SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\lambda (t) \leftarrow - \log \left(\frac {t}{\sigma_ {d} \sqrt {t ^ {2} + \sigma_ {d} ^ {2}}}\right) \qquad t _ {\lambda} (\lambda) \leftarrow \frac {\sigma_ {d}}{\sqrt {\frac {1}{\sigma_ {d} ^ {2} e ^ {- 2 \lambda}} - 1}}, \quad \sigma_ {d} \leftarrow \sigma_ {d a t a}
$$
$$
h \leftarrow \lambda_ {t} - \lambda_ {s}, \quad \lambda_ {s _ {1}} \leftarrow \lambda_ {s} + r h, \quad s _ {1} \leftarrow t _ {\lambda} \left(\lambda_ {s _ {1}}\right)
$$
$$
\epsilon_ {\theta} \left(\widetilde {\mathbf {x}} _ {s}, s\right) \gets \left[ D _ {\theta} \left(\frac {\widetilde {\mathbf {x}} _ {s}}{\alpha_ {s}}; \sigma_ {s}\right) - \frac {\sigma_ {d} ^ {2}}{s ^ {2} + \sigma_ {d} ^ {2}} \frac {\widetilde {\mathbf {x}} _ {s}}{\alpha_ {s}} \right] \cdot \frac {\sqrt {s ^ {2} + \sigma_ {d} ^ {2}}}{s \sigma_ {d}},
$$
$$
\widetilde {\mathbf {x}} _ {s _ {1}} \gets \frac {s _ {1} ^ {2} + \sigma_ {d} ^ {2}}{s ^ {2} + \sigma_ {d} ^ {2}} \widetilde {\mathbf {x}} _ {s} + 2 \frac {s _ {1} \sqrt {s _ {1} ^ {2} + \sigma_ {d} ^ {2}}}{\sigma_ {d}} (e ^ {r h} - 1) \epsilon_ {\theta} (\widetilde {\mathbf {x}} _ {s}, s) + \sqrt {\frac {(s _ {1} ^ {2} + \sigma_ {d} ^ {2}) (s ^ {2} - s _ {1} ^ {2})}{(s ^ {2} + \sigma_ {d} ^ {2})}} z ^ {1}
$$
$$
\widetilde {\mathbf {x}} _ {t} \gets \frac {t ^ {2} + \sigma_ {d} ^ {2}}{s ^ {2} + \sigma_ {d} ^ {2}} \widetilde {\mathbf {x}} _ {s} + 2 ^ {\sqrt {t ^ {2} + \sigma_ {d} ^ {2}}} (\epsilon^ {h} - 1) \epsilon_ {\theta} (\widetilde {\mathbf {x}} _ {s _ {1}}, s _ {1}) + \sqrt {\frac {(t ^ {2} + \sigma_ {d} ^ {2}) (s _ {1} ^ {2} - t ^ {2})}{(s _ {1} ^ {2} + \sigma_ {d} ^ {2})}} (\epsilon^ {r h} z ^ {1} + z ^ {2})
$$
Return  $\tilde{\mathbf{x}}_{t_M}\gets$  last-step  $(\tilde{\mathbf{x}}_{t_{M - 1}},t_{M - 1},t_M)$
Algorithm 6 SEEDS-3 (EDM NP)
Input: initial value  $\mathbf{x}_T$ , steps  $\{t_i\}_{i=0}^M$ , data prediction model  $D_\theta, r_1 = 1/3, r_2 = 2/3$
$$
(t, s) \leftarrow (t _ {i}, t _ {i - 1}), \quad (z ^ {1}, z ^ {2}, z ^ {3}) \leftarrow \mathcal {N} (0, \operatorname {I d}) ^ {\otimes 3}
$$
$$
\lambda (t) \gets - \log \left(\frac {t}{\sigma_ {d} \sqrt {t ^ {2} + \sigma_ {d} ^ {2}}}\right) t _ {\lambda} (\lambda) \gets \frac {\sigma_ {d}}{\sqrt {\frac {1}{\sigma_ {d} ^ {2}} e ^ {- 2 \lambda} - 1}}, \quad \sigma_ {d} \gets \sigma_ {d a t a}
$$
$$
