SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathbb {E} \left[ c _ {l _ {1} j _ {1}} c _ {l _ {2}, j _ {2}} \right] = \sum_ {\alpha = 1} ^ {n} p _ {l _ {1} \alpha} p _ {j _ {1} \alpha} p _ {l _ {2} \alpha} p _ {j _ {2} \alpha} \sigma^ {4} \mathcal {I} _ {1} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \alpha) \\ + \sum_ {\alpha \neq \beta} ^ {n} p _ {l _ {1} \alpha} p _ {j _ {1} \alpha} p _ {l _ {2} \beta} p _ {j _ {2} \beta} \sigma^ {4} \mathcal {I} _ {2} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \alpha , \beta) \\ + \sum_ {\alpha \neq \beta} ^ {n} p _ {l _ {1} \alpha} p _ {j _ {1} \beta} p _ {l _ {2} \alpha} p _ {j _ {2} \beta} \sigma^ {4} \mathcal {I} _ {3} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \alpha , \beta), \tag {34} \\ \end{array}
$$
$$
\begin{array}{l} \mathcal {I} _ {1} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \alpha) \\ = \mathbb {E} \left\{\left[ \frac {h}{M} \sum_ {i = 1} ^ {M} \sum_ {k = 0} ^ {N - 1} \left(\int_ {0} ^ {k h} e ^ {\lambda_ {l _ {1}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \left(\int_ {0} ^ {k h} e ^ {\lambda_ {j _ {1}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \right. \right. \\ - \int_ {0} ^ {T} \mathbb {E} \left[ \left(\int_ {0} ^ {t} e ^ {\lambda_ {l _ {1}} (t - s)} \mathrm {d} w _ {\alpha} (s)\right) \left(\int_ {0} ^ {t} e ^ {\lambda_ {j _ {1}} (t - s)} \mathrm {d} w _ {\alpha} (s)\right) \right] \mathrm {d} t \bigg ] \times \\ \left[ \frac {h}{M} \sum_ {i = 1} ^ {M} \sum_ {k = 0} ^ {N - 1} \left(\int_ {0} ^ {k h} e ^ {\lambda_ {l _ {2}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \left(\int_ {0} ^ {k h} e ^ {\lambda_ {j _ {2}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \right. \\ \left. - \int_ {0} ^ {T} \mathbb {E} \left[\left(\int_ {0} ^ {t} e ^ {\lambda_ {l _ {2}} (t - s)} \mathrm {d} w _ {\alpha} (s)\right)\left(\int_ {0} ^ {t} e ^ {\lambda_ {j _ {2}} (t - s)} \mathrm {d} w _ {\alpha} (s)\right)\right] \mathrm {d} t \right]\left. \right\}, \\ \end{array}
$$
$$
\begin{array}{l} \mathcal {I} _ {2} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \alpha , \beta) \\ = \mathbb {E} \left\{\left[ \frac {h}{M} \sum_ {i = 1} ^ {M} \sum_ {k = 0} ^ {N - 1} \left(\int_ {0} ^ {k h} e ^ {\lambda_ {l _ {1}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \left(\int_ {0} ^ {k h} e ^ {\lambda_ {j _ {1}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \right. \right. \\ - \int_ {0} ^ {T} \mathbb {E} \left[ \left(\int_ {0} ^ {t} e ^ {\lambda_ {l _ {1}} (t - s)} \mathrm {d} w _ {\alpha} (s)\right) \left(\int_ {0} ^ {t} e ^ {\lambda_ {j _ {1}} (t - s)} \mathrm {d} w _ {\alpha} (s)\right) \right] \mathrm {d} t \bigg ] \times \\ \left[ \frac {h}{M} \sum_ {i = 1} ^ {M} \sum_ {k = 0} ^ {N - 1} \left(\int_ {0} ^ {k h} e ^ {\lambda_ {l _ {2}} (k h - s)} d w _ {\beta} ^ {(i)} (s)\right) \left(\int_ {0} ^ {k h} e ^ {\lambda_ {j _ {2}} (k h - s)} d w _ {\beta} ^ {(i)} (s)\right) \right. \\ \left. - \int_ {0} ^ {T} \mathbb {E} \left[ \left(\int_ {0} ^ {t} e ^ {\lambda_ {l _ {2}} (t - s)} \mathrm {d} w _ {\beta} (s)\right) \left(\int_ {0} ^ {t} e ^ {\lambda_ {j _ {2}} (t - s)} \mathrm {d} w _ {\beta} (s)\right) \right] \mathrm {d} t \right] \Bigg \} \\ \end{array}
