappa e^{-\rho \tau }-v_{n+1}+s_{n+1}\kappa e^{-\rho \tau }+u_{n+1}\right) \\&\qquad +\,\gamma \left( 1+2\kappa c_{n+1}e^{-2\rho \tau }-g_{n+1}e^{-\rho \tau }+s_{n+1}e^{-\rho \tau }\right) +c_{n+1}e^{-2\rho \tau } \\&\quad =\, c_{n+1}e^{-2\rho \tau }+\gamma ^2\delta _{n+1}^{-1} +\gamma \left( 1+2\kappa c_{n+1}e^{-2\rho \tau }-g_{n+1}e^{-\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \\&\quad =\, c_{n+1}e^{-2\rho \tau }-\frac{1}{4}\delta _{n+1}\left( 1+2\kappa c_{n+1}e^{-2\rho \tau }-g_{n+1}e^{-\rho \tau }+s_{n+1}e^{-\rho \tau }\right) ^2. \end{aligned}$$

For $X_{t_n}^2$, we obtain:

$$\begin{aligned}& -\lambda \varphi +\frac{\varphi ^2}{2q}+b_{n+1}\left( 1-\varphi \right) ^2+c_{n+1}e^{-2\rho \tau }\kappa ^2\varphi ^2+g_{n+1}e^{-\rho \tau }\left( 1-\varphi \right) \kappa \varphi \\&\qquad -v_{n+1}\left( \varphi ^2-\varphi \right) +\,s_{n+1}e^{-\rho \tau }\kappa \varphi ^2+u_{n+1}\varphi ^2 \\&\quad =\, b_{n+1}+\varphi ^2\Big (\frac{1}{2q}+b_{n+1}+c_{n+1}e^{-2\rho \tau }\kappa ^2-g_{n+1}\kappa e^{-\rho \tau }\\&\qquad \quad -\,v_{n+1}+s_{n+1}e^{-\rho \tau }\kappa +u_{n+1}\Big ) +\varphi \Big (-\lambda -2b_{n+1}+g_{n+1}\kappa e^{-\rho \tau }+v_{n+1}\Big )\\&\quad =\, b_{n+1}+\varphi ^2\delta _{n+1}^{-1}+\varphi \left( -\lambda -2b_{n+1}+g_{n+1}\kappa e^{-\rho \tau }+v_{n+1}\right) \\&\quad =\, b_{n+1}-\frac{1}{4}\delta _{n+1}\left( -\lambda -2b_{n+1}+g_{n+1}\kappa e^{-\rho \tau }+v_{n+1}\right) ^2. \end{aligned}$$

For $F_{t_n}^2$, we obtain:

$$\begin{aligned}&\eta +\frac{\eta ^2}{2q}-a^{N-n}\eta +b_{n+1}\eta ^2+c_{n+1}e^{-2\rho \tau }\kappa ^2\eta ^2+d_{n+1}m-g_{n+1}e^{-\rho \tau }\kappa \eta ^2\\&\qquad -\,h_{n+1}a\eta +l_{n+1}e^{-\rho \tau }a\kappa \eta -v_{n+1}\eta ^2+s_{n+1}e^{-\rho \tau }\kappa \eta ^2+u_{n+1}\eta ^2+w_{n+1}a\eta \\&\quad =\, d_{n+1}m+\eta ^2\Bigg (\frac{1}{2q}+b_{n+1}+c_{n+1}e^{-2\rho \tau }\kappa ^2-g_{n+1}e^{-\rho \tau }\kappa -v_{n+1}\\&\qquad +\,s_{n+1}e^{-\rho \tau }\kappa +u_{n+1}\Bigg ) +\eta \left( 1-a^{N-n}-h_{n+1}a+l_{n+1}e^{-\rho \tau }a\kappa +w_{n+1}a\right) \\&\quad =\,d_{n+1}m+\eta ^2\delta _{n+1}^{-1}+\eta \left( 1-a^{N-n}-h_{n+1}a+l_{n+1}e^{-\rho \tau }a\kappa +w_{n+1}a\right) \\&\quad =\,d_{n+1}m-\frac{1}{4}\delta _{n+1}\left( 1-a^{N-n}-h_{n+1}a+l_{n+1}e^{-\rho \tau }a\kappa +w_{n+1}a\right) ^2. \end{aligned}$$