h \leftarrow \lambda_ {t} - \lambda_ {s}, \quad \lambda_ {s _ {1}} \leftarrow \lambda_ {s} + r _ {1} h, \quad \lambda_ {s _ {2}} \leftarrow \lambda_ {s} + r _ {2} h, \quad s _ {1} \leftarrow t _ {\lambda} (\lambda_ {s _ {1}}), \quad s _ {2} \leftarrow t _ {\lambda} (\lambda_ {s _ {2}})
$$
$$
\widetilde {\mathbf {x}} _ {s _ {1}} \leftarrow \frac {s _ {1} ^ {2} + \sigma_ {d} ^ {2}}{s ^ {2} + \sigma_ {d} ^ {2}} \widetilde {\mathbf {x}} _ {s} + 2 \frac {s _ {1} \sqrt {s _ {1} ^ {2} + \sigma_ {d} ^ {2}}}{\sigma_ {d}} (e ^ {r _ {1} h} - 1) \epsilon_ {\theta} (\widetilde {\mathbf {x}} _ {s}, s) + \sqrt {\frac {(s _ {1} ^ {2} + \sigma_ {d} ^ {2}) (s ^ {2} - s _ {1} ^ {2})}{(s ^ {2} + \sigma_ {d} ^ {2})}} z ^ {1}
$$
$$
\boldsymbol {A} \gets \sqrt {\frac {(s _ {2} ^ {2} + \sigma_ {d} ^ {2}) (s _ {1} ^ {2} - s _ {2} ^ {2})}{(s _ {1} ^ {2} + \sigma_ {d} ^ {2})}} (e ^ {r _ {1} h} z ^ {1} + z ^ {2}), \quad P _ {1} \gets \epsilon_ {\theta} (\widetilde {\mathbf {x}} _ {s _ {1}}, s _ {1}) - \epsilon_ {\theta} (\mathbf {x} _ {s}, s)
$$
$$
\widetilde {\mathbf {x}} _ {s _ {2}} \leftarrow \frac {s _ {2} ^ {2} + \sigma_ {d} ^ {2}}{s ^ {2} + \sigma_ {d} ^ {2}} \widetilde {\mathbf {x}} _ {s} + 2 \frac {s _ {2} \sqrt {s _ {2} ^ {2} + \sigma_ {d} ^ {2}}}{\sigma_ {d}} \left(e ^ {r _ {2} h} - 1\right) \epsilon_ {\theta} (\widetilde {\mathbf {x}} _ {s}, s) + \frac {2 r _ {2}}{r _ {1}} \frac {s _ {2} \sqrt {s _ {2} ^ {2} + \sigma_ {d} ^ {2}}}{\sigma_ {d}} \left(\frac {e ^ {r _ {2} h} - 1}{r _ {2} h} - 1\right) P _ {1} + \boldsymbol {A}
$$
$$
\boldsymbol {B} \gets \sqrt {\frac {(t ^ {2} + \sigma_ {d} ^ {2}) (s _ {2} ^ {2} - t ^ {2})}{(s _ {2} ^ {2} + \sigma_ {d} ^ {2})}} \left(e ^ {r _ {2} h} z ^ {1} + e ^ {r _ {1} h} z ^ {2} + z ^ {3}\right), \quad P _ {2} \gets \epsilon_ {\theta} (\widetilde {\mathbf {x}} _ {s _ {2}}, s _ {2}) - \epsilon_ {\theta} (\mathbf {x} _ {s}, s)
$$
$$
\widetilde {\mathbf {x}} _ {t} \leftarrow \frac {t ^ {2} + \sigma_ {d} ^ {2}}{s ^ {2} + \sigma_ {d} ^ {2}} \widetilde {\mathbf {x}} _ {s} + 2 \frac {t \sqrt {t ^ {2} + \sigma_ {d} ^ {2}}}{\sigma_ {d}} (e ^ {h} - 1) \epsilon_ {\theta} (\widetilde {\mathbf {x}} _ {s}, s) + \frac {2}{r _ {2}} \frac {t \sqrt {t ^ {2} + \sigma_ {d} ^ {2}}}{\sigma_ {d}} \left(\frac {e ^ {h} - 1}{h} - 1\right) P _ {2} + \boldsymbol {B}
$$
C Convergence Proofs
In this Section, we give detailed proofs of Theorem 4.1, Proposition 4.3 and Corollary 4.4 stated in the main part of this paper. Let us start recalling its framework. We start by considering the NP NRSDE (15) with VP coefficients:
$$