$$
$$
\begin{array}{l} \mathcal {I} _ {3} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \alpha , \beta) \\ = \mathbb {E} \left\{\left[ \frac {h}{M} \sum_ {i = 1} ^ {M} \sum_ {k = 0} ^ {N - 1} \left(\int_ {0} ^ {k h} e ^ {\lambda_ {l _ {1}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \left(\int_ {0} ^ {k h} e ^ {\lambda_ {j _ {1}} (k h - s)} d w _ {\beta} ^ {(i)} (s)\right) \right] \times \right. \\ \left. \left[ \frac {h}{M} \sum_ {i = 1} ^ {M} \sum_ {k = 0} ^ {N - 1} \left(\int_ {0} ^ {k h} e ^ {\lambda_ {l _ {2}} (k h - s)} \mathrm {d} w _ {\alpha} ^ {(i)} (s)\right) \left(\int_ {0} ^ {k h} e ^ {\lambda_ {j _ {2}} (k h - s)} \mathrm {d} w _ {\beta} ^ {(i)} (s)\right) \right] \right\}. \\ \end{array}
$$
Note that  $w_{\alpha}^{(i)}$  and  $w_{\beta}^{(i)}$  are independent for  $\alpha \neq \beta$ . By using the expectations Equations (31) and (33), we can further obtain  $\mathcal{I}_2(M,h,T,\lambda_{l_1},\lambda_{j_1},\lambda_{l_2},\lambda_{j_2},\alpha ,\beta)$  as
$$
\begin{array}{l} \mathcal {I} _ {2} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \alpha , \beta) \\ = \left[ \frac {h}{\left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right)} \left(\frac {1 - e ^ {\left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right) T}}{1 - e ^ {\left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right) h}} - \frac {T}{h}\right) - \frac {1}{\left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right) ^ {2}} \left(e ^ {\left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right) T} - 1 - \left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right) T\right) \right] \\ \times \left[ \frac {h}{\left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right)} \left(\frac {1 - e ^ {\left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right) T}}{1 - e ^ {\left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right) h}} - \frac {T}{h}\right) - \frac {1}{\left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right) ^ {2}} \left(e ^ {\left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right) T} - 1 - \left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right) T\right) \right]. \\ \end{array}
$$
In the following computations, we will use  $\bar{C}$  and  $C(\lambda_{l_1},\lambda_{j_1},\lambda_{l_2},\lambda_{j_2})$  to represent some constants that are not depending on  $h,T,B$ .
The expectation  $\mathcal{I}_1(M,h,T,\lambda_{l_1},\lambda_{j_1},\lambda_{l_2},\lambda_{j_2},\alpha)$  is computed exactly the same way as in the proof of Theorem 1 by using the expectation results Equation (31) and Equation (32). Notice that the expectation result Equation (31) (when  $s = t$ ) has the same order in  $t$  as the expectation Equation (15). Moreover, the two expectations Equation (32) and Equation (17) have the same orders in  $s$  and  $t$ . Thus,  $\mathcal{I}_1(M,h,T,\lambda_{l_1},\lambda_{j_1},\lambda_{l_2},\lambda_{j_2},\alpha)$  has the same orders in  $h,T,B$  as the scalar MSE, i.e.
