SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathbb {E} \left[ \sup  _ {t _ {0} \leqslant t \leqslant t _ {M}} | \mathbf {y} (t) | ^ {2} \right] \\ \leqslant 3 \left[ \left| \frac {\alpha_ {t}}{\alpha_ {t _ {0}}} \right| ^ {2} \mathbb {E} \left[ | \mathbf {y} _ {t _ {0}} | ^ {2} \right] + 4 \mathbb {E} \left[ \left| \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} e ^ {- \hat {\lambda}} \hat {\epsilon} _ {\theta} (\hat {\mathbf {y}} (\lambda), \hat {\lambda}) d \lambda \right| ^ {2} \right] + 2 \alpha_ {t} ^ {2} \mathbb {E} \left[ \left| \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} e ^ {- \lambda} d \omega_ {\lambda} \right| ^ {2} \right] \right] \\ \leqslant 3 \left[ \left| \frac {\alpha_ {t}}{\alpha_ {t _ {0}}} \right| ^ {2} \mathbb {E} \left[ | \mathbf {y} _ {t _ {0}} | ^ {2} \right] + 4 \hat {T} \mathbb {E} \left[ \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} | e ^ {- \hat {\lambda}} | ^ {2} \left| \hat {\epsilon} _ {\theta} (\widehat {\mathbf {y}} (\lambda), \hat {\lambda}) \right| ^ {2} d \lambda \right] + 2 \alpha_ {t} ^ {2} \mathbb {E} \left[ \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} | e ^ {- \lambda} | ^ {2} d \lambda \right] \right] \\ \leqslant 3 \left[ \left| \frac {\alpha_ {t}}{\alpha_ {t _ {0}}} \right| ^ {2} \mathbb {E} \left[ | \mathbf {y} _ {t _ {0}} | ^ {2} \right] + 4 \hat {T} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} | e ^ {- \hat {\lambda}} | ^ {2} \mathbb {E} \left[ \left| \hat {\epsilon} _ {\theta} (\widehat {\mathbf {y}} (\lambda), \hat {\lambda}) \right| ^ {2} \right] d \lambda + 2 \alpha_ {t} ^ {2} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} e ^ {- 2 \lambda} d \lambda \right] \\ \leqslant 3 K \left[ \mathbb {E} \left[ | \mathbf {y} _ {t _ {0}} | ^ {2} \right] + 4 \hat {T} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ \left| \hat {\epsilon} _ {\theta} (\widehat {\mathbf {y}} (\lambda), \hat {\lambda}) \right| ^ {2} \right] d \lambda + \left(e ^ {2 \left(\lambda_ {t} - \lambda_ {t _ {0}}\right)} - 1\right) \right] \\ \leqslant 3 K \left[ \mathbb {E} \left[ | \mathbf {y} _ {t _ {0}} | ^ {2} \right] + 4 \hat {T} L _ {2} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \left(1 + \mathbb {E} \left[ | \widehat {\mathbf {y}} (\lambda) | ^ {2} \right]\right) \mathrm {d} \lambda + (e ^ {2 \hat {T}} - 1) \right] \\ \leqslant 3 K \left[ \mathbb {E} \left[ | \mathbf {y} _ {t _ {0}} | ^ {2} \right] + 4 \hat {T} L _ {2} \hat {T} + e ^ {2 \hat {T}} - 1 + 4 \hat {T} L _ {2} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ | \widehat {\mathbf {y}} (\lambda) | ^ {2} \right] d \lambda \right] \\ \leqslant 3 K \left(\mathbb {E} \left[ | \mathbf {y} _ {t _ {0}} | ^ {2} \right] + 4 \hat {T} L _ {2} \hat {T} + e ^ {2 \hat {T}} - 1\right) + 3 K 4 \hat {T} L _ {2} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ | \widehat {\mathbf {y}} (\lambda) | ^ {2} \right] d \lambda \\ \leqslant 3 K \left(\mathbb {E} \left[ | \mathbf {x} (t _ {0}) | ^ {2} \right] + 4 \hat {T} ^ {2} L _ {2} + e ^ {2 \hat {T}} - 1\right) + 1 2 K \hat {T} L _ {2} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ \sup  _ {\lambda_ {0} \leqslant r \leqslant \lambda} | \mathbf {y} (r) | ^ {2} \right] d \lambda , \\ \end{array}
$$
where we used the linearity of expectation, Hölder's inequality, Doob's martingale inequality, Itô isometry, the linear growth condition of  $\hat{\epsilon}_{\theta}$ , and we set  $K = \max \left(\left|\frac{\alpha_t}{\alpha_{t_0}}\right|^2, |e^{L_0}|^2, \bar{\sigma}_t^2, 1\right)$ . As we know that  $\mathbb{E}\left[|\mathbf{x}(t_0)|^2\right] < \infty$ , we apply Gronwall's inequality in the last line to obtain
$$
\mathbb {E} \left[ \sup  _ {t _ {0} \leqslant t \leqslant t _ {M}} | \mathbf {y} (t) | ^ {2} \right] \leqslant C _ {1}, \quad C _ {1} := 3 K \left[ \mathbb {E} \left[ | \mathbf {x} (t _ {0}) | ^ {2} \right] + 4 \hat {T} ^ {2} L _ {2} + e ^ {2 \hat {T}} - 1 \right] e ^ {1 2 K \hat {T} L _ {2}}.
$$
Second, we have, for  $s = t_i$ ,  $u = t_{i + 1}$  and  $t\in [t_\lambda (u),t_\lambda (s)[$
$$
\mathbf {y} (t) - \widehat {\mathbf {y}} (t) = \left(\frac {\alpha_ {t}}{\alpha_ {s}} - 1\right) \mathbf {y} _ {s} - 2 \alpha_ {t} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} e ^ {- \widehat {\lambda}} \hat {\epsilon} _ {\theta} (\mathbf {y} _ {\lambda_ {s}}, \lambda_ {s}) \mathrm {d} \lambda ,
$$
so that, using Hölder's inequality, we get
$$
\left| \mathbf {y} (t) - \widehat {\mathbf {y}} (t) \right| ^ {2} \leqslant 3 \left[ \left| \frac {\alpha_ {t}}{\alpha_ {s}} - 1 \right| ^ {2} \left| \mathbf {y} _ {s} \right| ^ {2} + 4 h \int_ {\lambda_ {s}} ^ {\lambda_ {t}} \left| e ^ {- \hat {\lambda}} \right| ^ {2} \left| \hat {\epsilon} _ {\theta} \left(\mathbf {y} _ {\lambda_ {s}}, \lambda_ {s}\right) \right| ^ {2} d \lambda \right].
$$
Now using Itô isometry we obtain:
$$
\mathbb {E} \left[ | \mathbf {y} (t) - \widehat {\mathbf {y}} (t) | ^ {2} \right] \leqslant 3 \left[ \left| \frac {\alpha_ {t}}{\alpha_ {s}} - 1 \right| ^ {2} \mathbb {E} \left[ | \mathbf {y} _ {s} | ^ {2} \right] + 4 h \mathbb {E} \left[ \int_ {\lambda_ {s}} ^ {\lambda_ {t}} | e ^ {- \hat {\lambda}} | ^ {2} | \hat {\epsilon} _ {\theta} (\mathbf {y} _ {\lambda_ {s}}, \lambda_ {s}) | ^ {2} d \lambda \right] \right].