For $F_{t_n}D_{t_n}$, we obtain:

$$\begin{aligned}&\gamma +\frac{2\gamma \eta }{2q}+\eta -a^{N-n}\gamma +2b_{n+1}\gamma \eta +2c_{n+1}e^{-2\rho \tau }\left( 1+\kappa \gamma \right) \kappa \eta \\&\qquad -\,g_{n+1}e^{-\rho \tau }\left( \kappa \eta \gamma +\eta \left( 1+\kappa \gamma \right) \right) -h_{n+1}a\gamma +l_{n+1}e^{-\rho \tau }a\left( 1+\kappa \gamma \right) -v_{n+1}2\gamma \eta \\&\qquad +\,s_{n+1}e^{-\rho \tau }\left( 2\kappa \gamma \eta +\eta \right) +\,2u_{n+1}\gamma \eta +w_{n+1}a\gamma \\&\quad =\,\gamma \left( 1-a^{N-n}-h_{n+1}a+l_{n+1}\kappa e^{-\rho \tau }a+w_{n+1}a\right) \\&\qquad +\,\eta \left( 1+2c_{n+1}\kappa e^{-2\rho \tau }-g_{n+1}e^{-\rho \tau }+s_{n+1}e^{-\rho \tau }\right) \\&\qquad +\,\gamma \eta 2\left( \frac{1}{2q}{+}b_{n{+}1}{+}c_{n+1}\kappa ^2 e^{-2\rho \tau }-g_{n+1}\kappa e^{-\rho \tau }-v_{n{+}1}{+}s_{n{+}1}\kappa e^{-\rho \tau }+u_{n+1}\right) \\&\qquad +\, l_{n+1}e^{-\rho \tau }a \\&\quad =\, l_{n+1}e^{-\rho \tau }a -\,\frac{1}{2}\delta _{n+1}\left( 1-a^{N-n}-h_{n+1}a+l_{n+1}\kappa e^{-\rho \tau }a+w_{n+1}a\right) \\&\qquad \times \left( 1+2c_{n+1}\kappa e^{-2\rho \tau }-\,g_{n+1}e^{-\rho \tau }+s_{n+1}e^{-\rho \tau }\right) . \end{aligned}$$

For $F_{t_n}X_{t_n}$, we obtain:

$$\begin{aligned}&\varphi +\frac{2\varphi \eta }{2q}-\lambda \eta +a^{N-n}-a^{N-n}\varphi -2b_{n+1}\left( 1-\varphi \right) \eta +2c_{n+1}e^{-2\rho \tau }\kappa ^2\varphi \eta \\&\qquad +\,g_{n+1}e^{-\rho \tau }\left( \left( 1-\varphi \right) \kappa \eta -\eta \kappa \varphi \right) +h_{n+1}a\left( 1-\varphi \right) +l_{n+1}e^{-\rho \tau }a\kappa \varphi \\&\qquad -\,v_{n+1}\left( 2\varphi \eta -\eta \right) +2s_{n+1}e^{-\rho \tau }\kappa \varphi \eta +2u_{n+1}\varphi \eta +w_{n+1}a\varphi \\&\quad =\, a^{N-n}+h_{n+1}a -\,\frac{1}{2}\delta _{n+1}\left( -\lambda -2b_{n+1}+g_{n+1} e^{-\rho \tau }\kappa +v_{n+1}\right) \\&\qquad \times \left( 1-a^{N-n}-h_{n+1}a+\,l_{n+1}\kappa e^{-\rho \tau }a+w_{n+1}a\right) . \end{aligned}$$

For $D_{t_n}X_{t_n}$, we obtain:

$$\begin{aligned}&-\lambda \gamma +\frac{2\gamma \varphi }{2q}+\varphi -2b_{n+1}\left( 1-\varphi \right) \gamma +2c_{n+1}e^{-2\rho \tau }\left( 1+\kappa \gamma \right) \kappa \varphi +g_{n+1}e^{-\rho \tau } \\&\qquad \times \, \Big (\left( 1\!-\!\varphi \right) \left( 1\!+\!\kappa \gamma \right) \!-\!\gamma \kappa \varphi \Big )\!-\!v_{n+1}\left( 2\gamma \varphi -\gamma \right) \!+\!s_{n+1}e^{-\rho \tau }\left( 2\kappa \gamma \varphi \!+\!\varphi \right) \!+\!2u_{n+1}\gamma \varphi \\&\quad =\, g_{n+1}e^{-\rho \tau } -\frac{1}{2}\delta _{n+1}\left( -\lambda -2b_{n+1}+g_{n+1} e^{-\rho \tau }\kappa +v_{n+1}\right) \\&\quad \quad \times \left( 1+c_{n+1}2\kappa e^{-2\rho \tau }-\,g_{n+1} e^{-\rho \tau }+s_{n+1}e^{-\rho \tau }\right) . \end{aligned}$$