$$
\begin{array}{l} \mathcal {I} _ {1} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \alpha) = (\bar {C} _ {1} + C _ {1} (\lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}) \mathcal {O} (T)) T ^ {2} h ^ {2} + \mathcal {O} (h ^ {3}) \\ + \left(\bar {C} _ {2} + C _ {2} \left(\lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}\right) \mathcal {O} (T)\right) \frac {T ^ {5}}{h B} + \mathcal {O} \left(\frac {1}{B}\right) \\ \end{array}
$$
The expectation  $\mathcal{I}_2(M,h,T,\lambda_{l_1},\lambda_{j_1},\lambda_{l_2},\lambda_{j_2},\alpha ,\beta)$  can be computed directly and has the result:
$$
\begin{array}{l} \mathcal {I} _ {2} \left(M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \alpha , \beta\right) = \frac {\left(e ^ {\left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right) T} - 1\right) \left(e ^ {\left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right) T} - 1\right) h ^ {2}}{4 \left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right) \left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right)} + \mathcal {O} \left(h ^ {3}\right) \\ = \left(\frac {1}{4} T ^ {2} + C _ {3} \left(\lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}\right) \mathcal {O} \left(T ^ {3}\right)\right) h ^ {2} + \mathcal {O} \left(h ^ {3}\right). \\ \end{array}
$$
The expectation  $\mathcal{I}_3(M,h,T,\lambda_{l_1},\lambda_{j_1},\lambda_{l_2},\lambda_{j_2},\alpha ,\beta)$  can be computed as follows:
$$
\begin{array}{l} \mathcal {I} _ {3} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \alpha , \beta) \\ = \frac {h ^ {2}}{M} \sum_ {k = 0} ^ {n} \frac {\left(e ^ {\left(\lambda_ {l _ {1}} + \lambda_ {l _ {2}}\right) k h} - 1\right) \left(e ^ {\left(\lambda_ {j _ {1}} + \lambda_ {j _ {2}}\right) k h} - 1\right) h ^ {2}}{\left(\lambda_ {l _ {1}} + \lambda_ {l _ {2}}\right) \left(\lambda_ {j _ {1}} + \lambda_ {j _ {2}}\right)} + \\ \frac {h ^ {2}}{M} \sum_ {k <   q} \frac {e ^ {\lambda_ {l _ {1}} k h + \lambda_ {l _ {2}} q h + \lambda_ {j _ {1}} k h + \lambda_ {j _ {2}} q h}}{(\lambda_ {l _ {1}} + \lambda_ {l _ {2}}) (\lambda_ {j _ {1}} + \lambda_ {j _ {2}})} \left(1 - e ^ {- (\lambda_ {l _ {1}} + \lambda_ {l _ {2}}) k h}\right) \left(1 - e ^ {- (\lambda_ {j _ {1}} + \lambda_ {j _ {2}}) k h}\right) \\ \frac {h ^ {2}}{M} \sum_ {k <   q} \frac {e ^ {\lambda_ {l _ {1}} q h + \lambda_ {l _ {2}} k h + \lambda_ {j _ {1}} q h + \lambda_ {j _ {2}} k h}}{\left(\lambda_ {l _ {1}} + \lambda_ {l _ {2}}\right) \left(\lambda_ {j _ {1}} + \lambda_ {j _ {2}}\right)} \left(1 - e ^ {- \left(\lambda_ {l _ {1}} + \lambda_ {l _ {2}}\right) k h}\right) \left(1 - e ^ {- \left(\lambda_ {j _ {1}} + \lambda_ {j _ {2}}\right) k h}\right) \\ = \left(\bar {C} _ {4} + C _ {4} \left(\lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}\right) \mathcal {O} (T)\right) \frac {T ^ {5}}{h B} + \mathcal {O} \left(\frac {1}{B}\right). \\ \end{array}
$$
Thus, the final result is obtained by the expression of MSE in Equation (30), Equation (34) and the above computations. Again, we rely on symbolic computation to obtain the expression and corresponding Taylor approximations and include the notebooks of all derivations in the supplementary material.