\begin{array}{l} \mathrm {d} \mathbf {x} _ {t} = \left[ f (t) \mathbf {x} _ {t} + \frac {g ^ {2} (t)}{\bar {\sigma} _ {t}} \boldsymbol {\epsilon} _ {\theta} (\mathbf {x} _ {t}, t) \right] \mathrm {d} t + g (t) \mathrm {d} \bar {\omega} _ {t} \tag {42} \\ = \left[ f (t) \mathbf {x} _ {t} + \frac {2 \alpha_ {t} ^ {2} \dot {\sigma} _ {t} \sigma_ {t}}{\bar {\sigma} _ {t}} \boldsymbol {\epsilon} _ {\theta} (\mathbf {x} _ {t}, t) \right] d t + g (t) d \bar {\boldsymbol {\omega}} _ {t} \\ = \left[ \frac {\mathrm {d} \log \alpha_ {t}}{\mathrm {d} t} \mathbf {x} _ {t} + 2 \alpha_ {t} \dot {\sigma} _ {t} \boldsymbol {\epsilon} _ {\theta} (\mathbf {x} _ {t}, t) \right] \mathrm {d} t + \alpha_ {t} \sqrt {\frac {\mathrm {d} [ \sigma_ {t} ^ {2} ]}{\mathrm {d} t}} \mathrm {d} \bar {\boldsymbol {\omega}} _ {t}. \\ \end{array}
$$
Denote  $t_{\lambda}$  the inverse of  $\lambda_t \coloneqq -\log(\sigma_t)$  (which is a strictly decreasing function of  $t$ ) and denote  $\widehat{\mathbf{x}}_{\lambda} \coloneqq \mathbf{x}(t_{\lambda}(\lambda)), \widehat{\epsilon}_{\theta}(\widehat{\mathbf{x}}_{\lambda}, \lambda) \coloneqq \epsilon_{\theta}(\mathbf{x}(t_{\lambda}(\lambda)), t_{\lambda}(\lambda))$ . We consider a time discretization  $\{t_i\}_{i=0}^{M+1}$  going backwards in time starting from  $t_0 = T$  to  $t_{M+1} = 0$  and to ease the notation we will always
denote  $t < s$  for two consecutive time-steps  $t_i < t_{i-1}$ . The analytic solution at time  $t$  of the RSDE (2) with coefficients (7) and initial value  $\mathbf{x}_s$  reads
$$
\mathbf {x} _ {t} = \frac {\alpha_ {t}}{\alpha_ {s}} \mathbf {x} _ {s} - 2 \alpha_ {t} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} e ^ {- \lambda} \hat {\epsilon} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) d \lambda - \sqrt {2} \alpha_ {t} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} e ^ {- \lambda} d \bar {\omega} (\lambda). \tag {43}
$$
Given an initial condition  $\widetilde{\mathbf{x}}_{t_0} = \mathbf{x}_T$ , the SEEDS-1 iterates read, for  $h_i = \lambda_{t_i} - \lambda_{t_{i - 1}} = \lambda_t - \lambda_s$
$$
\widetilde {\mathbf {x}} _ {t} = \frac {\alpha_ {t}}{\alpha_ {s}} \widetilde {\mathbf {x}} _ {s} - 2 \bar {\sigma} _ {t} (e ^ {h _ {i}} - 1) \boldsymbol {\epsilon} _ {\theta} (\widetilde {\mathbf {x}} _ {s}, s) - \bar {\sigma} _ {t} \sqrt {e ^ {2 h _ {i}} - 1} \boldsymbol {\epsilon} \quad \boldsymbol {\epsilon} \sim \mathcal {N} (\mathbf {0}, \mathbf {I} _ {d}).
$$
Assumption C.1.
