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Optimal Pair–Trade Execution with Generalized Cross–Impact

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Abstract

We examine a discrete–time optimal pair–trade execution problem with generalized cross–impact. This research is an extension of Fukasawa et al. (2020b), which considers the price impact of aggregate random orders posed by small traders with a Markovian dependence. We focus on how a risk–averse large trader optimally executes two correlated assets to maximize his/her expected utility from the terminal wealth over a finite horizon. A Markov decision process modeling constitutes the basis for the formulation of the optimal pair–trade execution problem. Then, under some regularity conditions, the backward induction method of dynamic programming enables us to derive the optimal pair–trade execution strategy and its associated optimal value function. The trading orders of each risky asset posed by small traders do affect the optimal execution volume of both risky assets. Moreover, numerical results with simulation experiments show that the cross–impact affects the optimal execution strategy and a round–trip trade exists for the large trader to utilize a ‘statistical’ arbitrage and to increase his/her expected utility under our model setting of cross–impact.

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Notes

  1. The positive \(q_{t}^{i}\) for \(t \in \{ { 1, \ldots , T } \}\) stand for the acquisition and negative \(q_{t}^{i}\) the liquidation of the risky asset \(i \in \{ { 1, 2 } \}\). This setting allows us to establish a similar setup for a selling problem of a large trader.

  2. This assumption would be inconsistent with the situation observed in a real market. Cartea and Jaimungal (2016a) and Cartea and Jaimungal (2016b) conduct a linear regression of price changes on net order–flow using trading data obtained from Nasdaq to estimate the permanent and temporary price impact. Then they reveal that assuming linear price impact is compatible with the real stock market and that the price impact caused by both buy and sell trades are deemed as the same from a statistical analysis point of view.

  3. In the rest of this paper, we suppose that the two stochastic processes, \(\pmb {\omega }_{t}\) and \(\pmb {\varepsilon }_{t}\) for \(t \in \{ { 1, \ldots , T } \}\) are mutually independent for simplicity.

  4. We also draw the boxplot in the figures. The bold line in the center of the boxplot shows the median of the data. The top end of the box represents the third quartile, and the bottom end of the box represents the first quartile. The upper and lower whiskers are the largest and smallest data points in the range of (1st quartile − 1.5 \(\times \) (3rd quartile − 1.5 \(\times \) (3rd quartile − 1st quartile)) and above (3rd quartile \(+\) 1.5 \(\times \) (3rd quartile − 1st quartile)) and below, respectively. Circles represent data points that are larger or smaller than the whiskers, i.e., outliers.

  5. The feature of the optimal execution volume at maturity is also explained by the same reason in the rest.

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Acknowledgements

The authors would like to thank Dr. Rei Yamamoto for closely examining our preliminary draft and the comments at the Nippon Finance Conference 2020 Autumn. Also, we would like to thank two anonymous referees for their careful reading of earlier versions of our manuscript and their many constructive comments and suggestions.

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Authors and Affiliations

  1. Graduate School of Economics, Osaka University, Osaka, 560-0043, Japan

    Masamitsu Ohnishi

  2. Center for Mathematical Modeling and Data Science, Osaka University, Osaka, 560-0043, Japan

    Masamitsu Ohnishi

  3. Graduate School of Management, Tokyo Metropolitan University, Tokyo, 100-0005, Japan

    Makoto Shimoshimizu

Authors
  1. Masamitsu Ohnishi
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  2. Makoto Shimoshimizu
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Correspondence to Makoto Shimoshimizu.

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This work was supported by Japan Society for the Promotion of Science under KAKENHI [Grant Numbers 17K01255 and 19J10501].

Appendices

Appendix

Proof of Theorem 1

We derive the optimal execution volume \({\varvec{q}}_{t}^{*}\) at time \(t \in \{ { 1, \ldots , T } \}\) by backward induction method of dynamic programming from the maturity T via the following three steps.

Step 1:

From the assumption that the large trader must unwind all the remainder of his/her position at time \(t = T\),

$$\begin{aligned} {\overline{\varvec{Q}}}_{T + 1} = {\overline{\varvec{Q}}}_{T} - {\varvec{q}}_{T} = {\varvec{0}}, \end{aligned}$$
(A.1)

must hold, which yields \({\overline{\varvec{Q}}}_{T} = {\varvec{q}}_{T}\). Then, for \(t = T\),

$$\begin{aligned} V_{T} \big [ {\varvec{s}}_{T} \big ]&= \sup _{{\varvec{q}}_{T} \in {\mathbb {R}}^{2}} {\mathbb {E}} \Big [ V_{T + 1} \big [ {\varvec{s}}_{T + 1} \big ] \Big \vert {\varvec{s}}_{T} \Big ] \nonumber \\&= \sup _{{\varvec{q}}_{T} \in {\mathbb {R}}^{2}} {\mathbb {E}} \Big [ V_{T + 1} \big [ W_{T + 1}, {\varvec{P}}_{T + 1}, {\overline{\varvec{Q}}}_{T + 1}, {\varvec{R}}_{T + 1}, {\varvec{v}}_{T} \big ] \Big \vert W_{T}, {\varvec{P}}_{T}, {\overline{\varvec{Q}}}_{T}, {\varvec{R}}_{T}, {\varvec{v}}_{T - 1} \Big ] \nonumber \\&= \sup _{{\varvec{q}}_{T} \in {\mathbb {R}}^{2}} {\mathbb {E}} \Big [ - \exp \left\{ - \gamma W_{T + 1} \right\} \Big \vert W_{T}, {\varvec{P}}_{T}, {\overline{\varvec{Q}}}_{T}, {\varvec{R}}_{T}, {\varvec{v}}_{T - 1} \Big ] \nonumber \\&= \sup _{{\varvec{q}}_{T} \in {\mathbb {R}}^{2}} {\mathbb {E}} \Big [ - \exp \left\{ - \gamma \big [ W_{T} - \left[ {\varvec{P}}_{T} + \left( \pmb {\varLambda }_{T} {\varvec{q}}_{T} + \pmb {\kappa }_{T} {\varvec{v}}_{T} \right) \right] ^{\top } {\varvec{q}}_{T} \big ] \right\} \nonumber \\&\qquad \Big \vert W_{T}, {\varvec{P}}_{T}, {\overline{\varvec{Q}}}_{T}, {\varvec{R}}_{T}, {\varvec{v}}_{T - 1} \Big ] \nonumber \\&= - \exp \left\{ - \gamma \big [ W_{T} - {\varvec{P}}_{T}^{\top } {\overline{\varvec{Q}}}_{T} - {\overline{\varvec{Q}}}_{T}^{\top } \pmb {\varLambda }_{T} {\overline{\varvec{Q}}}_{T} \big ] \right\} \nonumber \\&\quad \times {\mathbb {E}} \Big [ \exp \left\{ \gamma {\varvec{v}}_{T}^{\top } \pmb {\kappa }_{T}^{\top } {\overline{\varvec{Q}}}_{T} \right\} \Big \vert W_{T}, {\varvec{P}}_{T}, {\overline{\varvec{Q}}}_{T}, {\varvec{R}}_{T}, {\varvec{v}}_{T - 1} \Big ]. \end{aligned}$$
(A.2)

Using the fact that \({\varvec{v}}_{T}^{\top } \pmb {\kappa }_{T}^{\top } {\overline{\varvec{Q}}}_{T} = {\overline{\varvec{Q}}}_{T}^{\top } \pmb {\kappa }_{T} {\varvec{v}}_{T} (\in {\mathbb {R}})\) and the moment–generating function with respect to \({\varvec{v}}_{T}\), we have

$$\begin{aligned}&{\mathbb {E}} \Big [ \exp \left\{ \gamma {\overline{\varvec{Q}}}_{T}^{\top } \pmb {\kappa }_{T} {\varvec{v}}_{T} \right\} \Big \vert W_{T}, {\varvec{P}}_{T}, {\overline{\varvec{Q}}}_{T}, {\varvec{R}}_{T}, {\varvec{v}}_{T - 1} \Big ] \nonumber \\&\quad = \exp \left\{ \gamma {\overline{\varvec{Q}}}_{T}^{\top } \pmb {\kappa }_{T} \left( {\varvec{a}}_{T}^{{\varvec{v}}} + \mathbf{b}_{T}^{{\varvec{v}}} {\varvec{v}}_{T - 1} \right) + \frac{1}{2} \left( \gamma {\overline{\varvec{Q}}}_{T}^{\top } \pmb {\kappa }_{T} \right) \pmb {\varSigma }_{T}^{{\varvec{v}}} \left( \gamma {\overline{\varvec{Q}}}_{T}^{\top } \pmb {\kappa }_{T} \right) ^{\top } \right\} . \end{aligned}$$
(A.3)

Thus, Eq. (A.2) becomes

$$\begin{aligned} V_{T} \big [ {\varvec{s}}_{T} \big ]&= - \exp \left\{ - \gamma \Big [ W_{T} - {\varvec{P}}_{T}^{\top } {\overline{\varvec{Q}}}_{T} - {\overline{\varvec{Q}}}_{T}^{T} \pmb {\varLambda }_{T} {\overline{\varvec{Q}}}_{T} \Big ] \right\} \nonumber \\&\quad \times \exp \left\{ \gamma {\overline{\varvec{Q}}}_{T}^{\top } \pmb {\kappa }_{T} \left( {\varvec{a}}_{T}^{{\varvec{v}}} + \mathbf{b}_{T}^{{\varvec{v}}} {\varvec{v}}_{T - 1} \right) + \frac{1}{2} \left( \gamma {\overline{\varvec{Q}}}_{T}^{\top } \pmb {\kappa }_{T} \right) \pmb {\varSigma }_{T}^{{\varvec{v}}} \left( \gamma {\overline{\varvec{Q}}}_{T}^{\top } \pmb {\kappa }_{T} \right) ^{\top } \right\} \nonumber \\&= - \exp \Big \{ - \gamma \Big [ W_{T} - {\varvec{P}}_{T}^{\top } {\overline{\varvec{Q}}}_{T} + {\overline{\varvec{Q}}}_{T}^{\top } \mathbf{G}_{T} {\overline{\varvec{Q}}}_{T} + {\varvec{H}}_{T}^{\top } {\overline{\varvec{Q}}}_{T} + {\overline{\varvec{Q}}}_{T}^{\top } \mathbf{M}_{T} {\varvec{v}}_{T - 1} \Big ] \Big \}, \end{aligned}$$
(A.4)

where we have used the fact that \({\overline{\varvec{Q}}}_{T}^{\top } \pmb {\kappa }_{T} {\varvec{a}}_{T}^{{\varvec{v}}} = \left( {\varvec{a}}_{T}^{{\varvec{v}}} \right) ^{\top } \pmb {\kappa }_{T}^{\top } {\overline{\varvec{Q}}}_{T}\) and

$$\begin{aligned} \mathbf{G}_{T}&:= - \pmb {\varLambda }_{T} - \frac{1}{2} \gamma \pmb {\kappa }_{T} \pmb {\varSigma }_{T}^{{\varvec{v}}} \pmb {\kappa }_{T}^{\top } (\in {\mathbb {R}}^{2 \times 2}); \nonumber \\ {\varvec{H}}_{T}^{\top }&:= - \left( {\varvec{a}}_{T}^{{\varvec{v}}} \right) ^{\top } \pmb {\kappa }_{T} (\in {\mathbb {R}}^{1 \times 2}); \nonumber \\ \mathbf{M}_{T}&:= - \pmb {\kappa }_{T} \mathbf{b}_{T}^{{\varvec{v}}} (\in {\mathbb {R}}^{2 \times 2}). \end{aligned}$$
(A.5)

Note that \(\mathbf{G}_{T}\) is negative definite (by Assumptions 1 and 2).