The extension from Theorem 3.4 to the discounted finite-horizon results can be done in the same way as in the above proof (add the discount factor  $\gamma$  in  $\hat{V}_M$ ) by using the expectation cost for any  $\lambda_1$  and  $\lambda_2$ :
$$
\begin{array}{l} \int_ {0} ^ {T} \gamma^ {t} \mathbb {E} \left[ \int_ {0} ^ {t} e ^ {\lambda_ {1} (t - u)} \mathrm {d} w (u) \int_ {0} ^ {s} e ^ {\lambda_ {2} (- u)} \mathrm {d} w (u) \right] \mathrm {d} t \\ = \frac {1}{\left(\lambda_ {1} + \lambda_ {2}\right)} \left(\frac {\gamma^ {T} e ^ {\left(\lambda_ {1} + \lambda_ {2}\right) T} - 1}{\log (\gamma) + \left(\lambda_ {1} + \lambda_ {2}\right)} - \frac {\gamma^ {T} - 1}{\log (\gamma)}\right). \\ \end{array}
$$
C.2 Proof of Corollary 3.7
Proof. We shall follow the similar proof as in the proof of Theorem 3.4 and the proof of Theorem 3.6.
Continuing from Equation (34), in infinite-horizon discounted setting, we have
$$
\begin{array}{l} \mathcal {I} _ {1} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \gamma , \alpha) \\ = \mathbb {E} \left\{\left[ \frac {h}{M} \sum_ {i = 1} ^ {M} \sum_ {k = 0} ^ {N - 1} \gamma^ {k h} \left(\int_ {0} ^ {k h} e ^ {\lambda_ {l _ {1}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \left(\int_ {0} ^ {k h} e ^ {\lambda_ {j _ {1}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \right. \right. \\ - \int_ {0} ^ {\infty} \gamma^ {t} \mathbb {E} \left[ \left(\int_ {0} ^ {t} e ^ {\lambda_ {l _ {1}} (t - s)} \mathrm {d} w _ {\alpha} (s)\right) \left(\int_ {0} ^ {t} e ^ {\lambda_ {j _ {1}} (t - s)} \mathrm {d} w _ {\alpha} (s)\right) \right] \mathrm {d} t \bigg ] \times \\ \left[ \frac {h}{M} \sum_ {i = 1} ^ {M} \sum_ {k = 0} ^ {N - 1} \gamma^ {k h} \left(\int_ {0} ^ {k h} e ^ {\lambda_ {l _ {2}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \left(\int_ {0} ^ {k h} e ^ {\lambda_ {j _ {2}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \right. \\ \left. - \int_ {0} ^ {\infty} \gamma^ {t} \mathbb {E} \left[\left(\int_ {0} ^ {t} e ^ {\lambda_ {l _ {2}} (t - s)} \mathrm {d} w _ {\alpha} (s)\right)\left(\int_ {0} ^ {t} e ^ {\lambda_ {j _ {2}} (t - s)} \mathrm {d} w _ {\alpha} (s)\right)\right] \mathrm {d} t \right]\left. \right\}, \\ \end{array}
$$
$$
\begin{array}{l} \mathcal {I} _ {2} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \gamma , \alpha , \beta) \\ = \left[ \frac {h}{\left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right)} \left(\frac {1 - \gamma^ {T} e ^ {\left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right) T}}{1 - \gamma^ {h} e ^ {\left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right) h}} - \frac {1 - \gamma^ {T}}{1 - \gamma^ {h}}\right) - \frac {1}{\left(\lambda_ {l _ {1}} + \lambda_ {j _ {1}}\right)} \left(\frac {1}{\log (\gamma)} - \frac {1}{\log (\gamma) + \lambda_ {l _ {1}} + \lambda_ {j _ {1}}}\right) \right] \\ \times \left[ \frac {h}{\left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right)} \left(\frac {1 - \gamma^ {T} e ^ {\left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right) T}}{1 - \gamma^ {h} e ^ {\left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right) h}} - \frac {1 - \gamma^ {T}}{1 - \gamma^ {h}}\right) - \frac {1}{\left(\lambda_ {l _ {2}} + \lambda_ {j _ {2}}\right)} \left(\frac {1}{\log (\gamma)} - \frac {1}{\log (\gamma) + \lambda_ {l _ {2}} + \lambda_ {j _ {2}}}\right) \right], \\ \end{array}
$$
$$