$$
Now, using the bound  $\mathbb{E}\left[\max |\mathbf{y}_t|^2\right] \leqslant C_1$ , the fact that  $\epsilon_{\theta}$  is bounded and the same arguments as above, we get
$$
\begin{array}{l} \mathbb {E} \left[ | \mathbf {y} (t) - \widehat {\mathbf {y}} (t) | ^ {2} \right] \leqslant 3 \left[ \left| \frac {\alpha_ {t}}{\alpha_ {s}} - 1 \right| ^ {2} C _ {1} + 4 K h \mathbb {E} \left[ \int_ {\lambda_ {s}} ^ {\lambda_ {t}} | \hat {\epsilon} _ {\theta} (\mathbf {y} _ {\lambda_ {s}}, \lambda_ {s}) | ^ {2} d \lambda \right] \right] \\ \leqslant 3 \left[ \left| \frac {\alpha_ {t}}{\alpha_ {s}} - 1 \right| ^ {2} C _ {1} + 4 K h L _ {2} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} (1 + C _ {1}) \mathrm {d} \lambda \right] \\ \leqslant 3 \left[ \left| \frac {\alpha_ {t}}{\alpha_ {s}} - 1 \right| ^ {2} C _ {1} + 4 K h ^ {2} L _ {2} (1 + C _ {1}) \right]. \\ \end{array}
$$
Finally, as we have
$$
e ^ {h} = \frac {\sigma_ {s}}{\sigma_ {t}} = \frac {\alpha_ {t}}{\alpha_ {s}} \frac {\sqrt {1 - \alpha_ {s} ^ {2}}}{\sqrt {1 - \alpha_ {t} ^ {2}}},
$$
and  $1 < \frac{\sqrt{1 - \alpha_s^2}}{\sqrt{1 - \alpha_t^2}} \underset{M \to \infty}{\longrightarrow} 1$ , we obtain  $\left|\frac{\alpha_t}{\alpha_s} - 1\right|^2 \sim |e^h - 1|^2 \sim \mathcal{O}(h^2)$ . Now, by denoting  $|e^h - 1|^2 \leqslant K_2 h^2$ , we conclude that
$$
\mathbb {E} \left[ | \mathbf {y} (t) - \widehat {\mathbf {y}} (t) | ^ {2} \right] \leqslant C _ {2} h ^ {2},
$$
with  $C_2 \coloneqq 3K_2C_1 + 4KL_2(1 + C_1)$ . This finishes the proof.
C.2.3 Proof of Theorem 4.1
Let's now take a look at the approximation given  $\mathbf{y}_{t_0} = \mathbf{x}_{t_0}$ . We have
$$
\mathbf {y} _ {t} - \mathbf {x} _ {t} = 2 \alpha_ {t} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \left[ e ^ {- \lambda} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - e ^ {- \hat {\lambda}} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {y}} _ {\lambda}, \hat {\lambda}) \right] \mathrm {d} \lambda .
$$
Using the inequality  $\alpha_{t} \leq 1$ , the Lipschitz property of  $\widehat{\pmb{\epsilon}}_{\theta}$ , Assumption (C.1), and Hölder's inequality we deduce the bound:
$$
\begin{array}{l} \left| \mathbf {y} _ {t} - \mathbf {x} _ {t} \right| ^ {2} \leqslant 2 \left[ 4 \left| \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \left[ e ^ {- \lambda} \widehat {\epsilon} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - e ^ {- \hat {\lambda}} \widehat {\epsilon} _ {\theta} (\widehat {\mathbf {y}} _ {\lambda}, \hat {\lambda}) \right] d \lambda \right| ^ {2} \right] \\ \leqslant 8 \hat {T} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \left| e ^ {- \lambda} \widehat {\epsilon} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - e ^ {- \hat {\lambda}} \widehat {\epsilon} _ {\theta} (\widehat {\mathbf {y}} _ {\lambda}, \hat {\lambda}) \right| ^ {2} d \lambda . \\ \end{array}
$$
Taking the expectation yields
$$
\begin{array}{l} \mathbb {E} \left[ \sup  _ {t _ {0} \leqslant s \leqslant t} \left| \mathbf {y} _ {s} - \mathbf {x} _ {s} \right| ^ {2} \right] \\ \leqslant 8 \hat {T} \mathbb {E} \left[ \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \left| e ^ {- \lambda} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - e ^ {- \hat {\lambda}} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {y}} _ {\lambda}, \hat {\lambda}) \right| ^ {2} d \lambda \right]. \\ \end{array}