Step 2:

For \(t = T - 1\), we have

$$\begin{aligned}&V_{T - 1} \big [ {\varvec{s}}_{T - 1} \big ] \nonumber \\&\quad = \sup _{{\varvec{q}}_{T - 1} \in {\mathbb {R}}^{2}} {\mathbb {E}} \Big [ V_{T} \big [ {\varvec{s}}_{T} \big ] \Big \vert {\varvec{s}}_{T - 1} \Big ] \nonumber \\&\quad = \sup _{{\varvec{q}}_{T - 1} \in {\mathbb {R}}^{2}} {\mathbb {E}} \Big [ - \exp \Big \{ - \gamma \Big [ W_{T} - {\varvec{P}}_{T}^{\top } {\overline{\varvec{Q}}}_{T} + {\overline{\varvec{Q}}}_{T}^{\top } \mathbf{G}_{T} {\overline{\varvec{Q}}}_{T} + {\varvec{H}}_{T}^{\top } {\overline{\varvec{Q}}}_{T}\nonumber \\&\qquad + {\overline{\varvec{Q}}}_{T}^{\top } \mathbf{M}_{T} {\varvec{v}}_{T - 1} \Big ] \Big \} \Big \vert {\varvec{s}}_{T - 1} \Big ] \nonumber \\&\quad = \sup _{{\varvec{q}}_{T - 1} \in {\mathbb {R}}^{2}} {\mathbb {E}} \Big [ -\exp \Big \{ - \gamma \Big [ W_{T - 1} - \big \{ {\varvec{P}}_{T - 1} + \big ( \pmb {\varLambda }_{T - 1} {\varvec{q}}_{T - 1} + \pmb {\kappa }_{T - 1} {\varvec{v}}_{T - 1} \big ) \big \}^{\top } {\varvec{q}}_{T - 1} \nonumber \\&\qquad - \big \{ {\varvec{P}}_{T - 1} - ( 1 - \mathrm {e}^{- \rho } ) {\varvec{R}}_{T - 1} + \left( \mathrm {e}^{- \rho } \mathbf{A}_{T - 1} + \mathbf{B}_{T - 1} \right) \left( \pmb {\varLambda }_{T - 1} {\varvec{q}}_{T - 1} + \pmb {\kappa }_{T - 1} {\varvec{v}}_{T - 1} \right) \nonumber \\&\qquad + \pmb {\varepsilon }_{T - 1} \big \}^{\top } \left( {\overline{\varvec{Q}}}_{T - 1} - {\varvec{q}}_{T - 1} \right) \nonumber \\&\qquad + \left( {\overline{\varvec{Q}}}_{T - 1} - {\varvec{q}}_{T - 1} \right) ^{\top } \mathbf{G}_{T} \left( {\overline{\varvec{Q}}}_{T - 1} - {\varvec{q}}_{T - 1} \right) + {\varvec{H}}_{T}^{\top } \left( {\overline{\varvec{Q}}}_{T - 1} - {\varvec{q}}_{T - 1} \right) \nonumber \\&\qquad + \left( {\overline{\varvec{Q}}}_{T - 1} - {\varvec{q}}_{T - 1} \right) ^{\top } \mathbf{M}_{T} {\varvec{v}}_{T - 1} \Big ] \Big \} \Big \vert {\varvec{s}}_{T - 1} \Big ] \nonumber \\&\quad = \sup _{{\varvec{q}}_{T - 1} \in {\mathbb {R}}^{2}} \Bigg [ - \exp \Big \{ - \gamma \Big [ {\varvec{q}}_{T - 1}^{\top } \left\{ \pmb {\varLambda }_{T - 1}^{\top } \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \right\} - \mathbf{G}_{T} \right\} {\varvec{q}}_{T - 1}\nonumber \\&\qquad + {\overline{\varvec{Q}}}_{T - 1}^{\top } \left\{ \pmb {\varLambda }_{T - 1}^{\top } \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \right\} - 2 \mathbf{G}_{T} \right\} {\varvec{q}}_{T - 1} \nonumber \\&\quad - ( 1 - \mathrm {e}^{-\rho } ) {\varvec{R}}_{T - 1}^{\top } {\varvec{q}}_{T - 1} - {\varvec{H}}_{T - 1}^{\top } {\varvec{q}}_{T - 1} \nonumber \\&\qquad + W_{T - 1} - {\varvec{P}}_{T - 1} {\overline{\varvec{Q}}}_{T - 1} + {\overline{\varvec{Q}}}_{T - 1}^{\top } \mathbf{G}_{T} {\overline{\varvec{Q}}}_{T - 1} + {\varvec{H}}_{T - 1}^{\top } {\overline{\varvec{Q}}}_{T - 1} + ( 1 - \mathrm {e}^{-\rho } ) {\overline{\varvec{Q}}}_{T - 1}^{\top } {\varvec{R}}_{T - 1} \Big ] \Big \} \nonumber \\&\qquad \times {\mathbb {E}} \left[ \exp \Big \{ \gamma \left[ {\varvec{q}}_{T - 1}^{\top } \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \pmb {\kappa }_{T - 1} + \mathbf{M}_{T - 1} \right\} \nonumber \right. \right. \\&\left. \left. \qquad + {\overline{\varvec{Q}}}_{T - 1}^{\top } \left( \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right) \right] {\varvec{v}}_{T - 1} \Big \} \Big \vert {\varvec{s}}_{T - 1} \right] \nonumber \\&\qquad \times {\mathbb {E}} \left[ \exp \Big \{ \gamma \left( {\overline{\varvec{Q}}}_{T - 1} - {\varvec{q}}_{T - 1} \right) \pmb {\varepsilon }_{T - 1} \Big \} \Big \vert {\varvec{s}}_{T - 1} \right] \Bigg ], \end{aligned}$$
(A.6)

where \(\pmb {\varPi }_{T - 1} := \mathrm {e}^{-\rho } \mathbf{A}_{T - 1} + \mathbf{B}_{T - 1}\). Using the moment generating function, we obtain

  1. 1.

    First expectation in Eq. (A.6):

    $$\begin{aligned}&{\mathbb {E}} \left[ \exp \Big \{ \gamma \left[ {\varvec{q}}_{T - 1}^{\top } \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \pmb {\kappa }_{T - 1} + \mathbf{M}_{T - 1} \right\} \nonumber \right. \right. \\&\left. \left. \qquad + {\overline{\varvec{Q}}}_{T - 1}^{\top } \left( \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right) \right] {\varvec{v}}_{T - 1} \Big \} \Big \vert {\varvec{s}}_{T - 1} \right] \nonumber \\&\quad = \exp \Big \{ \gamma \left[ {\varvec{q}}_{T - 1}^{\top } \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \pmb {\kappa }_{T - 1}\nonumber \right. \right. \\&\left. \left. \qquad + \mathbf{M}_{T - 1} \right\} + {\overline{\varvec{Q}}}_{T - 1}^{\top } \left( \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right) \right] \left( {\varvec{a}}_{T - 1}^{{\varvec{v}}} + \mathbf{b}_{T - 1}^{{\varvec{v}}} {\varvec{v}}_{T - 2} \right) \nonumber \\&\qquad + \frac{1}{2} \gamma ^{2} \left[ {\varvec{q}}_{T - 1}^{\top } \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \pmb {\kappa }_{T - 1} + \mathbf{M}_{T - 1} \right\} + {\overline{\varvec{Q}}}_{T - 1}^{\top } \left( \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right) \right] \pmb {\varSigma }_{T - 1}^{{\varvec{v}}} \nonumber \\&\qquad \left[ {\varvec{q}}_{T - 1}^{\top } \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \pmb {\kappa }_{T - 1} + \mathbf{M}_{T - 1} \right\} + {\overline{\varvec{Q}}}_{T - 1}^{\top } \left( \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right) \right] ^{\top } \Big \}; \end{aligned}$$
    (A.7)
  2. 2.

    Second expectation in Eq. (A.6):

    $$\begin{aligned}&{\mathbb {E}} \left[ \exp \Big \{ \gamma \left( {\overline{\varvec{Q}}}_{T - 1} - {\varvec{q}}_{T - 1} \right) \pmb {\varepsilon }_{T - 1} \Big \} \Big \vert {\varvec{s}}_{T - 1} \right] \nonumber \\&\quad = \exp \Big \{ \gamma \left( {\overline{\varvec{Q}}}_{T - 1} - {\varvec{q}}_{T - 1} \right) \pmb {\mu }_{T - 1}^{\pmb {\varepsilon }}\nonumber \\&\qquad + \frac{1}{2} \gamma ^{2} \left( {\overline{\varvec{Q}}}_{T - 1} - {\varvec{q}}_{T - 1} \right) ^{\top } \pmb {\varSigma }_{T - 1}^{\pmb {\varepsilon }} \left( {\overline{\varvec{Q}}}_{T - 1} - {\varvec{q}}_{T - 1} \right) \Big \}. \end{aligned}$$
    (A.8)