\begin{array}{l} \mathcal {I} _ {3} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, r, \alpha , \beta) \\ = \mathbb {E} \left\{\left[ \frac {h}{M} \sum_ {i = 1} ^ {M} \sum_ {k = 0} ^ {N - 1} \gamma^ {k h} \left(\int_ {0} ^ {k h} e ^ {\lambda_ {l _ {1}} (k h - s)} d w _ {\alpha} ^ {(i)} (s)\right) \left(\int_ {0} ^ {k h} e ^ {\lambda_ {j _ {1}} (k h - s)} d w _ {\beta} ^ {(i)} (s)\right) \right] \times \right. \\ \left. \left[ \frac {h}{M} \sum_ {i = 1} ^ {M} \sum_ {k = 0} ^ {N - 1} \gamma^ {k h} \left(\int_ {0} ^ {k h} e ^ {\lambda_ {l _ {2}} (k h - s)} \mathrm {d} w _ {\alpha} ^ {(i)} (s)\right) \left(\int_ {0} ^ {k h} e ^ {\lambda_ {j _ {2}} (k h - s)} \mathrm {d} w _ {\beta} ^ {(i)} (s)\right) \right] \right\}. \\ \end{array}
$$
Similar arguments as in proof of Theorem 3.4, we can conclude  $\mathcal{I}_1(M,h,T,\lambda_{l_1},\lambda_{j_1},\lambda_{l_2},\lambda_{j_2},\alpha)$  has the same orders in  $h,B,T$  as the MSE result in Theorem 3.6.
Moreover, let  $C_i(\lambda_{l_1}, \lambda_{j_1}, \lambda_{l_2}, \lambda_{j_2}, \gamma, T)$ 's are some constants that depend on  $\lambda_{l_1}, \lambda_{j_1}, \lambda_{l_2}, \lambda_{j_2}, \gamma, T$ , then
$$
\begin{array}{l} \mathcal {I} _ {2} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \gamma , \alpha , \beta) \\ = \sigma^ {4} \gamma^ {2 T} \left(C _ {1} \left(\lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \gamma , T\right) + C _ {2} \left(\lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \gamma , T\right) h\right) \\ + \sigma^ {4} \gamma^ {T} \left(C _ {3} \left(\lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \gamma , T\right) h ^ {2} + C _ {4} \left(\lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \gamma , T\right) h ^ {3}\right) \\ + \sigma^ {4} \left(\frac {1}{1 4 4} + \gamma^ {T} C _ {5} \left(\lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \gamma , T\right)\right) h ^ {4} + \mathcal {O} \left(h ^ {5}\right), \\ \end{array}
$$
$$
\begin{array}{l} \mathcal {I} _ {3} (M, h, T, \lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \gamma , \alpha , \beta) \\ = \frac {h ^ {2}}{M} \sum_ {k = 0} ^ {N - 1} \frac {\left(e ^ {\left(\lambda_ {l _ {1}} + \lambda_ {l _ {2}}\right) k h} - 1\right) \left(e ^ {\left(\lambda_ {j _ {1}} + \lambda_ {j _ {2}}\right) k h} - 1\right) h ^ {2} \gamma^ {2 k h}}{\left(\lambda_ {l _ {1}} + \lambda_ {l _ {2}}\right) \left(\lambda_ {j _ {1}} + \lambda_ {j _ {2}}\right)} + \\ \frac {h ^ {2}}{M} \sum_ {k <   q} \frac {e ^ {\lambda_ {l _ {1}} k h + \lambda_ {l _ {2}} q h + \lambda_ {j _ {1}} k h + \lambda_ {j _ {2}} q h}}{(\lambda_ {l _ {1}} + \lambda_ {l _ {2}}) (\lambda_ {j _ {1}} + \lambda_ {j _ {2}})} \left(1 - e ^ {- (\lambda_ {l _ {1}} + \lambda_ {l _ {2}}) k h}\right) \left(1 - e ^ {- (\lambda_ {j _ {1}} + \lambda_ {j _ {2}}) k h}\right) \gamma^ {(k + q) h} \\ \frac {h ^ {2}}{M} \sum_ {k <   q} \frac {e ^ {\lambda_ {l _ {1}} q h + \lambda_ {l _ {2}} k h + \lambda_ {j _ {1}} q h + \lambda_ {j _ {2}} k h}}{(\lambda_ {l _ {1}} + \lambda_ {l _ {2}}) (\lambda_ {j _ {1}} + \lambda_ {j _ {2}})} \left(1 - e ^ {- (\lambda_ {l _ {1}} + \lambda_ {l _ {2}}) k h}\right) \left(1 - e ^ {- (\lambda_ {j _ {1}} + \lambda_ {j _ {2}}) k h}\right) \gamma^ {(k + q) h} \\ = C _ {6} \left(\lambda_ {l _ {1}}, \lambda_ {j _ {1}}, \lambda_ {l _ {2}}, \lambda_ {j _ {2}}, \gamma , T\right) \frac {T ^ {5}}{h B} + \mathcal {O} \left(\frac {1}{B}\right). \\ \end{array}
$$
The result in this Corollary is obtained by combining the above results. And we include the notebooks of all derivations in the supplementary material.