1. The function  $\epsilon_{\theta}(\mathbf{x},t)$  is continuous (and hence bounded) on  $[0,T]$ , Lipschitz with respect to  $\mathbf{x}$  and there is a constant  $C$  such that, for  $t,s\in [0,T]$  with  $t < s$ , we have
$$
\left| \boldsymbol {\epsilon} _ {\theta} (\mathbf {x}, t) - \boldsymbol {\epsilon} _ {\theta} (\mathbf {y}, t) \right| ^ {2} \leqslant L _ {1} | \mathbf {x} - \mathbf {y} | ^ {2} \tag {44}
$$
$$
\left| \boldsymbol {\epsilon} _ {\theta} (\mathbf {x}, t) \right| ^ {2} \vee | g (t) | ^ {2} \leqslant L _ {2} \left(1 + \left| \mathbf {x} \right| ^ {2}\right) \tag {45}
$$
$$
\left| \boldsymbol {\epsilon} _ {\theta} (\mathbf {x}, t) - \boldsymbol {\epsilon} _ {\theta} (\mathbf {x}, s) \right| ^ {2} \leqslant L _ {3} (1 + | \mathbf {x} | ^ {2}) | t - s | ^ {2} \tag {46}
$$
2.  $h = \max_{1\leq i\leq M}|h_i|\sim O(1 / M)$  , where  $h_i = \lambda_{t_i} - \lambda_{t_{i - 1}}$
Let  $\mathcal{C}_P^l (\mathbb{R}^d,\mathbb{R})$  denote the family of  $l$  times continuously differentiable real-valued functions on  $\mathbb{R}^d$  whose partial derivatives of order  $\leqslant l$  have polynomial growth and let  $\mathcal{C}_P^{k,l}(I\times \mathbb{R}^d,\mathbb{R})$  be the space of functions  $g(\cdot ,\cdot)$  such that, for all  $(t,x)\in I\times \mathbb{R}^d$ ,  $g(\cdot ,x)\in \mathcal{C}^k (I,\mathbb{R})$  and  $g(t,\cdot)\in \mathcal{C}_P^l (\mathbb{R}^d,\mathbb{R})$ .
Assumption C.2. In addition to Assumption C.1, assume that all the components of  $\epsilon_{\theta}$  belong to  $\mathcal{C}_P^{4,2}(\mathbb{R}^d\times [0,1],\mathbb{R})$
Before going into the proofs, we give some context that lead us to necessitate such assumptions.
For an interval  $I = [t_0, T]$ , let  $\mathbf{x} = (\mathbf{x}(t))_I$  the solution of the following SDE
$$
\mathrm {d} \mathbf {x} (t) = f (\mathbf {x} (t), t) \mathrm {d} t + g (t) \mathrm {d} \omega (t), \tag {47}
$$
where  $g(t) = \hat{g}(t) \cdot \mathrm{Id}_d$  is considered here as a diagonal matrix with identical diagonal entries  $\hat{g}(t)$ . Suppose that  $f, g$  are continuous, and satisfy a linear growth and Lipschitz condition so that the conditions of the Existence and Uniqueness Theorem are fulfilled for the SDE (47).
Let  $I_{h} = \{t_{0},\dots ,t_{M}\}$  be a time discretization of  $I$  with step sizes  $h_n = t_{n + 1} - t_n$  for  $n = 0,\ldots ,M - 1$  and let  $h = \max_{0\leqslant n < M}h_n$ . A time discrete approximation scheme  $\widehat{\mathbf{x}} = (\widehat{\mathbf{x}}_n)_{I_h}$  will be defined as a sequence  $\widehat{\mathbf{x}}_0 = \mathbf{x}(t_0)$  and
$$
\widehat {\mathbf {x}} _ {n + 1} = \Phi (\widehat {\mathbf {x}} _ {n}, h _ {n}, I _ {n}), \qquad n = 0, \dots , M - 1,
$$
where  $I_0$  is independent of  $\widehat{\mathbf{x}}_0$ , with  $I_n = \omega(t_{n+1} - t_n)$  Wiener increments drawn from the normal distributions with zero mean and variance  $h_n$  and which are independent of  $\widehat{\mathbf{x}}_0, \ldots, \widehat{\mathbf{x}}_n$  and  $I_0, \ldots, I_{n-1}$ .