Therefore, by substituting Eq. (A.7) and (A.8) into Eq. (A.6) and rearranging the terms, we have

$$\begin{aligned}&V_{T - 1} \big [ {\varvec{s}}_{T - 1} \big ] \nonumber \\&\quad = \sup _{{\varvec{q}}_{T - 1} \in {\mathbb {R}}^{2}} - \exp \Big \{ - \gamma \Big [ - {\varvec{q}}_{T - 1}^{\top } \pmb {\tilde{\varOmega }}_{T - 1} {\varvec{q}}_{T - 1}\nonumber \\&\qquad + \left( {\overline{\varvec{Q}}}_{T - 1}^{\top } \pmb {\varTheta }_{T - 1} + {\varvec{R}}_{T - 1}^{\top } \pmb {\varXi }_{T - 1} + {\varvec{v}}_{T - 2}^{\top } \pmb {\varPhi }_{T - 1} + \pmb {\varPsi }_{T - 1}^{\top } \right) {\varvec{q}}_{T - 1} \nonumber \\&\qquad + W_{T - 1} - {\varvec{P}}_{T - 1}^{\top } {\overline{\varvec{Q}}}_{T - 1}\nonumber \\&\qquad + {\overline{\varvec{Q}}}_{T - 1}^{\top } \left\{ \mathbf{G}_{T}\nonumber \right. \\&\left. \qquad - \frac{1}{2} \gamma \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} \pmb {\varSigma }_{T - 1}^{{\varvec{v}}} \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} ^{\top } - \frac{1}{2} \gamma \pmb {\varSigma }_{T - 1}^{\pmb {\varepsilon }} \right\} {\overline{\varvec{Q}}}_{T - 1} \nonumber \\&\qquad + \left\{ {\varvec{H}}_{T}^{\top } - \left( {\varvec{a}}_{T - 1}^{\mathbf{v}} \right) ^{\top } \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} ^{\top }\nonumber \right. \\&\left. \qquad - \left( \pmb {\mu }_{T - 1}^{\pmb {\varepsilon }} \right) ^{\top } \right\} {\overline{\varvec{Q}}}_{T - 1} + ( 1 - \mathrm {e}^{- \rho } ) {\overline{\varvec{Q}}}_{T - 1}^{\top } {\varvec{R}}_{T - 1} \nonumber \\&\qquad - {\overline{\varvec{Q}}}_{T - 1}^{\top } \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} \mathbf{b}_{T - 1}^{{\varvec{v}}} {\varvec{v}}_{T - 2} \Big ] \Big \}, \end{aligned}$$
(A.9)

with the following relations:

$$\begin{aligned} \pmb {\tilde{\varOmega }}_{T - 1}&:= \frac{1}{2} \left( \pmb {\varOmega }_{T - 1} + \pmb {\varOmega }_{T - 1}^{\top } \right) \nonumber \\&= \frac{1}{2} \pmb {\varLambda }_{T - 1} \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \right\} + \frac{1}{2} \left\{ \pmb {\varLambda }_{T - 1} \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \right\} \right\} ^{\top } - \mathbf{G}_{T} \nonumber \\&\quad + \frac{1}{2} \gamma \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \pmb {\kappa }_{T - 1} + \mathbf{M}_{T} \right\} \pmb {\varSigma }_{T - 1}^{{\varvec{v}}} \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \pmb {\kappa }_{T - 1} + \mathbf{M}_{T} \right\} ^{\top }\nonumber \\&\quad + \frac{1}{2} \gamma \pmb {\varSigma }_{T - 1}^{\pmb {\varepsilon }} \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varTheta }_{T - 1}&:= - \pmb {\varPi }_{T - 1} \pmb {\varLambda }_{T - 1} - 2 \mathbf{G}_{T}\nonumber \\&\quad - \gamma \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} \pmb {\varSigma }_{T - 1}^{{\varvec{v}}} \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \pmb {\kappa }_{T - 1} + \mathbf{M}_{T} \right\} ^{\top } \nonumber \\&\quad + \gamma \pmb {\varSigma }_{T - 1}^{\pmb {\varepsilon }} \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varXi }_{T - 1}&:= - ( 1 - \mathrm {e}^{-\rho } ) \mathbf{I}_{2} \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varPhi }_{T - 1}&:= - \left( \mathbf{b}_{T - 1}^{{\varvec{v}}} \right) ^{\top } \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \pmb {\kappa }_{T - 1} + \mathbf{M}_{T} \right\} ^{\top } \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varPsi }_{T - 1}^{\top }&:= - {\varvec{H}}_{T}^{\top } - \left( {\varvec{a}}_{T - 1}^{{\varvec{v}}} \right) ^{\top } \left\{ \left( \mathbf{I}_{2} - \pmb {\varPi }_{T - 1} \right) \pmb {\kappa }_{T - 1} + \mathbf{M}_{T} \right\} ^{\top } + \left( \pmb {\mu }_{T - 1}^{\pmb {\varepsilon }} \right) ^{\top } \ ( \in {\mathbb {R}}^{1 \times 2} ). \end{aligned}$$
(A.10)

By the assumption 2.1, \(\pmb {\tilde{\varOmega }}_{T - 1}\) becomes a (symmetric and) positive definite matrix. Finding the optimal execution volume \({\varvec{q}}_{T - 1}^{*}\) which attains the supremum of Eq. (A.6) is equivalent to finding the one which yields the maximum of the following function \(K_{T - 1} ( {\varvec{q}}_{T - 1} )\) defined as

$$\begin{aligned}&K_{T - 1} ( {\varvec{q}}_{T - 1} ) := - {\varvec{q}}_{T - 1}^{\top } \pmb {\tilde{\varOmega }}_{T - 1} {\varvec{q}}_{T - 1}\nonumber \\&\quad + \left( {\overline{\varvec{Q}}}_{T - 1}^{\top } \pmb {\varTheta }_{T - 1} + {\varvec{R}}_{T - 1}^{\top } \pmb {\varXi }_{T - 1} + {\varvec{v}}_{T - 2}^{\top } \pmb {\varPhi }_{T - 1} + \pmb {\varPsi }_{T - 1}^{\top } \right) {\varvec{q}}_{T - 1} \nonumber \\&\quad + W_{T - 1} - {\varvec{P}}_{T - 1} {\overline{\varvec{Q}}}_{T - 1} + {\overline{\varvec{Q}}}_{T - 1} \left\{ \mathbf{G}_{T}\nonumber \right. \\&\left. \quad - \frac{1}{2} \gamma \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} \pmb {\varSigma }_{T - 1}^{{\varvec{v}}} \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} ^{\top }\nonumber \right. \\&\left. \quad - \frac{1}{2} \gamma \pmb {\varSigma }_{T - 1}^{\pmb {\varepsilon }} \right\} {\overline{\varvec{Q}}}_{T - 1} \nonumber \\&\quad + \left\{ {\varvec{H}}_{T}^{\top } - \left( {\varvec{a}}_{T - 1}^{{\varvec{v}}} \right) ^{\top } \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} ^{\top } - \left( \pmb {\mu }_{T - 1}^{\pmb {\varepsilon }} \right) ^{\top } \right\} {\overline{\varvec{Q}}}_{T - 1} + ( 1 - \mathrm {e}^{- \rho } ) {\overline{\varvec{Q}}}_{T - 1} {\varvec{R}}_{T - 1} \nonumber \\&\quad + {\overline{\varvec{Q}}}_{T - 1} \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} \mathbf{b}_{T - 1}^{{\varvec{v}}} {\varvec{v}}_{T - 2}, \end{aligned}$$
(A.11)

since both Eq. (A.6) and Eq. (A.11) are concave functions with respect to \({\varvec{q}}_{T - 1}\). Thus, by completing the square of \(K_{T - 1}\), we obtain the optimal execution volume \(\mathbf{q}_{T - 1}^{*}\) as

$$\begin{aligned} {\varvec{q}}_{T - 1}^{*}&= \pmb {\tilde{\varOmega }}_{T - 1}^{-1} \left\{ \pmb {\varTheta }_{T - 1}^{\top } {\overline{\varvec{Q}}}_{T - 1} + \pmb {\varXi }_{T - 1}^{\top } {\varvec{R}}_{T - 1} + \pmb {\varPhi }_{T - 1}^{\top } {\varvec{v}}_{T - 2} + \pmb {\varPsi }_{T - 1} \right\} \nonumber \\&\left( =: {\varvec{a}}_{T - 1} + \mathbf{b}_{T - 1} {\overline{\varvec{Q}}}_{T - 1} + \mathbf{c}_{T - 1} {\varvec{R}}_{T - 1} + \mathbf{d}_{T - 1} {\varvec{v}}_{T - 2} \right) . \end{aligned}$$
(A.12)

Thus, the optimal value function at time \(T - 1\) becomes a functional form as follows:

$$\begin{aligned}&V_{T - 1} \big [ {\varvec{s}}_{T - 1} \big ] \nonumber \\&\quad = - \exp \Big \{ - \gamma \Big [ W_{T - 1} - {\varvec{P}}_{T - 1}^{\top } {\overline{\varvec{Q}}}_{T - 1} + {\overline{\varvec{Q}}}_{T - 1}^{\top } \mathbf{G}_{T - 1} {\overline{\varvec{Q}}}_{T - 1}\nonumber \\&\qquad + {\varvec{H}}_{T - 1}^{\top } {\overline{\varvec{Q}}}_{T - 1} + {\overline{\varvec{Q}}}_{T - 1}^{\top } \mathbf{I}_{T - 1} {\varvec{R}}_{T - 1} + {\varvec{R}}_{T - 1}^{\top } \mathbf{J}_{T - 1} {\varvec{R}}_{T - 1} \nonumber \\&\qquad + {\varvec{L}}_{T - 1}^{\top } {\varvec{R}}_{T - 1} + {\overline{\varvec{Q}}}_{T - 1}^{\top } \mathbf{M}_{T - 1} {\varvec{v}}_{T - 2} + {\varvec{R}}_{T - 1}^{\top } \mathbf{N}_{T - 1} {\varvec{v}}_{T - 2}\nonumber \\&\qquad + {\varvec{v}}_{T - 2}^{\top } \mathbf{X}_{T - 1} {\varvec{v}}_{T - 2} + {\varvec{Y}}_{T - 1}^{\top } {\varvec{v}}_{T - 2} + Z_{T - 1} \Big ] \Big \}, \end{aligned}$$
(A.13)

where

$$\begin{aligned} \mathbf{G}_{T - 1}&:= \mathbf{G}_{T} - \frac{1}{2} \gamma \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} \pmb {\varSigma }_{T - 1}^{{\varvec{v}}} \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} ^{\top }\nonumber \\&\quad - \frac{1}{2} \gamma \pmb {\varSigma }_{T - 1}^{\pmb {\varepsilon }} + \frac{1}{4} \pmb {\varTheta }_{T - 1} \pmb {\tilde{\varOmega }}_{T - 1}^{-1} \pmb {\varTheta }_{T - 1}^{\top }; \nonumber \\ {\varvec{H}}_{T - 1}^{\top }&:= {\varvec{H}}_{T}^{\top } - \left( {\varvec{a}}_{T - 1}^{{\varvec{v}}} \right) ^{\top } \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} ^{\top } - \left( \pmb {\mu }_{T - 1}^{\pmb {\varepsilon }} \right) ^{\top } + \frac{1}{2} \pmb {\varPsi }_{T - 1}^{\top } \pmb {\tilde{\varOmega }}_{T - 1}^{-1} \pmb {\varTheta }_{T - 1}^{\top }; \nonumber \\ \mathbf{I}_{T - 1}&:= ( 1 - \mathrm {e}^{- \rho } ) \mathbf{I}_{2} + \frac{1}{2} \pmb {\varTheta }_{T - 1} \pmb {\tilde{\varOmega }}_{T - 1}^{-1} \pmb {\varXi }_{T - 1}^{\top }; \quad \mathbf{J}_{T - 1}\nonumber \\&:= \frac{1}{4} \pmb {\varXi }_{T - 1} \pmb {\tilde{\varOmega }}_{T - 1}^{-1} \pmb {\varXi }_{T - 1}^{\top }; \quad {\varvec{L}}_{T - 1}^{\top } := \frac{1}{2} \pmb {\varPsi }_{T - 1}^{\top } \pmb {\tilde{\varOmega }}_{T - 1}^{-1} \pmb {\varXi }_{T - 1}^{\top }; \nonumber \\ \mathbf{M}_{T - 1}&:= \left\{ \pmb {\varPi }_{T - 1} \pmb {\kappa }_{T - 1} - \mathbf{M}_{T} \right\} \mathbf{b}_{T - 1}^{{\varvec{v}}} + \frac{1}{2} \pmb {\varTheta }_{T - 1} \pmb {\tilde{\varOmega }}_{T - 1}^{-1} \pmb {\varPhi }_{T - 1}^{\top }; \quad \mathbf{N}_{T - 1}\nonumber \\&:= \frac{1}{2} \pmb {\varXi }_{T - 1} \pmb {\tilde{\varOmega }}_{T - 1}^{-1} \pmb {\varPhi }_{T - 1}^{\top }; \nonumber \\ \mathbf{X}_{T - 1}&:= \frac{1}{4} \pmb {\varPhi }_{T - 1} \pmb {\tilde{\varOmega }}_{T - 1}^{-1} \pmb {\varPhi }_{T - 1}^{\top }; \quad {\varvec{Y}}_{T - 1}^{\top } := \frac{1}{2} \pmb {\varPsi }_{T - 1}^{\top } \pmb {\tilde{\varOmega }}_{T - 1}^{-1} \pmb {\varPhi }_{T - 1}^{\top }; \quad \nonumber \\ Z_{T - 1}&:= \frac{1}{4} \pmb {\varPsi }_{T - 1}^{\top } \pmb {\tilde{\varOmega }}_{T - 1}^{-1} \pmb {\varPsi }_{T - 1}. \end{aligned}$$
(A.14)
Step 3:

For \(t \in \{ { T - 2, \ldots , 2 } \}\), we can assume from the above results that, at time \(t + 1\), the optimal value function has the following functional form:

$$\begin{aligned}&V_{t + 1} \big [ {\varvec{s}}_{t + 1} \big ] \nonumber \\&\quad = - \exp \Big \{ - \gamma \Big [ W_{t + 1} - {\varvec{P}}_{t + 1}^{\top } {\overline{\varvec{Q}}}_{t + 1} + {\overline{\varvec{Q}}}_{t + 1}^{\top } \mathbf{G}_{t + 1} {\overline{\varvec{Q}}}_{t + 1} + {\varvec{H}}_{t + 1}^{\top } {\overline{\varvec{Q}}}_{t + 1}\nonumber \\&\qquad + {\overline{\varvec{Q}}}_{t + 1}^{\top } \mathbf{I}_{t + 1} {\varvec{R}}_{t + 1} + {\varvec{R}}_{t + 1}^{\top } \mathbf{J}_{t + 1} {\varvec{R}}_{t + 1} \nonumber \\&\qquad + {\varvec{L}}_{t + 1}^{\top } {\varvec{R}}_{t + 1} + {\overline{\varvec{Q}}}_{t + 1}^{\top } \mathbf{M}_{t + 1} {\varvec{v}}_{t} + {\varvec{R}}_{t + 1}^{\top } \mathbf{N}_{t + 1} {\varvec{v}}_{t} + {\varvec{v}}_{t}^{\top } \mathbf{X}_{t + 1} {\varvec{v}}_{t} + {\varvec{Y}}_{t + 1}^{\top } {\varvec{v}}_{t} + Z_{t + 1} \Big ] \Big \}. \end{aligned}$$
(A.15)

Then, we can obtain the following calculation by substituting the dynamics of \(W_{t + 1}, {\varvec{P}}_{t + 1}, {\overline{\varvec{Q}}}_{t + 1}, {\varvec{R}}_{ t + 1}, {\varvec{v}}_{t}\) into the equation above:

$$\begin{aligned} V_{t} \big [ {\varvec{s}}_{t} \big ]&= \sup _{{\varvec{q}}_{t} \in {\mathbb {R}}^{2}} - \exp \Big \{ - \gamma \Big [ {\varvec{q}}_{t}^{\top } \left( - \pmb {\varLambda }_{t} + \pmb {\varPi }_{t} \pmb {\varLambda }_{t} + \mathbf{G}_{t + 1}\nonumber \right. \\&\left. \quad - \mathrm {e}^{- \rho } \mathbf{I}_{t + 1} \mathbf{A}_{t} \pmb {\varLambda }_{t} + \mathrm {e}^{- 2 \rho } \pmb {\varLambda }_{t}^{\top } \mathbf{A}_{t}^{\top } \mathbf{J}_{t + 1} \mathbf{A}_{t} \pmb {\varLambda }_{t} \right) {\varvec{q}}_{t} \nonumber \\&\quad + \Big [ {\overline{\varvec{Q}}}_{t}^{\top } \left( - \pmb {\varPi }_{t} \pmb {\varLambda }_{t} - 2 \mathbf{G}_{t + 1} + \mathrm {e}^{- \rho } \mathbf{I}_{t + 1} \mathbf{A}_{t} \pmb {\varLambda }_{t} \right) \nonumber \\&\quad + {\varvec{R}}_{t}^{\top } \left( - ( 1 - \mathrm {e}^{- \rho } ) \mathbf{I}_{2} - \mathrm {e}^{- \rho } \mathbf{I}_{t + 1} + 2 \mathrm {e}^{- 2 \rho } \mathbf{J}_{t + 1} \mathbf{A}_{t} \pmb {\varLambda }_{t} \right) \nonumber \\&\quad + \left( - {\varvec{H}}_{t + 1}^{\top } + \mathrm {e}^{- \rho } {\varvec{L}}_{t + 1}^{\top } \mathbf{A}_{t} \pmb {\varLambda }_{t} \right) \Big ] {\varvec{q}}_{t} + W_{t} - {\varvec{P}}_{t}^{\top } {\overline{\varvec{Q}}}_{t} + {\overline{\varvec{Q}}}_{t}^{\top } \mathbf{G}_{t + 1} {\overline{\varvec{Q}}}_{t} + {\varvec{H}}_{t + 1}^{\top } {\overline{\varvec{Q}}}_{t} \nonumber \\&\quad + {\overline{\varvec{Q}}}_{t}^{\top } \left( ( 1 - \mathrm {e}^{- \rho } ) \mathbf{I}_{2} + \mathrm {e}^{- \rho } \mathbf{I}_{t} \right) {\varvec{R}}_{t} + \mathrm {e}^{- 2 \rho } {\varvec{R}}_{t}^{\top } \mathbf{J}_{t + 1} {\varvec{R}}_{t} + \mathrm {e}^{- \rho } {\varvec{L}}_{t + 1}^{\top } {\varvec{R}}_{t} + Z_{t + 1} \Big ] \Big \} \nonumber \\&\quad \times {\mathbb {E}} \left[ \exp \Big \{ - \gamma \left[ {\varvec{v}}_{t}^{\top } \pmb {\xi }_{t} {\varvec{v}}_{t} + \left( {\varvec{q}}_{t}^{\top } \pmb {\delta }_{t} + {\overline{\varvec{Q}}}_{t} \pmb {\eta }_{t} + {\varvec{R}}_{t}^{\top } \pmb {\theta }_{t} + \pmb {\phi }_{t}^{\top } \right) {\varvec{v}}_{t} \right] \Big \} \Big \vert {\varvec{s}}_{t} \right] \nonumber \\&\quad \times {\mathbb {E}} \left[ \exp \Big \{ \gamma \left( {\overline{\varvec{Q}}}_{t} - {\varvec{q}}_{t} \right) ^{\top } \pmb {\varepsilon }_{t} \Big \} \Big \vert {\varvec{s}}_{t} \right] , \end{aligned}$$
(A.16)

where \(\pmb {\varPi }_{t} := \mathrm {e}^{-\rho } \mathbf{A}_{t} + \mathbf{B}_{t}\) and

$$\begin{aligned} \pmb {\xi }_{t}&:= \mathrm {e}^{-2\rho } \pmb {\kappa }_{t}^{\top } \mathbf{A}_{t} \mathbf{J}_{t + 1} \mathbf{A}_{t} \pmb {\kappa }_{t} + \mathrm {e}^{-\rho } \pmb {\kappa }_{t} \mathbf{A}_{t} \mathbf{N}_{t + 1} + \mathbf{X}_{t + 1} \ (\in {\mathbb {R}}^{2 \times 2}); \\ \pmb {\delta }_{t}&:= - \left( \mathbf{I}_{2} - \pmb {\varPi }_{t} \right) \pmb {\kappa }_{t} - \mathrm {e}^{-\rho } \mathbf{I}_{t + 1} \mathbf{A}_{t} \pmb {\kappa }_{t}\nonumber \\&\quad + 2 \mathrm {e}^{-2\rho } \pmb {\varLambda }_{t} \mathbf{A}_{t} \mathbf{J}_{t + 1} \mathbf{A}_{t} \pmb {\kappa }_{t} - \mathbf{M}_{t + 1} + \mathrm {e}^{-\rho } \pmb {\varLambda }_{t} \mathbf{A}_{t} \mathbf{N}_{t + 1} \ (\in {\mathbb {R}}^{2 \times 2}); \\ \pmb {\eta }_{t}&:= - \pmb {\varPi }_{t} \pmb {\kappa }_{t} + \mathrm {e}^{-\rho } \mathbf{I}_{t + 1} \mathbf{A}_{t} \pmb {\kappa }_{t} + \mathbf{M}_{t + 1} \ (\in {\mathbb {R}}^{2 \times 2}); \\ \pmb {\theta }_{t}&:= 2 \mathrm {e}^{-2\rho } \mathbf{J}_{t + 1} \mathbf{A}_{t} \pmb {\kappa }_{t} + \mathrm {e}^{-\rho } \mathbf{N}_{t + 1} \ (\in {\mathbb {R}}^{2 \times 2}); \\ \pmb {\phi }_{t}^{\top }&:= \mathrm {e}^{-\rho } {\varvec{L}}_{t + 1}^{\top } \mathbf{A}_{t} \pmb {\kappa }_{t} + {\varvec{Y}}_{t + 1}^{\top } \ (\in {\mathbb {R}}^{1 \times 2}). \end{aligned}$$

The direct calculation leads to the following equations:

  1. 1.