C.3 The case when  $A$  is a general stable matrix
Lemma C.1 (MSE when  $A$  is a general stable matrix). Let  $A$  be a stable  $n \times n$  matrix with distinct eigenvalues  $\lambda_1, \dots, \lambda_m$  and corresponding multiplicities  $q_1, \dots, q_m$ . There exist some constants  $\{\bar{C}_i\}_{i=1}^m$ ,  $\bar{C}_0$  and  $C_j(\lambda_1, \dots, \lambda_m, \gamma, T)'s$ , such that the mean-squared error of the Monte-Carlo estimator in different settings satisfies
(1) Finite-Horizon undiscounted setting:
$$
\begin{array}{l} M S E _ {T} \in \left[ \sum_ {i = 1} ^ {m} q _ {i} \bar {C} _ {i} M S E _ {T} (h, B, \lambda_ {i}), \quad C _ {1} (\lambda_ {1}, \dots , \lambda_ {m}, T) \sigma^ {4} T ^ {2} h ^ {2} \right. \\ \left. + \frac {\left(\bar {C} _ {2} + C _ {3} \left(\lambda_ {1} , \cdots , \lambda_ {m} , T\right) \mathcal {O} (T)\right) \sigma^ {4} T ^ {2 n + 3}}{B h} + \mathcal {O} \left(h ^ {3}\right) + \mathcal {O} \left(\frac {1}{B}\right) \right], \tag {35} \\ \end{array}
$$
where  $MSE_T(h, B, \lambda_i)$  is the mean-squared error of the Monte-Carlo estimator in Theorem 3.1 by replacing the drift  $a$  by  $\lambda_i$ .
(2) Finite-Horizon discounted setting:
$$
\begin{array}{l} M S E _ {T} \in \left[ \sum_ {i = 1} ^ {m} q _ {i} \bar {C} _ {i} M S E _ {T} (h, B, \gamma , \lambda_ {i}), \quad C _ {4} (\lambda_ {1}, \dots , \lambda_ {m}, \gamma , T) \sigma^ {4} \gamma^ {2 T} T ^ {2} h ^ {2} \right. \\ + C _ {5} \left(\lambda_ {1}, \dots , \lambda_ {m}, \gamma , T\right) \sigma^ {4} \gamma^ {T} h ^ {3} + C _ {6} \left(\lambda_ {1}, \dots , \lambda_ {m}, T\right) \sigma^ {4} h ^ {4} \\ \left. + \frac {\left(C _ {7} \left(\lambda_ {1} , \cdots , \lambda_ {m} , \gamma , T\right)\right) \sigma^ {4} T ^ {2 n - 1}}{B h} + \mathcal {O} \left(h ^ {5}\right) + \mathcal {O} \left(\frac {1}{B}\right) \right], \tag {36} \\ \end{array}
$$
where  $MSE_T(h, B, \gamma, \lambda_i)$  is the mean-squared error of the Monte-Carlo estimator in Lemma B.1 by replacing the drift  $a$  by  $\lambda_i$ .