$$
Now, for the first integral, by writing:
$$
\begin{array}{l} e ^ {- \lambda} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - e ^ {- \bar {\lambda}} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {y}} _ {\lambda}, \hat {\lambda}) \\ = e ^ {- \lambda} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - e ^ {- \widehat {\lambda}} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) + e ^ {- \widehat {\lambda}} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - e ^ {- \widehat {\lambda}} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \widehat {\lambda}) \\ + e ^ {- \hat {\lambda}} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \hat {\lambda}) - e ^ {- \hat {\lambda}} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\mathbf {y} _ {\lambda}, \hat {\lambda}) + e ^ {- \hat {\lambda}} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\mathbf {y} _ {\lambda}, \hat {\lambda}) - e ^ {- \hat {\lambda}} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {y}} _ {\lambda}, \hat {\lambda}) \\ = \left(e ^ {- \lambda} - e ^ {- \hat {\lambda}}\right) \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) + e ^ {- \hat {\lambda}} \left(\widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \hat {\lambda})\right) \\ + e ^ {- \hat {\lambda}} \left(\widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \hat {\lambda}) - \widehat {\boldsymbol {\epsilon}} _ {\theta} (\mathbf {y} _ {\lambda}, \hat {\lambda}) + \widehat {\boldsymbol {\epsilon}} _ {\theta} (\mathbf {y} _ {\lambda}, \hat {\lambda}) - \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {y}} _ {\lambda}, \hat {\lambda})\right), \\ \end{array}
$$
we can state the following inequalities:
$$
\begin{array}{l} \mathbb {E} \left[ \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \left| e ^ {- \lambda} \widehat {\pmb {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - e ^ {- \hat {\lambda}} \widehat {\pmb {\epsilon}} _ {\theta} (\widehat {\mathbf {y}} _ {\lambda}, \hat {\lambda}) \right| ^ {2} \mathrm {d} \lambda \right] \\ \leqslant 3 \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} | e ^ {- \lambda} - e ^ {- \hat {\lambda}} | ^ {2} \mathbb {E} \left[ | \widehat {\epsilon} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) | ^ {2} \right] d \lambda + 3 \mathbb {E} \left[ \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} | e ^ {- \hat {\lambda}} | ^ {2} \left| \widehat {\epsilon} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - \widehat {\epsilon} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \hat {\lambda}) \right| ^ {2} d \lambda \right] \\ + 3 \mathbb {E} \left[ \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} | e ^ {- \hat {\lambda}} | ^ {2} \left| \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \hat {\lambda}) - \widehat {\boldsymbol {\epsilon}} _ {\theta} (\mathbf {y} _ {\lambda}, \hat {\lambda}) + \widehat {\boldsymbol {\epsilon}} _ {\theta} (\mathbf {y} _ {\lambda}, \hat {\lambda}) - \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {y}} _ {\lambda}, \hat {\lambda}) \right| ^ {2} d \lambda \right] \\ \leqslant 3 \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} | e ^ {- \lambda} - e ^ {- \hat {\lambda}} | ^ {2} \mathbb {E} \left[ | \widehat {\epsilon} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) | ^ {2} \right] d \lambda + 3 K L _ {3} \mathbb {E} \left[ \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} (1 + | \widehat {\mathbf {x}} _ {\lambda} | ^ {2}) | (\lambda - \hat {\lambda}) | ^ {2} d \lambda \right] \\ + 3 \mathbb {E} \left[ \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} | e ^ {- \hat {\lambda}} | ^ {2} \left| \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \hat {\lambda}) - \widehat {\boldsymbol {\epsilon}} _ {\theta} (\mathbf {y} _ {\lambda}, \hat {\lambda}) + \widehat {\boldsymbol {\epsilon}} _ {\theta} (\mathbf {y} _ {\lambda}, \hat {\lambda}) - \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {y}} _ {\lambda}, \hat {\lambda}) \right| ^ {2} \mathrm {d} \lambda \right] \\ \leqslant 3 L _ {2} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} | e ^ {- \hat {\lambda}} | ^ {2} | e ^ {\lambda - \hat {\lambda}} - 1 | ^ {2} (1 + \mathbb {E} [ | \widehat {\mathbf {x}} _ {\lambda} | ^ {2} ]) \mathrm {d} \lambda + 3 K L _ {3} h ^ {2} \mathbb {E} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} (1 + | \widehat {\mathbf {x}} _ {\lambda} | ^ {2}) \mathrm {d} \lambda \\ + 3 K L _ {1} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ | \widehat {\mathbf {x}} _ {\lambda} - \mathbf {y} _ {\lambda} + \mathbf {y} _ {\lambda} - \widehat {\mathbf {y}} _ {\lambda} | ^ {2} \right] \mathrm {d} \lambda . \\ \end{array}
$$
Thus, we obtain
$$
\begin{array}{l} \mathbb {E} \left[ \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \left| e ^ {- \lambda} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - e ^ {- \hat {\lambda}} \widehat {\boldsymbol {\epsilon}} _ {\theta} (\widehat {\mathbf {y}} _ {\lambda}, \hat {\lambda}) \right| ^ {2} d \lambda \right] \\ \leqslant 3 L _ {2} K \hat {T} | e ^ {h} - 1 | ^ {2} (1 + C _ {1}) + 3 L _ {3} h ^ {2} K (1 + C _ {1}) \hat {T} \\ + 3 K L _ {1} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ | \widehat {\mathbf {x}} _ {\lambda} - \mathbf {y} _ {\lambda} + \mathbf {y} _ {\lambda} - \widehat {\mathbf {y}} _ {\lambda} | ^ {2} \right] d \lambda \\ \leqslant 3 L _ {2} K \hat {T} | e ^ {h} - 1 | ^ {2} (1 + C _ {1}) + 3 L _ {3} h ^ {2} K (1 + C _ {1}) \hat {T} \\ + 6 K L _ {1} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ | \widehat {\mathbf {x}} _ {\lambda} - \mathbf {y} _ {\lambda} | ^ {2} + | \mathbf {y} _ {\lambda} - \widehat {\mathbf {y}} _ {\lambda} | ^ {2} \right] d \lambda \\ \end{array}
$$
$$
\begin{array}{l} \leqslant 3 L _ {2} K \hat {T} | e ^ {h} - 1 | ^ {2} (1 + C _ {1}) + 3 L _ {3} h ^ {2} K (1 + C _ {1}) \hat {T} \\ + 6 K L _ {1} \hat {T} C _ {2} h ^ {2} + 6 K L _ {1} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ | \widehat {\mathbf {x}} _ {\lambda} - \mathbf {y} _ {\lambda} | ^ {2} \right] d \lambda \\ \end{array}
$$
$$
\begin{array}{l} \leqslant 3 L _ {2} K \hat {T} | e ^ {h} - 1 | ^ {2} (1 + C _ {1}) + 3 L _ {3} h ^ {2} K (1 + C _ {1}) \hat {T} \\ + 6 K L _ {1} \hat {T} C _ {2} h ^ {2} + 6 K L _ {1} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ \sup  _ {\lambda_ {0} \leqslant r \leqslant \lambda} | \widehat {\mathbf {x}} (r) - \mathbf {y} (r) | ^ {2} \right] d \lambda \\ \end{array}
$$
$$
\begin{array}{l} \leqslant 3 K \hat {T} \left(1 + C _ {1}\right) \left[ L _ {2} \left| e ^ {h} - 1 \right| ^ {2} + L _ {3} h ^ {2} \right] + 6 K L _ {1} \hat {T} C _ {2} h ^ {2} \\ + 6 K L _ {1} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ \sup  _ {\lambda_ {0} \leqslant r \leqslant \lambda} | \widehat {\mathbf {x}} (r) - \mathbf {y} (r) | ^ {2} \right] d \lambda . \\ \end{array}