    First expectation in Eq. (A.16):

    $$\begin{aligned}&{\mathbb {E}} \left[ \exp \Big \{ - \gamma \left[ {\varvec{v}}_{t}^{\top } \pmb {\xi }_{t} {\varvec{v}}_{t} + \left( {\varvec{q}}_{t}^{\top } \pmb {\delta }_{t} + {\overline{\varvec{Q}}}_{t} \pmb {\eta }_{t} + {\varvec{R}}_{t}^{\top } \pmb {\theta }_{t} + \pmb {\phi }_{t}^{\top } \right) {\varvec{v}}_{t} \right] \Big \} \Big \vert {\varvec{s}}_{t} \right] \nonumber \\&\quad = \frac{\left|\left( \pmb {\varSigma }_{t}^{*} \right) \right|}{\left|\pmb {\varSigma }_{t}^{{\varvec{v}}} \right|} \exp \Big \{ \frac{1}{2} \Big [ \left( {\varvec{a}}_{t}^{{\varvec{v}}} + \mathbf{b}_{t}^{{\varvec{v}}} {\varvec{v}}_{t - 1} \right) ^{\top } \left\{ \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1}\nonumber \right. \\&\left. \qquad - \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \right\} \left( {\varvec{a}}_{t}^{{\varvec{v}}} + \mathbf{b}_{t}^{{\varvec{v}}} {\varvec{v}}_{t - 1} \right) \nonumber \\&\quad - 2 \gamma \left[ {\varvec{q}}_{t}^{\top } \pmb {\delta }_{t} + {\overline{\varvec{Q}}}_{t}^{\top } \pmb {\eta }_{t} + {\varvec{R}}_{t}^{\top } \pmb {\theta }_{t} + \pmb {\phi }_{t}^{\top } \right] \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} ( \pmb {\varSigma }_{t}^{{\varvec{v}}} )^{-1} \left( {\varvec{a}}_{t}^{{\varvec{v}}} + \mathbf{b}_{t}^{{\varvec{v}}} {\varvec{v}}_{t - 1} \right) \nonumber \\&\quad + \gamma ^{2} \left[ {\varvec{q}}_{t}^{\top } \pmb {\delta }_{t} + {\overline{\varvec{Q}}}_{t}^{\top } \pmb {\eta }_{t} + {\varvec{R}}_{t}^{\top } \pmb {\theta }_{t} + \pmb {\phi }_{t}^{\top } \right] \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \left[ {\varvec{q}}_{t}^{\top } \pmb {\delta }_{t} + {\overline{\varvec{Q}}}_{t}^{\top } \pmb {\eta }_{t} + {\varvec{R}}_{t}^{\top } \pmb {\theta }_{t} + \pmb {\phi }_{t}^{\top } \right] ^{\top } \Big ] \Big \}; \end{aligned}$$
    (A.17)
  2. 2.

    Second expectation in Eq. (A.16):

    $$\begin{aligned}&{\mathbb {E}} \left[ \exp \Big \{ \gamma \left( \mathbf{{\overline{Q}}}_{t} - \mathbf{q}_{t} \right) \pmb {\varepsilon }_{t} \Big \} \Big \vert s_{t} \right] \nonumber \\&\quad = \exp \Big \{ \gamma \left( {\overline{\varvec{Q}}}_{t} - {\varvec{q}}_{t} \right) \pmb {\mu }_{t}^{\pmb {\varepsilon }} + \frac{1}{2} \gamma ^{2} \left( {\overline{\varvec{Q}}}_{t} - {\varvec{q}}_{t} \right) ^{\top } \pmb {\varSigma }_{t}^{\pmb {\varepsilon }} \left( {\overline{\varvec{Q}}}_{t} - {\varvec{q}}_{t} \right) \Big \}. \end{aligned}$$
    (A.18)

To derive Eq. (A.17), the following lemma has been used. Although this lemma is a straightforward result, we here note the result as a lemma for this paper to be self–contained.

Lemma 1

For an n–dimensional normally distributed random variable \({\varvec{X}}\) with mean \(\pmb {\mu } \in {\mathbb {R}}^{n}\) and variance \(\pmb {\varSigma } \in {\mathbb {R}}^{n \times n}\) (where \(\pmb {\varSigma }\) is a symmetric positive definite matrix), we have

$$\begin{aligned}&{\mathbb {E}} \Big [ \exp \Big \{ {\varvec{s}}^{\top } {\varvec{X}} + {\varvec{X}}^{\top } \mathbf{V} {\varvec{X}} \Big \} \Big ]\nonumber \\&\quad = \frac{|\left( \pmb {\varSigma }^{*} \right) ^{-1} |^{\frac{1}{2}}}{|\pmb {\varSigma } |^{\frac{1}{2}}} \exp \left\{ \frac{1}{2} \left[ \left( \pmb {\mu }^{*} \right) ^{\top } \pmb {\varSigma }^{-1} \left( \pmb {\varSigma }^{*} \right) ^{-1} \pmb {\varSigma }^{-1} \pmb {\mu }^{*} - \pmb {\mu }^{\top } \pmb {\varSigma }^{-1} \pmb {\mu } \right] \right\} , \end{aligned}$$
(A.19)

where \({\varvec{s}} \in {\mathbb {R}}^{n}\), \(\mathbf{V} \in {\mathbb {R}}^{n \times n}\), \(\pmb {\mu }^{*} := \pmb {\mu } + \pmb {\varSigma } {\varvec{s}}\) and \(\pmb {\varSigma }^{*} := \pmb {\varSigma }^{-1} - 2 \mathbf{V}\), under the assumption that \(\pmb {\varSigma }^{*}\) is a non–singular matrix. Remind that when \(\mathbf{V} = \mathbf{0}\), the result is consistent with the one obtained from the moment generating function.

Proof

We can assume, without loss of generality, that \(\mathbf{V}\) is symmetric, since \({\varvec{X}}^{\top } \mathbf{V} {\varvec{X}} \in {\mathbb {R}}\) we have a symmetric matrix \({\tilde{\mathbf {V}}} (:= \frac{\mathbf{V} + \mathbf{V}^{\top }}{2})\) which satisfies \({\varvec{X}}^{\top } \mathbf{\tilde{V}} {\varvec{X}} = {\varvec{X}}^{\top } \mathbf{V} {\varvec{X}}\) even if \(\mathbf{V}\) is not symmetric. Define \({\varvec{x}} := ( x_{1}, \ldots , x_{n} )^{\top } \in {\mathbb {R}}^{n}\). Then, direct calculation yields

$$\begin{aligned}&{\mathbb {E}} \Big [ \exp \Big \{ {\varvec{s}}^{\top } {\varvec{X}} + {\varvec{X}}^{\top } \mathbf{V} {\varvec{X}} \Big \} \Big ] \nonumber \\&\quad = \frac{1}{\left( 2 \pi \right) ^{\frac{n}{2}} |\pmb {\varSigma } |^{\frac{1}{2}}} \int _{{\mathbb {R}}^{n}}^{} \exp \left\{ - \frac{1}{2} \left[ {\varvec{x}}^{\top } \left( \pmb {\varSigma }^{-1} - 2 \mathbf{V} \right) {\varvec{x}}\nonumber \right. \right. \\&\left. \left. \qquad - 2 \left( \pmb {\mu } + \pmb {\varSigma } {\varvec{s}} \right) ^{\top } \pmb {\varSigma }^{-1} {\varvec{x}} + \pmb {\mu }^{\top } \pmb {\varSigma }^{-1} \pmb {\mu } \right] \right\} \mathrm {d}{\varvec{x}}, \end{aligned}$$
(A.20)

where \(\mathrm {d}{\varvec{x}} := \mathrm {d}x_{1} \cdots \mathrm {d}x_{n}\). If we set \(\pmb {\varSigma }^{*} := \pmb {\varSigma }^{-1} - 2 \mathbf{V}\) and \(\pmb {\mu }^{*} := \pmb {\mu } + \pmb {\varSigma } {\varvec{s}}\), then by the assumption that \(\pmb {\varSigma }^{*} \in {\mathbb {R}}^{n \times n}\) is a non–singular matrix, Eq. (A.20) results in

$$\begin{aligned}&{\mathbb {E}} \Big [ \exp \Big \{ {\varvec{s}}^{\top } {\varvec{X}} + {\varvec{X}}^{\top } \mathbf{V} {\varvec{X}} \Big \} \Big ] \nonumber \\&\quad = \frac{1}{\left( 2 \pi \right) ^{\frac{n}{2}} |\pmb {\varSigma } |^{\frac{1}{2}}} \int _{{\mathbb {R}}^{n}}^{} \exp \left\{ - \frac{1}{2} \left[ {\varvec{x}}^{\top } \pmb {\varSigma }^{*} {\varvec{x}} - 2 \left( \pmb {\mu }^{*} \right) ^{\top } \pmb {\varSigma }^{-1} {\varvec{x}} + \pmb {\mu }^{\top } \pmb {\varSigma }^{-1} \pmb {\mu } \right] \right\} \mathrm {d}{\varvec{x}} \nonumber \\&\quad = \frac{\left( 2 \pi \right) ^{\frac{n}{2}} |\left( \pmb {\varSigma }^{*} \right) ^{-1} |^{\frac{1}{2}}}{\left( 2 \pi \right) ^{\frac{n}{2}} |\pmb {\varSigma } |^{\frac{1}{2}}} \exp \left\{ \frac{1}{2} \left[ \left( \pmb {\mu }^{*} \right) ^{\top } \pmb {\varSigma }^{-1} \left( \pmb {\varSigma }^{*} \right) ^{-1} \pmb {\varSigma }^{-1} \pmb {\mu }^{*} - \pmb {\mu }^{\top } \pmb {\varSigma }^{-1} \pmb {\mu } \right] \right\} \nonumber \\&\qquad \times \underbrace{\int _{{\mathbb {R}}^{n}}^{} \frac{1}{\left( 2 \pi \right) ^{\frac{n}{2}} |\left( \pmb {\varSigma }^{*} \right) ^{-1} |^{\frac{1}{2}}} \exp \left\{ - \frac{1}{2} \left( {\varvec{x}}^{\top } - \left( \pmb {\varSigma }^{*} \right) ^{-1} \pmb {\varSigma }^{-1} \pmb {\mu } \right) \left\{ \left( \pmb {\varSigma }^{*} \right) ^{-1} \right\} ^{-1} \left( {\varvec{x}}^{\top } - \left( \pmb {\varSigma }^{*} \right) ^{-1} \pmb {\varSigma }^{-1} \pmb {\mu } \right) \right\} \mathrm {d}{\varvec{x}}}_{= 1} \nonumber \\&\quad = \frac{|\left( \pmb {\varSigma }^{*} \right) ^{-1} |^{\frac{1}{2}}}{|\pmb {\varSigma } |^{\frac{1}{2}}} \exp \left\{ \frac{1}{2} \left[ \left( \pmb {\mu }^{*} \right) ^{\top } \pmb {\varSigma }^{-1} \left( \pmb {\varSigma }^{*} \right) ^{-1} \pmb {\varSigma }^{-1} \pmb {\mu }^{*} - \pmb {\mu }^{\top } \pmb {\varSigma }^{-1} \pmb {\mu } \right] \right\} . \end{aligned}$$
(A.21)

\(\square \)

Eq. (A.21) can be rewritten as follows:

$$\begin{aligned}&\frac{|\left( \pmb {\varSigma }^{*} \right) ^{-1} |^{\frac{1}{2}}}{|\pmb {\varSigma } |^{\frac{1}{2}}} \exp \left\{ \frac{1}{2} \left[ \left( \pmb {\mu }^{*} \right) ^{\top } \pmb {\varSigma }^{-1} \left( \pmb {\varSigma }^{*} \right) ^{-1} \pmb {\varSigma }^{-1} \pmb {\mu }^{*} - \pmb {\mu }^{\top } \pmb {\varSigma }^{-1} \pmb {\mu } \right] \right\} \nonumber \\&\quad = \frac{\left|\left( \pmb {\varSigma }^{*} \right) ^{-1} \right|}{\left|\pmb {\varSigma } \right|} \exp \left\{ \frac{1}{2} \left[ \pmb {\mu }^{\top } \left\{ \pmb {\varSigma } \left( \pmb {\varSigma }^{*} \right) ^{-1} \pmb {\varSigma }^{-1} - \pmb {\varSigma }^{-1} \right\} \pmb {\mu }\nonumber \right. \right. \\&\left. \left. \quad + 2 {\varvec{s}}^{\top } \left( \pmb {\varSigma }^{*} \right) ^{-1} ( \pmb {\varSigma } )^{-1} \pmb {\mu } + {\varvec{s}}^{\top } \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} {\varvec{s}} \right] \right\} . \end{aligned}$$
(A.22)

Therefore, setting \({\varvec{s}}^{\top } := - \gamma \left[ {\varvec{q}}_{t}^{\top } \pmb {\delta }_{t} + {\overline{\varvec{Q}}}_{t}^{\top } \pmb {\eta }_{t} + {\varvec{R}}_{t}^{\top } \pmb {\theta }_{t} + \pmb {\phi }_{t}^{\top } \right] \) and \(\mathbf{V} := - \frac{\gamma }{2} \left( \pmb {\xi }_{t} + \pmb {\xi }_{t}^{\top } \right) \) in Eq. (A.21) yields Eq. (A.17).