For $D_{t_n}$ and $D_{t_n}x_{t_{n-1}}$, we obtain:

$$\begin{aligned}&x_{t_{n-1}}\left( \gamma \frac{\alpha }{2}\right) +\gamma \frac{\beta }{2}+\lambda X_0 \gamma +\frac{2\gamma \varpi }{2q}+\varpi -\gamma \frac{\beta }{2}-\lambda X_0 \gamma +2b_{n+1}\gamma \varpi \\&\qquad +\,2c_{n+1}e^{-2\rho \tau }\left( 1+\kappa \gamma \right) \kappa \varpi -g_{n+1}e^{-\rho \tau }\left( \kappa \varpi \gamma +\varpi \left( 1+\kappa \gamma \right) \right) -v_{n+1}2\gamma \varpi \\&\qquad +\,s_{n+1}e^{-\rho \tau }\left( 2\kappa \gamma \varpi +\varpi \right) +2u_{n+1}\gamma \varpi \\&\quad = -\frac{1}{2}\delta _{n+1}\frac{\alpha x_{t_{n-1}}}{2}\left( 1+c_{n+1}2\kappa e^{-2\rho \tau }-g_{n+1} e^{-\rho \tau }+s_{n+1}e^{-\rho \tau }\right) . \end{aligned}$$

For $X_{t_n}$ and $X_{t_n}x_{t_{n-1}}$, we obtain:

$$\begin{aligned}&x_{t_{n-1}} \left( \varphi \frac{\alpha }{2}\right) +\varphi \frac{\beta }{2}+\lambda X_0\varphi +\frac{2\varphi \varpi }{2q}-\lambda \varpi +\frac{\beta }{2}-\varphi \frac{\beta }{2}+\lambda X_0-\lambda X_0\varphi \\&\qquad -\,2b_{n+1}\left( 1-\varphi \right) \varpi +2c_{n+1}e^{-2\rho \tau }\kappa ^2\varphi \varpi +\,g_{n+1}e^{-\rho \tau }\left( \left( 1-\varphi \right) \kappa \varpi -\varpi \kappa \varphi \right) \\&\qquad -\,v_{n+1}\left( 2\varphi \varpi -\varpi \right) +2s_{n+1}e^{-\rho \tau }\kappa \varphi \varpi +2u_{n+1}\varphi \varpi \\&\quad =\, \lambda X_0+\frac{\beta }{2}-\frac{1}{2}\delta _{n+1}\frac{\alpha x_{t_{n-1}}}{2}\left( -\lambda -2b_{n+1}+g_{n+1} e^{-\rho \tau }\kappa +v_{n+1}\right) . \end{aligned}$$

For $F_{t_n}$ and $F_{t_n}x_{t_{n-1}}$, we obtain:

$$\begin{aligned}&x_{t_{n-1}} \left( \eta \frac{\alpha }{2}\right) +\eta \frac{\beta }{2}+\lambda X_0 \eta +\frac{2\eta \varpi }{2q}+\varpi -a^{N-n}\varpi -\eta \frac{\beta }{2}-\lambda X_0 \eta \\&\qquad +\,2b_{n+1}\eta \varpi \!+\!2c_{n+1}e^{-2\rho \tau }\kappa ^2\eta \varpi {-}2g_{n{+}1}e^{-\rho \tau }\kappa \eta \varpi -h_{n+1}a\varpi \!+\!l_{n{+}1}e^{-\rho \tau }a\kappa \varpi \\&\qquad -\,v_{n+1}2\eta \varpi +2s_{n+1}e^{-\rho \tau }\kappa \eta \varpi +2u_{n+1}\eta \varpi +w_{n+1}a\varpi \\&=\, -\frac{1}{2}\delta _{n+1}\frac{\alpha x_{t_{n-1}}}{2}\left( 1-a^{N-n}-h_{n+1}a+l_{n+1}\kappa e^{-\rho \tau }a+w_{n+1}a\right) . \end{aligned}$$

For the remainder, we obtain:

$$\begin{aligned}&\Bigg (\varpi \frac{\beta }{2}+\lambda X_0 \varpi +\frac{\varpi ^2}{2q}-\varpi \frac{\beta }{2}-\lambda X_0 \varpi +b_{n+1}\varpi ^2+c_{n+1}e^{-2\rho \tau }\kappa ^2\varpi ^2\\&\qquad -\,g_{n+1}e^{-\rho \tau }\kappa \varpi ^2-v_{n+1}\varpi ^2+s_{n+1}e^{-\rho \tau }\kappa \varpi ^2+u_{n+1}\varpi ^2\Bigg )+x_{t_{n-1}} \left( \varpi \frac{\alpha }{2}\right) \\&\quad =\, \frac{1}{4}\delta _{n+1}\left( \frac{\alpha ^2 x_{t_{n-1}}^2}{4}\right) -\frac{1}{2}\delta _{n+1}\left( \frac{\alpha ^2 x_{t_{n-1}}^2}{4}\right) \\&\quad = -\frac{1}{4}\delta _{n+1}\left( \frac{\alpha ^2 x_{t_{n-1}}^2}{4}\right) . \end{aligned}$$

This proves Eq. 6 and concludes the proof.