$$
Putting everything together yields
$$
\begin{array}{l} \mathbb {E} \left[ \sup  _ {t _ {0} \leqslant s \leqslant t} \left| \mathbf {y} _ {s} - \mathbf {x} _ {s} \right| ^ {2} \right] \\ \leqslant 8 \hat {T} [ 3 K \hat {T} (1 + C _ {1}) [ L _ {2} | e ^ {h} - 1 | ^ {2} + L _ {3} h ] + 6 K L _ {1} \hat {T} C _ {2} h ^ {2} ] \\ + 8 \hat {T} 6 K L _ {1} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ \sup  _ {\lambda_ {0} \leqslant r \leqslant \lambda} | \widehat {\mathbf {x}} (r) - \mathbf {y} (r) | ^ {2} \right] d \lambda \\ \leqslant 2 4 K \hat {T} ^ {2} \left(1 + C _ {1}\right) \left[ L _ {2} \left| e ^ {h} - 1 \right| ^ {2} + L _ {3} h ^ {2} \right] + 4 8 K L _ {1} \hat {T} ^ {2} C _ {2} h ^ {2} \\ + 4 8 \hat {T} K L _ {1} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} \mathbb {E} \left[ \sup  _ {\lambda_ {0} \leqslant r \leqslant \lambda} | \widehat {\mathbf {x}} (r) - \mathbf {y} (r) | ^ {2} \right] d \lambda . \\ \end{array}
$$
Now, by denoting  $|e^h - 1|^2 \leqslant K_2 h^2$ , we apply the continuous version of the Gronwall Lemma to obtain:
$$
\mathbb {E} \left[ \sup  _ {t _ {0} \leqslant s \leqslant t} | \mathbf {y} _ {t} - \mathbf {x} _ {t} | ^ {2} \right] \leqslant C _ {0} h ^ {2},
$$
$$
C _ {0} = \left[ 2 4 K \hat {T} ^ {2} \left(1 + C _ {1}\right) \left[ L _ {2} K _ {2} + L _ {3} \right] + 4 8 K \hat {T} ^ {2} L _ {1} C _ {2} \right] e ^ {4 8 \hat {T} K L _ {1}}.
$$
Finally, using Lyapunov's inequality we obtain, for  $C = \sqrt{C_0}$
$$
\mathbb {E} \left[ \left| \mathbf {y} _ {T} - \mathbf {x} _ {T} \right| \right] \leqslant \left(\mathbb {E} \left[ \left| \mathbf {y} _ {T} - \mathbf {x} _ {T} \right| ^ {2} \right]\right) ^ {1 / 2} \leqslant C h.
$$
In other words, for  $C$  as stated above, we have the following inequality
$$
\sqrt {\mathbb {E} \left[ \sup  _ {t _ {0} \leqslant t \leqslant t _ {M}} | \widetilde {\mathbf {x}} _ {t} - \mathbf {x} _ {t} | ^ {2} \right]} \leqslant C h, \quad \text {a s} h \longrightarrow 0.
$$
Remark C.9. From the above, it is easy to induce that the following order for the one step error
$$
\mathbb {E} [ | \mathbf {x} (t _ {1}) - \mathbf {y} _ {1} | ^ {2} ] = \mathcal {O} (h ^ {3}).
$$
Now, since  $G_{t}g(t) = 0$  in this case (additive noise), it is easy to see from the truncated Ito-Taylor expansion of  $\mathbf{x}$  that  $|\mathbb{E}[\mathbf{x}(t_1) - \mathbf{y}_1]| = \mathcal{O}(h^2)$ . As such, we apply Theorem C.4 to conclude that SEEDS-1 has strong convergence of global order 1.0.
C.2.4 Discrete-time approximation
By Theorem 4.1, we know that SEEDS-1, being of strong order 1.0, it is immediately also of weak order 1. Nonetheless, let's give a discrete approach of this statement that we will use for the proofs of convergence for the remaining solvers as stated in Corollary 4.4. Define the discrete time process:
$$
\mathbf {y} _ {t _ {0}} \gets \mathbf {x} _ {T}, \mathbf {y} _ {t _ {i}} \gets \frac {\alpha_ {t _ {i}}}{\alpha_ {t _ {i - 1}}} \mathbf {y} _ {t _ {i - 1}} - 2 \sigma_ {t _ {i}} (e ^ {h _ {i}} - 1) \boldsymbol {\epsilon} (\mathbf {y} _ {t _ {i - 1}}, t _ {i - 1}) - \sqrt {2} \alpha_ {t _ {i}} \int_ {\lambda_ {t _ {i - 1}}} ^ {\lambda_ {t _ {i}}} e ^ {- s} \mathrm {d} \bar {\boldsymbol {\omega}} (s).