Substituting Eq. (A.17) and (A.18) into Eq. (A.16) and rearranging results in

$$\begin{aligned}&V_{t} \big [ {\varvec{s}}_{t} \big ]\nonumber \\&\quad = \sup _{{\varvec{q}}_{t} \in {\mathbb {R}}^{2}} - \exp \Big \{ - \gamma \Big [- {\varvec{q}}_{t}^{\top } \pmb {\tilde{\varOmega }}_{t} {\varvec{q}}_{t}\nonumber \\&\qquad + \Big [ {\overline{\varvec{Q}}}_{t}^{\top } \pmb {\varTheta }_{t} + {\varvec{R}}_{t}^{\top } \pmb {\varXi }_{t} + {\varvec{v}}_{t - 1}^{\top } \pmb {\varPhi }_{t} + \pmb {\varPsi }_{t}^{\top } \Big ] {\varvec{q}}_{t} + W_{t} - {\varvec{P}}_{t}^{\top } {\overline{\varvec{Q}}}_{t} \nonumber \\&\qquad + {\overline{\varvec{Q}}}_{t}^{\top } \Big [ \mathbf{G}_{t + 1} - \frac{1}{2} \gamma \pmb {\eta }_{t} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\eta }_{t}^{\top } - \frac{1}{2} \gamma \pmb {\varSigma }_{t}^{\pmb {\varepsilon }} \Big ] {\overline{\varvec{Q}}}_{t} + \Big [ {\varvec{H}}_{t + 1}^{\top } + \left( {\varvec{a}}_{t}^{{\varvec{v}}} \right) ^{\top } \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\eta }_{t}^{\top } \nonumber \\&\qquad - \gamma \pmb {\phi }_{t}^{\top } \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\eta }_{t}^{\top } - \left( \pmb {\mu }_{t}^{\pmb {\varepsilon }} \right) ^{\top } \Big ] {\overline{\varvec{Q}}}_{t} + {\overline{\varvec{Q}}}_{t}^{\top } \Big [ ( 1 - \mathrm {e}^{- \rho } ) \mathbf{I}_{2} + \mathrm {e}^{- \rho } \mathbf{I}_{t} - \gamma \pmb {\eta }_{t} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\theta }_{t}^{\top } \Big ] {\varvec{R}}_{t} \nonumber \\&\qquad + {\varvec{R}}_{t}^{\top } \Big [ \mathrm {e}^{- 2 \rho } \mathbf{J}_{t + 1} - \frac{1}{2} \gamma \pmb {\theta }_{t} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\theta }_{t}^{\top } \Big ] {\varvec{R}}_{t}\nonumber \\&\qquad + \Big [ \mathrm {e}^{- \rho } {\varvec{L}}_{t + 1}^{\top } + \left( {\varvec{a}}_{t}^{{\varvec{v}}} \right) ^{\top } \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\theta }_{t}^{\top } - \gamma \left( \pmb {\phi }_{t} \right) ^{\top } \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\theta }_{t}^{\top } \Big ] {\varvec{R}}_{t} \nonumber \\&\qquad + {\overline{\varvec{Q}}}_{t}^{\top } \pmb {\eta }_{t} \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \mathbf{b}_{t}^{{\varvec{v}}} {\varvec{v}}_{t - 1} +{\varvec{R}}_{t}^{\top } \pmb {\theta }_{t} \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \mathbf{b}_{t}^{{\varvec{v}}} {\varvec{v}}_{t - 1}\nonumber \\&\quad - \frac{1}{2 \gamma } {\varvec{v}}_{t - 1}^{\top } \left( \mathbf{b}_{t}^{{\varvec{v}}} \right) ^{\top } \pmb {\varSigma }_{t}^{**} \mathbf{b}_{t}^{{\varvec{v}}} {\varvec{v}}_{t - 1} \nonumber \\&\qquad + \Big [ - \frac{1}{\gamma } \left( {\varvec{a}}_{t}^{{\varvec{v}}} \right) ^{\top } \pmb {\varSigma }_{t}^{**} \mathbf{b}_{t}^{{\varvec{v}}} + \pmb {\phi }_{t}^{\top } \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \mathbf{b}_{t}^{{\varvec{v}}} \Big ] {\varvec{v}}_{t - 1} + Z_{t + 1} - \frac{1}{2 \gamma } \left( {\varvec{a}}_{t}^{{\varvec{v}}} \right) ^{\top } \pmb {\varSigma }_{t}^{**} {\varvec{a}}_{t}^{{\varvec{v}}} \nonumber \\&\qquad + \left( {\varvec{a}}_{t}^{{\varvec{v}}} \right) ^{\top } \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\phi }_{t} - \frac{1}{2 \gamma } \pmb {\phi }_{t}^{\top } \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\phi }_{t} + x_{t} \Big ] \Big \}, \end{aligned}$$
(A.23)

where \(\pmb {\varSigma }_{t}^{*} := \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} + \gamma \left( \pmb {\xi }_{t} + \pmb {\xi }_{t}^{\top } \right) \), \(\pmb {\varSigma }_{t}^{**} := \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} - \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1}\), \(\displaystyle {x_{t} := \log \frac{\left|\left( \pmb {\varSigma }_{t}^{*} \right) \right|}{\left|\pmb {\varSigma }_{t}^{{\varvec{v}}} \right|}}\), and

$$\begin{aligned} \pmb {\tilde{\varOmega }}_{t}&:= \frac{1}{2} \left( \pmb {\varOmega }_{t} + \pmb {\varOmega }_{t}^{\top } \right) \nonumber \\&= \frac{1}{2} \pmb {\varLambda }_{t} \left( \mathbf{I}_{2} - \pmb {\varPi }_{t} \right) + \frac{1}{2} \left\{ \pmb {\varLambda }_{t} \left( \mathbf{I}_{2} - \pmb {\varPi }_{t} \right) \right\} ^{\top } - \mathbf{G}_{t + 1} + \frac{1}{2} \mathrm {e}^{-\rho } \mathbf{I}_{t + 1} \mathbf{A}_{t} \pmb {\varLambda }_{t} + \frac{1}{2} \left\{ \mathrm {e}^{-\rho } \mathbf{I}_{t + 1} \mathbf{A}_{t} \pmb {\varLambda }_{t} \right\} ^{\top } \nonumber \\&\quad - \mathrm {e}^{-2\rho } \pmb {\varLambda }_{t} \mathbf{A}_{t} \mathbf{J}_{t + 1} \mathbf{A}_{t} \pmb {\varLambda }_{t} + \frac{1}{2} \gamma \pmb {\delta }_{t} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\delta }_{t}^{\top } + \frac{1}{2} \gamma \pmb {\varSigma }_{t}^{\pmb {\varepsilon }} \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varTheta }_{t}&:= - \pmb {\varPi }_{t} \pmb {\varLambda }_{t} - 2 \mathbf{G}_{t + 1} + \mathrm {e}^{-\rho } \mathbf{I}_{t + 1} \mathbf{A}_{t} \pmb {\varLambda }_{t} - \gamma \pmb {\eta }_{t}\left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\delta }_{t}^{\top } + \gamma \pmb {\varSigma }_{t}^{\pmb {\varepsilon }} \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varXi }_{t}&:= - ( 1 - \mathrm {e}^{-\rho } ) \mathbf{I}_{2} - \mathrm {e}^{-\rho } \mathbf{I}_{t + 1} + 2 \mathrm {e}^{- 2\rho } \mathbf{J}_{t + 1} \mathbf{A}_{t} \pmb {\varLambda }_{t} - \gamma \pmb {\theta }_{t} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\delta }_{t}^{\top } \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varPhi }_{t}&:= \left( \mathbf{b}_{t}^{{\varvec{v}}} \right) ^{\top } \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\delta }_{t}^{\top } \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varPsi }_{t}^{\top }&:= - {\varvec{H}}_{t + 1}^{\top } + \mathrm {e}^{-\rho } {\varvec{L}}_{t + 1}^{\top } \mathbf{A}_{t} \pmb {\varLambda }_{t} + \left( {\varvec{a}}_{t}^{{\varvec{v}}} \right) ^{\top } \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\delta }_{t}^{\top }\nonumber \\&\quad - \gamma \pmb {\phi }_{t}^{\top } \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\delta }_{t}^{\top } + \left( \pmb {\mu }_{t}^{\pmb {\varepsilon }} \right) ^{\top } \ ( \in {\mathbb {R}}^{1 \times 2} ). \end{aligned}$$
(A.24)

Thus, under the following assumptions (or regularity conditions):

  1. 1.

    \(\pmb {\varSigma }_{t}^{*} := \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} + \gamma \left( \pmb {\xi }_{t} + \pmb {\xi }_{t}^{\top } \right) \) is non-singular;

  2. 2.