(3) Infinite-Horizon discounted setting:
$$
\begin{array}{l} M S E _ {\infty} \in \left[ \sum_ {i = 1} ^ {m} q _ {i} \bar {C} _ {i} M S E _ {\infty} (h, B, \gamma , \lambda_ {i}), \right. \\ \left(C _ {8} \left(\lambda_ {1}, \dots , \lambda_ {m}, \gamma , T\right) + C _ {9} \left(\lambda_ {1}, \dots , \lambda_ {m}, \gamma , T\right) h\right) \sigma^ {4} \gamma^ {2 T} \\ + \left(C _ {1 0} \left(\lambda_ {1}, \dots , \lambda_ {m}, \gamma , T\right) h ^ {2} + C _ {1 1} \left(\lambda_ {1}, \dots , \lambda_ {m}, \gamma , T\right) h ^ {3}\right) \sigma^ {4} \gamma^ {T} \\ + C _ {1 2} \left(\lambda_ {1}, \dots , \lambda_ {m}, T\right) \sigma^ {4} h ^ {4} + \frac {\left(C _ {1 3} \left(\lambda_ {1} , \cdots , \lambda_ {m} , \gamma , T\right)\right) \sigma^ {4} T ^ {2 n - 1}}{B h} \\ \left. + \mathcal {O} \left(h ^ {5}\right) + \mathcal {O} \left(\frac {1}{B}\right) \right], \tag {37} \\ \end{array}
$$
where  $MSE_{\infty}(h, B, \gamma, \lambda_i)$  is the mean-squared error of the Monte-Carlo estimator in Theorem 3.6 by replacing the drift  $a$  by  $\lambda_i$ .
Proof. As we can see the proof of Lemma B.1 is based on the proof of Theorem 3.1 with adding a discount factor  $\gamma$ , and the proof of Theorem 3.6 is based on the proof of Lemma B.1 with the decomposition Equation (12). By using the same flow direction, it is sufficient to show the result in case (1) and the results in case (2) and (3) follow.
Consider the decomposition of  $\mathrm{MSE}_T$  in finite-horizon undiscounted setting:
$$
\begin{array}{l} \mathrm {M S E} _ {T} = \mathbb {E} \left[ \left(\hat {V} _ {M} - V _ {T}\right) ^ {2} \right] \\ = \mathbb {E} \left[ \left(\hat {V} _ {M} - \mathbb {E} \left[ \hat {V} _ {M} \right] + \mathbb {E} \left[ \hat {V} _ {M} \right] - V _ {T}\right) ^ {2} \right] \\ = \underbrace {\mathbb {E} \left[ \hat {V} _ {M} ^ {2} \right] - \mathbb {E} \left[ \hat {V} _ {M} \right] ^ {2}} _ {\text {P a r t 1}} + \underbrace {\left(\mathbb {E} \left[ \hat {V} _ {M} \right] - V _ {T}\right) ^ {2}} _ {\text {P a r t 2}} \\ \end{array}
$$
Before the analysis of part 1 and part 2, we will introduce the following mean-squared error notations for the finite-horizon undiscounted scalar case with drift  $\lambda_{i}$ :
$$
\operatorname {M S E} _ {T} (h, B, \lambda_ {i}) = \operatorname {V a r} (h, \lambda_ {i}) + \text {A p p r o x i m a t i o n} (h, B, \lambda_ {i}), \tag {38}
$$
where  $\operatorname{Var}(h, \lambda_i) = \mathbb{E}\left[\hat{V}_M^2\right] - \mathbb{E}\left[\hat{V}_M\right]^2$  and Approximation  $(h, B, \lambda_i) = \left(\mathbb{E}\left[\hat{V}_M\right] - V_T\right)^2$ .