$$
We will prove that  $\mathbb{E}[|\mathbf{y}_{t_M} - \mathbf{x}_{t_M}|]$  is of order  $h$  as  $h\longrightarrow 0$ . Note that  $(\mathbf{y}_{t_i})_i$  has the same distribution as  $(\widetilde{\mathbf{x}}_{t_i})_i$  described in Algorithm 2 since the stochastic integrals  $\left(\int_{\lambda_{t_i - 1}}^{\lambda_{t_i}}e^{-s}\mathrm{d}\bar{\omega} (s)\right)_i$  are independent and each  $\int_{\lambda_{t_i - 1}}^{\lambda_{t_i}}e^{-s}\mathrm{d}\bar{\omega} (s)$  is distributed as  $\frac{1}{\sqrt{2}}\frac{\sigma_{t_i}}{\alpha_{t_i}}\sqrt{e^{2h_i} - 1}\epsilon$  with  $h_i = \lambda_{t_i} - \lambda_{t_i - 1},\quad \epsilon \sim \mathcal{N}(\mathbf{0},\mathbf{I}_d)$ . We have:
$$
\mathbf {y} _ {t _ {i}} - \mathbf {x} _ {t _ {i}} = \frac {\alpha_ {t _ {i}}}{\alpha_ {t _ {i - 1}}} (\mathbf {y} _ {t _ {i - 1}} - \mathbf {x} _ {t _ {i - 1}}) + 2 \alpha_ {t _ {i}} \int_ {\lambda_ {t _ {i - 1}}} ^ {\lambda_ {t _ {i}}} e ^ {- \lambda} (\widehat {\epsilon} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) - \epsilon (\mathbf {y} _ {t _ {i - 1}}, t _ {i - 1})) d \lambda .
$$
For simplicity, in what follows  $C$  will denote a constant not dependent on the subdivision of  $[0,T]$  that may change from one line to the next by systematically denoting the maximum value of the
different constants appearing in the line before. Using the inequality  $\alpha_{t} \leq 1$ , the Lipschitz property of  $\epsilon$ , we deduce the bound:
$$
| \mathbf {y} _ {t _ {i}} - \mathbf {x} _ {t _ {i}} | \leq \frac {\alpha_ {t _ {i}}}{\alpha_ {t _ {i - 1}}} | \mathbf {y} _ {t _ {i - 1}} - \mathbf {x} _ {t _ {i - 1}} | + C \left[ \int_ {\lambda_ {t _ {i - 1}}} ^ {\lambda_ {t _ {i}}} e ^ {- \lambda} | \widehat {\mathbf {x}} _ {\lambda} - \mathbf {y} _ {t _ {i - 1}} | \mathrm {d} \lambda + \int_ {\lambda_ {t _ {i - 1}}} ^ {\lambda_ {t _ {i}}} e ^ {- \lambda} | t _ {\lambda} (\lambda) - t _ {i - 1} | \mathrm {d} \lambda \right].
$$
$$
\begin{array}{l} \int_ {\lambda_ {t _ {i - 1}}} ^ {\lambda_ {t _ {i}}} e ^ {- \lambda} | \widehat {\mathbf {x}} _ {\lambda} - \mathbf {y} _ {t _ {i - 1}} | \mathrm {d} \lambda \leq \int_ {\lambda_ {t _ {i - 1}}} ^ {\lambda_ {t _ {i}}} e ^ {- \lambda} | \widehat {\mathbf {x}} _ {\lambda} - \mathbf {x} _ {t _ {i - 1}} | \mathrm {d} \lambda + \int_ {\lambda_ {t _ {i - 1}}} ^ {\lambda_ {t _ {i}}} e ^ {- \lambda} | \mathbf {x} _ {t _ {i - 1}} - \mathbf {y} _ {t _ {i - 1}} | \mathrm {d} \lambda \\ \leq \int_ {\lambda_ {t _ {i - 1}}} ^ {\lambda_ {t _ {i}}} e ^ {- \lambda} | \widehat {\mathbf {x}} _ {\lambda} - \mathbf {x} _ {t _ {i - 1}} | d \lambda + C h | \mathbf {x} _ {t _ {i - 1}} - \mathbf {y} _ {t _ {i - 1}} |, \\ \end{array}
$$
and recall that  $\widehat{\mathbf{x}}_u = \mathbf{x}_{t_\lambda (u)}$ . Using the fact that  $t_\lambda$  is increasing and Lemma C.11, we have:
$$
\mathbb {E} \left[ \int_ {\lambda_ {t _ {i - 1}}} ^ {\lambda_ {t _ {i}}} e ^ {- \lambda} | \widehat {\mathbf {x}} _ {\lambda} - \mathbf {x} _ {t _ {i - 1}} | \mathrm {d} \lambda \right] \leq C \int_ {\lambda_ {t _ {i - 1}}} ^ {\lambda_ {t _ {i}}} e ^ {- u} \sqrt {| t _ {\lambda} - t _ {i - 1} |} \mathrm {d} \lambda \leq C h ^ {3 / 2}.