    \(\pmb {\tilde{\varOmega }}_{t} := \frac{1}{2} \left( \pmb {\varOmega }_{t} + \pmb {\varOmega }_{t}^{\top } \right) \) is positive (semi)definite,

we can derive the optimal execution volume at time t via the same method used in step 2 and obtain as follows:

$$\begin{aligned} {\varvec{q}}_{t}^{*}&= \pmb {\tilde{\varOmega }}_{t}^{-1} \left\{ \pmb {\varTheta }_{t}^{\top } {\overline{\varvec{Q}}}_{t} + \pmb {\varXi }_{t}^{\top } {\varvec{R}}_{t} + \pmb {\varPhi }_{t}^{\top } {\varvec{v}}_{t - 1} + \pmb {\varPsi }_{t} \right\} \nonumber \\&\quad \left( =: {\varvec{a}}_{t} + \mathbf{b}_{t} {\overline{\varvec{Q}}}_{t} + \mathbf{c}_{t} {\varvec{R}}_{t} + \mathbf{d}_{t} {\varvec{v}}_{t - 1} \right) . \end{aligned}$$
(A.25)

Finally, by substituting this into Eq. (A.23) the optimal value function becomes

$$\begin{aligned} V_{t} \big [ {\varvec{s}}_{t} \big ]&= - \exp \Big \{ - \gamma \Big [ W_{t} - {\varvec{P}}_{t}^{\top } {\overline{\varvec{Q}}}_{t} + {\overline{\varvec{Q}}}_{t}^{\top } \mathbf{G}_{t} {\overline{\varvec{Q}}}_{t} + {\varvec{H}}_{t}^{\top } {\overline{\varvec{Q}}}_{t}\nonumber \\&\quad + {\overline{\varvec{Q}}}_{t}^{\top } \mathbf{I}_{t} {\varvec{R}}_{t} + {\varvec{R}}_{t}^{\top } \mathbf{J}_{t} {\varvec{R}}_{t} + {\varvec{L}}_{t}^{\top } {\varvec{R}}_{t} \nonumber \\&\quad + {\overline{\varvec{Q}}}_{t}^{\top } \mathbf{M}_{t} {\varvec{v}}_{t - 1} + {\varvec{R}}_{t}^{\top } \mathbf{N}_{t} {\varvec{v}}_{t - 1} + {\varvec{v}}_{t - 1}^{\top } \mathbf{X}_{t} {\varvec{v}}_{t - 1} + {\varvec{Y}}_{t}^{\top } {\varvec{v}}_{t - 1} + Z_{t} \Big ] \Big \}, \end{aligned}$$
(A.26)

where

$$\begin{aligned} \mathbf{G}_{t}&:= \mathbf{G}_{t + 1} - \frac{1}{2} \gamma \pmb {\eta }_{t} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\eta }_{t}^{\top } - \frac{1}{2} \gamma \pmb {\varSigma }_{t}^{\pmb {\varepsilon }} + \frac{1}{4} \pmb {\varTheta }_{t} \pmb {\tilde{\varOmega }}_{t}^{-1} \pmb {\varTheta }_{t}^{\top }; \nonumber \\ {\varvec{H}}_{t}^{\top }&:= {\varvec{H}}_{t + 1}^{\top } + \left( {\varvec{a}}_{t}^{{\varvec{v}}} \right) ^{\top } \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\eta }_{t}^{\top } - \gamma \pmb {\varphi }_{t}^{\top } \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\eta }_{t}^{\top } - \left( \pmb {\mu }_{t}^{\pmb {\varepsilon }} \right) ^{\top }\nonumber \\&\quad + \frac{1}{2} \pmb {\varPsi }_{t}^{\top } \pmb {\tilde{\varOmega }}_{t}^{-1} \pmb {\varTheta }_{t}^{\top }; \nonumber \\ \mathbf{I}_{t}&:= ( 1 - \mathrm {e}^{- \rho } ) \mathbf{I}_{2} + \mathrm {e}^{- \rho } \mathbf{I}_{t + 1} - \gamma \pmb {\eta }_{t} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\theta }_{t}^{\top } + \frac{1}{2} \pmb {\varTheta }_{t} \pmb {\tilde{\varOmega }}_{t}^{-1} \pmb {\varXi }_{t}^{\top }; \nonumber \\ \mathbf{J}_{t}&:= \mathrm {e}^{- 2 \rho } \mathbf{J}_{t + 1} - \frac{1}{2} \gamma \pmb {\theta }_{t} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\theta }_{t}^{\top } + \frac{1}{4} \pmb {\varXi }_{t} \pmb {\tilde{\varOmega }}_{t}^{-1} \pmb {\varXi }_{t}^{\top }; \nonumber \\ {\varvec{L}}_{t}^{\top }&:= \mathrm {e}^{- \rho } {\varvec{L}}_{t + 1}^{\top } + \left( {\varvec{a}}_{t}^{{\varvec{v}}} \right) ^{\top } \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\theta }_{t}^{\top } - \gamma \left( \pmb {\phi }_{t} \right) ^{\top } \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\theta }_{t}^{\top }\nonumber \\&\quad + \frac{1}{2} \pmb {\varPsi }_{t}^{\top } \pmb {\tilde{\varOmega }}_{t}^{-1} \pmb {\varXi }_{t}^{\top }; \nonumber \\ \mathbf{M}_{t}&:= \pmb {\eta }_{t} \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \mathbf{b}_{t}^{{\varvec{v}}} + \frac{1}{2} \pmb {\varTheta }_{t} \pmb {\tilde{\varOmega }}_{t}^{-1} \pmb {\varPhi }_{t}^{\top }; \nonumber \\ \mathbf{N}_{t}&:= \pmb {\theta }_{t} \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \mathbf{b}_{t}^{{\varvec{v}}} +\frac{1}{2} \pmb {\varXi }_{t} \pmb {\tilde{\varOmega }}_{t}^{-1} \pmb {\varPhi }_{t}^{\top }; \nonumber \\ \mathbf{X}_{t}&:= - \frac{1}{2 \gamma } \left( \mathbf{b}_{t}^{{\varvec{v}}} \right) ^{\top } \pmb {\varSigma }_{t}^{**} \mathbf{b}_{t}^{{\varvec{v}}} + \frac{1}{4} \pmb {\varPhi }_{t} \pmb {\tilde{\varOmega }}_{t}^{-1} \pmb {\varPhi }_{t}^{\top }; \nonumber \\ {\varvec{Y}}_{t}^{\top }&:= - \frac{1}{\gamma } \left( {\varvec{a}}_{t}^{{\varvec{v}}} \right) ^{\top } \pmb {\varSigma }_{t}^{**} \mathbf{b}_{t}^{{\varvec{v}}} + \pmb {\phi }_{t}^{\top } \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \mathbf{b}_{t}^{{\varvec{v}}} + \frac{1}{2} \pmb {\varPsi }_{t}^{\top } \pmb {\tilde{\varOmega }}_{t}^{-1} \pmb {\varPhi }_{t}^{\top }; \nonumber \\ Z_{t}&:= Z_{t + 1} - \frac{1}{2 \gamma } \left( {\varvec{a}}_{t}^{{\varvec{v}}} \right) ^{\top } \pmb {\varSigma }_{t}^{**} {\varvec{a}}_{t}^{{\varvec{v}}} + \left( {\varvec{a}}_{t}^{{\varvec{v}}} \right) ^{\top } \left( \pmb {\varSigma }_{t}^{{\varvec{v}}} \right) ^{-1} \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\phi }_{t} - \frac{1}{2 \gamma } \pmb {\phi }_{t}^{\top } \left( \pmb {\varSigma }_{t}^{*} \right) ^{-1} \pmb {\phi }_{t} \nonumber \\&\quad + x_{t} + \frac{1}{4} \pmb {\varPsi }_{t}^{\top } \pmb {\tilde{\varOmega }}_{t}^{-1} \pmb {\varPsi }_{t}. \end{aligned}$$
(A.27)

\(\square \)

Proof of Theorem 2

The value function at maturity becomes

$$\begin{aligned} V_{T + 1} \big [ s_{T + 1} \big ]&= - \exp \Big \{ - \gamma \Big [ W_{T + 1} - \left( {\varvec{P}}_{T + 1} + \pmb {\chi }_{T + 1} {\overline{\varvec{Q}}}_{T + 1} \right) ^{\top } {\overline{\varvec{Q}}}_{T + 1} \Big ] \Big \}, \end{aligned}$$
(B.1)

and thereby, the optimal value function at time T is calculated as follows:

$$\begin{aligned} V_{T} \big [ {\varvec{s}}_{T} \big ]&= \sup _{{\varvec{q}}_{T} \in {\mathbb {R}}^{2}} {\mathbb {E}} \Big [ V_{T + 1} \big [ W_{T + 1}, {\varvec{P}}_{T + 1}, {\overline{\varvec{Q}}}_{T + 1}, {\varvec{R}}_{T + 1}, {\varvec{v}}_{T} \big ] \Big \vert W_{T}, {\varvec{P}}_{T}, {\overline{\varvec{Q}}}_{T}, {\varvec{R}}_{T}, {\varvec{v}}_{T - 1} \Big ] \nonumber \\&= \sup _{{\varvec{q}}_{T} \in {\mathbb {R}}^{2}}{\mathbb {E}} \Big [ - \exp \Big \{ - \gamma \Big [ W_{T + 1} - \left( {\varvec{P}}_{T + 1} + \pmb {\chi }_{T + 1} {\overline{\varvec{Q}}}_{T + 1} \right) {\overline{\varvec{Q}}}_{T + 1} \Big ] \Big \} \Big \vert {\varvec{s}}_{T} \Big ] \nonumber \\&= \sup _{{\varvec{q}}_{T} \in {\mathbb {R}}^{2}} - \exp \Big \{ - \gamma \Big [ - {\varvec{q}}_{T}^{\top } \pmb {\tilde{\varOmega }}_{T} {\varvec{q}}_{T}\nonumber \\&\quad + \Big [ \overline{{\varvec{Q}}}_{T}^{\top } \pmb {\varTheta }_{T} + {\varvec{R}}_{T}^{\top } \pmb {\varXi }_{T} + {\varvec{v}}_{T - 1}^{\top } \pmb {\varPhi }_{T} + \pmb {\varPsi }_{T}^{\top } \Big ] {\varvec{q}}_{T} + W_{T} - {\varvec{P}}_{T}^{\top } {\overline{\varvec{Q}}}_{t} \nonumber \\&\quad + \overline{{\varvec{Q}}}_{T}^{\top } \Big [ - \left( \pmb {\chi }_{T + 1} + \pmb {\chi }_{T + 1}^{\top } \right) - \frac{1}{2} \gamma \pmb {\varPi }_{T} \pmb {\kappa }_{T} \pmb {\varSigma }_{T}^{{\varvec{v}}} \pmb {\kappa }_{T} \pmb {\varPi }_{T} - \frac{1}{2} \gamma \pmb {\varSigma }_{T}^{\pmb {\varepsilon }} \Big ] \overline{{\varvec{Q}}}_{T} \nonumber \\&\quad + \Big [ - \left( {\varvec{a}}_{T}^{{\varvec{v}}} \right) ^{\top } \pmb {\kappa }_{T} \pmb {\varPi }_{T} - \left( \pmb {\mu }_{T}^{\pmb {\varepsilon }} \right) ^{\top } \Big ] \overline{{\varvec{Q}}}_{T} + ( 1 - \mathrm {e}^{-\rho } ) \overline{{\varvec{Q}}}_{T}^{\top } {\varvec{R}}_{T} - {\overline{\varvec{Q}}}_{T}^{\top } \pmb {\varPi }_{T} \pmb {\kappa }_{T} \mathbf{b}_{T}^{{\varvec{v}}} {\varvec{v}}_{T - 1} \Big ] \Big \}, \end{aligned}$$
(B.2)

where \(\pmb {\varPi }_{T} := \mathrm {e}^{-\rho } \mathbf{A}_{T} + \mathbf{B}_{T}\), and