For part 1:
$$
\begin{array}{l} \mathbb {E} \left[ \hat {V} _ {M} ^ {2} \right] = \frac {h ^ {2}}{M} \sum_ {i, j, k, l} \mathbb {E} \left[ X _ {i} (k h) ^ {\top} Q X _ {i} (k h) X _ {j} (l h) ^ {\top} Q X _ {j} (l h) \right] \\ = \frac {h ^ {2}}{M ^ {2}} \sum_ {i, j, k, l} \left[ \mathbb {E} \left[ X _ {i} (k h) ^ {\top} Q X _ {i} (k h) \right] \mathbb {E} \left[ X _ {j} (l h) ^ {\top} Q X _ {j} (l h) \right] + 2 \operatorname {t r} \left\{Q \mathbb {E} \left[ X _ {i} (k h) X _ {j} (l h) ^ {\top} \right] \right\} ^ {2} \right] \\ = h ^ {2} \sum_ {k, l} \mathbb {E} \left[ X (k h) ^ {\top} Q X (k h) \right] \mathbb {E} \left[ X _ {j} (l h) ^ {\top} Q X (l h) \right] + \\ \frac {2 h ^ {2}}{M} \sum_ {k} \operatorname {t r} \left\{Q \mathbb {E} [ X (k h) X (k h) ^ {\top} ] \right\} ^ {2} + \frac {4 h ^ {2}}{M} \sum_ {k <   l} \operatorname {t r} \left\{Q \mathbb {E} [ X (k h) X (l h) ^ {\top} ] \right\} ^ {2}, \tag {39} \\ \end{array}
$$
where the second equality is based on Isserlis' theorem and the trace operation.
Notice that  $\mathbb{E}\left[\hat{V}_M\right]^2 = h^2\sum_{k,l}\mathbb{E}\left[X(kh)^\top QX(kh)\right]\mathbb{E}\left[X(lh)^\top QX(lh)\right]$ , thus
$$
\begin{array}{l} \mathbb {E} \left[ \hat {V} _ {M} ^ {2} \right] - \mathbb {E} \left[ \hat {V} _ {M} \right] ^ {2} \\ = \frac {2 h ^ {2}}{M} \sum_ {k} \operatorname {t r} \left\{Q \mathbb {E} [ X (k h) X (k h) ^ {\top} ] \right\} ^ {2} + \frac {4 h ^ {2}}{M} \sum_ {k <   l} \operatorname {t r} \left\{Q \mathbb {E} [ X (k h) X (l h) ^ {\top} ] \right\} ^ {2} \\ \end{array}
$$
To analyze the above form, we decompose the matrix  $A$  by it Jordan form, i.e.  $A = P^{-1}JP$  for some inevitable matrix  $P$  and  $J = \mathrm{diag}(J_i,\dots ,J_m)$ , where  $J_{i}$  is the Jordan block corresponding to the eigenvalue  $\lambda_{i}$ .
Notice that  $e^{J(kh - s)} = \mathrm{diag}(e^{J_1(kh - s)},\dots ,e^{J_m(kh - s)})$  , where
$$
e ^ {J _ {i} (k h - s)} = e ^ {\lambda_ {i} (k h - s)} \left( \begin{array}{c c c c c} 1 & k h - s & \frac {(k h - s) ^ {2}}{2 !} & \dots & \frac {(k h - s) ^ {q _ {i} - 1}}{(q _ {i} - 1) !} \\ & 1 & k h - s & \dots & \frac {(k h - s) ^ {q _ {i} - 2}}{(q _ {i} - 2) !} \\ & & \ddots & & \vdots \\ & & & & 1 \end{array} \right).
$$
Combining with the fact that for any  $k,l$
$$
\begin{array}{l} \mathbb {E} \left[ X (k h) X (l h) ^ {\top} \right] = \int_ {0} ^ {k h \wedge l h} e ^ {A (k h - s)} e ^ {A ^ {\top} (l h - s)} d s \\ = \int_ {0} ^ {k h \wedge l h} P ^ {- 1} e ^ {J (k h - s)} P P ^ {\top} e ^ {J ^ {\top} (l h - s)} P ^ {- \top} d s, \\ \end{array}
$$
we can conclude that for any  $k \leq l$ , tr  $\{Q\mathbb{E}\left[X(kh)X(lh)^{\top}\right]\}$  is a linear combination of  $\mathcal{L}_{1,i,j}$  and  $\mathcal{L}_{2,i,j}$  for all  $i,j$ , where