$$
Introduce  $U_{i} = \mathbb{E}[|\mathbf{y}_{t_{i}} - \mathbf{x}_{t_{i}}|]$ . Since  $\int_{\lambda_{t_{i-1}}}^{\lambda_{t_{i}}} e^{-\lambda} |t_{\lambda}(\lambda) - t_{i-1}| \, \mathrm{d}\lambda \leq C h^{2}$ :
$$
U _ {i} \leq \left(\frac {\alpha_ {t _ {i}}}{\alpha_ {t _ {i - 1}}} + C h\right) U _ {i - 1} + C h ^ {2}.
$$
Let  $a_{i} = \frac{\alpha_{t_{i}}}{\alpha_{t_{i - 1}}} + Ch$  and  $b_{i} = Ch^{2}$ . By applying Lemma C.12, we have:  $U_{M} \leq A_{M}U_{0} + \sum_{k = 1}^{M}A_{k,n}b_{k}$  with  $A_{M} = \prod_{k = 1}^{M}a_{k}$  and  $A_{k,M} = A_M / A_k = \prod_{i = k + 1}^{M}a_{i}$ . Note that  $U_{0} = 0$  since  $\mathbf{y}_{t_0} = \mathbf{x}_T$  and so:
$$
U _ {M} \leq C h ^ {2} \sum_ {k = 0} ^ {M - 1} \left(\sup  _ {1 \leq i \leq M} a _ {i}\right) ^ {k}.
$$
Using our hypothesis, we can bound:
$$
\sum_ {k = 0} ^ {M - 1} \left(\sup  _ {1 \leq i \leq M} a _ {i}\right) ^ {k} \leq \sum_ {k = 0} ^ {M - 1} (\exp (C h) + C h) ^ {k} = \frac {(\exp (C h) + C h) ^ {1 / h} - 1}{(\exp (C h) + C h) - 1}.
$$
The quantity on the right is of order  $C / h$ . Indeed, as  $h$  goes to 0,  $\exp (Ch) + Ch - 1$  is equivalent to  $2Ch$  and  $(\exp (Ch) + Ch)^{1 / h}$  converges to a constant. This gives the bound  $U_{M} \leq Ch$ , when  $h \to 0$  and by using Proposition C.8 and Theorem C.6 we conclude that SEEDS-1 is convergent of weak order 1.
C.2.5 Useful Lemmas
Lemma C.10. (Continuous Gronwall Lemma) Let  $I = [a,b]$  denote a compact interval of the real line with  $a < b$ . Let  $\alpha, \beta, u$  be continuous real-valued functions defined on  $I$ . Assume  $\beta$  is non-negative,  $\alpha$  is non-decreasing and  $u$  satisfies the integral inequality
$$
u (t) \leq \alpha (t) + \int_ {a} ^ {t} \beta (s) u (s) \mathrm {d} s, \quad \forall t \in I.
$$
$$
u (t) \leq \alpha (t) \exp \left(\int_ {a} ^ {t} \beta (s) \mathrm {d} s\right), \quad \forall t \in I.
$$