$$\begin{aligned} \pmb {\tilde{\varOmega }}_{T}&:= \frac{1}{2} \left( \pmb {\varOmega }_{T} + \pmb {\varOmega }_{T}^{\top } \right) \nonumber \\&= \frac{1}{2} \pmb {\varLambda }_{T} \left( \mathbf{I}_{2} - \pmb {\varPi }_{T} \right) \nonumber \\&\quad + \frac{1}{2} \left\{ \pmb {\varLambda }_{T} \left( \mathbf{I}_{2} - \pmb {\varPi }_{T} \right) \right\} ^{\top } + \frac{1}{2} \left( \pmb {\chi }_{T + 1} + \pmb {\chi }_{T + 1}^{\top } \right) + \frac{1}{2} \gamma \left( \mathbf{I}_{2} - \pmb {\varPi }_{T} \right) \pmb {\kappa }_{T} \pmb {\varSigma }_{T}^{{\varvec{v}}} \pmb {\kappa }_{T} \left( \mathbf{I}_{2} - \pmb {\varPi }_{T} \right) \nonumber \\&\quad + \frac{1}{2} \gamma \pmb {\varSigma }_{T}^{\pmb {\varepsilon }} \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varTheta }_{T}&:= - \pmb {\varPi }_{T} \pmb {\varLambda }_{T} + \left( \pmb {\chi }_{T + 1} + \pmb {\chi }_{T + 1}^{\top } \right) - \gamma \pmb {\varPi }_{T}\pmb {\kappa }_{T} \pmb {\varSigma }_{T}^{{\varvec{v}}} \pmb {\kappa }_{T} \left( \mathbf{I}_{2} - \pmb {\varPi }_{T} \right) + \gamma \pmb {\varSigma }_{T}^{\pmb {\varepsilon }} \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varXi }_{T}&:= - ( 1 - \mathrm {e}^{-\rho } ) \mathbf{I}_{2} \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varPhi }_{T}&:= - \left( \mathbf{b}_{T}^{{\varvec{v}}} \right) ^{\top } \pmb {\kappa }_{T} \left( \mathbf{I}_{2} - \pmb {\varPi }_{T} \right) \ ( \in {\mathbb {R}}^{2 \times 2} ); \nonumber \\ \pmb {\varPsi }_{T}^{\top }&:= - \left( {\varvec{a}}_{T}^{{\varvec{v}}} \right) ^{\top } \pmb {\kappa }_{T} \left( \mathbf{I}_{2} - \pmb {\varPi }_{T} \right) + \left( \pmb {\mu }_{T}^{\pmb {\varepsilon }} \right) ^{\top } \ ( \in {\mathbb {R}}^{1 \times 2} ). \end{aligned}$$
(B.3)

We can derive the optimal execution volume satisfying Eq. (B.2) by obtaining the optimal execution volume \({\varvec{q}}_{t}^{*}\) which attains the maximum of

$$\begin{aligned}&K_{T} \big [ {\varvec{q}}_{T} \big ] := - {\varvec{q}}_{T}^{\top } \pmb {\tilde{\varOmega }}_{T} {\varvec{q}}_{T} + \Big [ {\overline{\varvec{Q}}}_{T}^{\top } \pmb {\varTheta }_{T} + {\varvec{R}}_{T}^{\top } \pmb {\varXi }_{T} + {\varvec{v}}_{T - 1}^{\top } \pmb {\varPhi }_{T} + \pmb {\varPsi }_{T}^{\top } \Big ] {\varvec{q}}_{T} + W_{T} - {\varvec{P}}_{T}^{\top } \overline{{\varvec{Q}}}_{T} \nonumber \\&\quad + {\overline{\varvec{Q}}}_{T}^{\top } \Big [ - \left( \pmb {\chi }_{T + 1} + \pmb {\chi }_{T + 1}^{\top } \right) - \frac{1}{2} \gamma \pmb {\varPi }_{T} \pmb {\kappa }_{T} \pmb {\varSigma }_{T}^{{\varvec{v}}} \pmb {\kappa }_{T} \pmb {\varPi }_{T} - \frac{1}{2} \gamma \pmb {\varSigma }_{T}^{\pmb {\varepsilon }} \Big ] {\overline{\varvec{Q}}}_{T} \nonumber \\&\quad + \Big [ - \left( {\varvec{a}}_{T}^{{\varvec{v}}} \right) ^{\top } \pmb {\kappa }_{T} \pmb {\varPi }_{T} - \left( \pmb {\mu }_{T}^{\pmb {\varepsilon }} \right) ^{\top } \Big ] {\overline{\varvec{Q}}}_{T} + ( 1 - \mathrm {e}^{-\rho } ) {\overline{\varvec{Q}}}_{T}^{\top } {\varvec{R}}_{T} - \overline{{\varvec{Q}}}_{T}^{\top } \pmb {\varPi }_{T} \pmb {\kappa }_{T} \mathbf{b}_{T}^{{\varvec{v}}} {\varvec{v}}_{T - 1}. \end{aligned}$$
(B.4)

Eq. (B.4) is a quadratic function with a negative definite matrix \(\pmb {\varOmega }_{T} + \pmb {\varOmega }_{T}^{\top }\) with respect to \({\varvec{q}}_{T}\), and thereby a concave function with respect to \({\varvec{q}}_{T}\), which leads to the concavity of Eq. (B.2) with respect to \({\varvec{q}}_{T}\). Therefore, by completing the square of \(K_{T} \big [ {\varvec{q}}_{T} \big ]\) with respect to \({\varvec{q}}_{T}\), we obtain the optimal execution volume at time \(t = T\):

$$\begin{aligned} {\varvec{q}}_{T}^{*}&=: \mathbf{f} ( {\varvec{s}}_{T} ) = \pmb {\tilde{\varOmega }}_{T}^{-1} \left\{ \pmb {\varTheta }_{T}^{\top } \overline{{\varvec{Q}}}_{T} + \pmb {\varXi }_{T}^{\top } {\varvec{R}}_{T} + \pmb {\varPhi }_{T}^{\top } {\varvec{v}}_{T - 1} + \pmb {\varPsi }_{T} \right\} \nonumber \\&\quad \left( =: {\varvec{a}}_{T - 1}^{*} + \mathbf{b}_{T - 1}^{*} {\overline{\varvec{Q}}}_{T - 1} + \mathbf{c}_{T - 1}^{*} {\varvec{R}}_{T - 1} + \mathbf{d}_{T - 1}^{*} {\varvec{v}}_{T - 1} \right) , \end{aligned}$$
(B.5)

where

$$\begin{aligned} {\varvec{a}}_{T}^{*} := \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varPsi }_{T}; \quad \mathbf{b}_{T}^{*} := \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varTheta }_{T}^{\top }; \quad \mathbf{c}_{T}^{*} := \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varXi }_{T}^{\top }; \quad \mathbf{d}_{T}^{*} := \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varPhi }_{T}^{\top }, \end{aligned}$$
(B.6)

and by substituting this into Eq. (B.2) the optimal value function becomes

$$\begin{aligned} V_{T} \big [ {\varvec{s}}_{T} \big ]&= - \exp \Big \{ - \gamma \Big [ W_{T} - {\varvec{P}}_{T}^{\top } {\overline{\varvec{Q}}}_{T} + {\overline{\varvec{Q}}}_{T}^{\top } \mathbf{G}_{T}^{*} {\overline{\varvec{Q}}}_{T} + {\varvec{H}}_{T}^{*\top } {\overline{\varvec{Q}}}_{T}\nonumber \\&\quad + {\overline{\varvec{Q}}}_{T}^{\top } \mathbf{I}_{T}^{*} {\varvec{R}}_{T} + {\varvec{R}}_{T}^{\top } \mathbf{J}_{T}^{*} {\varvec{R}}_{T} + {\varvec{L}}_{T}^{*\top } {\varvec{R}}_{T} \nonumber \\&\quad + {\overline{\varvec{Q}}}_{T}^{\top } \mathbf{M}_{T}^{*} {\varvec{v}}_{T - 1} + {\varvec{R}}_{T}^{\top } \mathbf{N}_{T}^{*} {\varvec{v}}_{T - 1} + {\varvec{v}}_{T - 1}^{\top } \mathbf{X}_{T}^{*} {\varvec{v}}_{T - 1} + {\varvec{Y}}_{T}^{*\top } {\varvec{v}}_{T - 1} + Z_{T}^{*} \Big ] \Big \}, \end{aligned}$$
(B.7)

where

$$\begin{aligned} \mathbf{G}_{T}^{*}&:= - \left( \pmb {\chi }_{T + 1} + \pmb {\chi }_{T + 1}^{\top } \right) - \frac{1}{2} \gamma \pmb {\varPi }_{T} \pmb {\kappa }_{T} \pmb {\varSigma }_{T}^{{\varvec{v}}} \pmb {\kappa }_{T} \pmb {\varPi }_{T} - \frac{1}{2} \gamma \pmb {\varSigma }_{T}^{\pmb {\varepsilon }} + \frac{1}{4} \pmb {\varTheta }_{T} \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varTheta }_{T}^{\top }; \nonumber \\ {\varvec{H}}_{T}^{*\top }&:= - \left( {\varvec{a}}_{T}^{{\varvec{v}}} \right) ^{\top } \pmb {\kappa }_{T} \pmb {\varPi }_{T} - \left( \pmb {\mu }_{T}^{\pmb {\varepsilon }} \right) ^{\top } + \frac{1}{2} \pmb {\varPsi }_{T - 1}^{\top } \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varTheta }_{T - 1}^{\top }; \nonumber \\ \mathbf{I}_{T}^{*}&:= ( 1 - \mathrm {e}^{- \rho } ) \mathbf{I}_{2} + \frac{1}{2} \pmb {\varTheta }_{T} \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varXi }_{T}^{\top }; \quad \mathbf{J}_{T}^{*} := \frac{1}{4} \pmb {\varXi }_{T} \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varXi }_{T}^{\top }; \quad {\varvec{L}}_{T}^{*\top } := \frac{1}{2} \pmb {\varPsi }_{T}^{\top } \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varXi }_{T}^{\top }; \nonumber \\ \mathbf{M}_{T}^{*}&:= - \pmb {\varPi }_{T} \pmb {\kappa }_{T} \mathbf{b}_{T}^{{\varvec{v}}} + \frac{1}{2} \pmb {\varTheta }_{T} \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varPhi }_{T}^{\top }; \quad \mathbf{N}_{T}^{*} := \frac{1}{2} \pmb {\varXi }_{T} \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varPhi }_{T}^{\top }; \nonumber \\ \mathbf{X}_{T}^{*}&:= \frac{1}{4} \pmb {\varPhi }_{T} \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varPhi }_{T}^{\top }; \quad {\varvec{Y}}_{T}^{*\top } := \frac{1}{2} \pmb {\varPsi }_{T}^{\top } \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varPhi }_{T}^{\top }; \quad Z_{T}^{*} := \frac{1}{4} \pmb {\varPsi }_{T}^{\top } \pmb {\tilde{\varOmega }}_{T}^{-1} \pmb {\varPsi }_{T}. \end{aligned}$$
(B.8)

For \(t \in \{ { T - 1, \ldots , 1 } \}\), we can recursively derive the optimal execution volume and optimal value function at each time by a similar derivation which we use to obtain the optimal execution volume in the last subsection for time \(t \in \{ { T - 2, \ldots , 1 } \}\). \(\square \)

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Ohnishi, M., Shimoshimizu, M. Optimal Pair–Trade Execution with Generalized Cross–Impact. Asia-Pac Financ Markets 29, 253–289 (2022). https://doi.org/10.1007/s10690-021-09349-1

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