A scheme  $\widehat{\mathbf{x}}$  converges in the strong (resp. weak) sense, with global order  $p > 0$ , to the solution  $\mathbf{x}$  of the SDE (47) if there is a constant  $C > 0$ , independent of  $h$  and  $\delta > 0$ , such that, for each  $h \in ]0, \delta]$ , we have
$$
\left(\mathbb {E} \left[ | \mathbf {x} (T) - \widehat {\mathbf {x}} _ {M} | ^ {2} \right]\right) ^ {1 / 2} \leq C h ^ {p}, \quad \left(\text {r e s p .} | \mathbb {E} [ G (\mathbf {x} (T)) ] - \mathbb {E} [ G (\widehat {\mathbf {x}} _ {M}) ] | \leq C h ^ {p}, \forall G \in \mathcal {C} _ {P} ^ {2 (p + 1)} (\mathbb {R} ^ {d}, \mathbb {R})\right).
$$
Notice that if  $(\mathbb{E}[||\mathbf{x}(T) - \widehat{\mathbf{x}}_M|^2])^{1 / 2} = \mathcal{O}(h^p)$  then for every function  $f$  satisfying a Lipschitz condition, we have  $|\mathbb{E}[f(\mathbf{x}(T))] - \mathbb{E}[f(\widehat{\mathbf{x}}_M)]| = \mathcal{O}(h^p)$ . Nevertheless, this is not enough to infer the optimal weak order of convergence of such method.
Strong convergence is concerned with the precision of the path, while the weak convergence is with the precision of the moments. As, for DPMs, the center of attention is the evolution of the probability densities rather than that of the noising process of single data samples, weak convergence is enough to guarantee the well-conditioning of our numerical schemes. Moreover, when the diffusion coefficient vanishes, then both strong and weak convergence (with the choice  $G = \mathrm{id}$ ) reduce to the usual deterministic convergence criterion for ODEs.
Let us now state some useful results that will be used later on.
Assumption C.3. All moments of the initial value  $\widehat{\mathbf{x}}_0$  exist,  $f$  is continuous, satisfy a linear growth and globally Lipschitz condition.
In particular, since  $I$  is a closed finite interval in  $\mathbb{R}$ ,  $f(\cdot, x)$  and  $g$  are bounded by some constant.
Theorem C.4 ([25]). In addition to C.3, suppose that
$$
\left| \mathbb {E} \left[ \mathbf {x} \left(t _ {1}\right) - \widehat {\mathbf {x}} _ {1} \right] \right| \leqslant C h ^ {p _ {1}}
$$
$$
\left(\mathbb {E} \left[ \left| \mathbf {x} \left(t _ {1}\right) - \widehat {\mathbf {x}} _ {1} \right| ^ {2} \right]\right) ^ {1 / 2} \leqslant C h ^ {p _ {2}}
$$
with  $p_2 \geqslant 1/2$  and  $p_1 \geqslant p_2 + 1/2$ . Then  $\widehat{\mathbf{x}}$  is of strong global order  $p = p_2 - 1/2$ .
Assumption C.5. All moments of the initial value  $\widehat{\mathbf{x}}_0$  exist,  $f$  is continuous, satisfy a linear growth and Lipschitz condition with all their components belonging to  $\mathcal{C}_P^{p + 1,2(p + 1)}(I\times \mathbb{R}^d,\mathbb{R})$  and  $g\in$ $\mathcal{C}^{p + 1}(I,\mathbb{R})$
Theorem C.6 ([25]). In addition to C.5, suppose that
1. for large enough  $r \in \mathbb{N}$ , the moments  $\mathbb{E}[|\widehat{\mathbf{x}}_n|^{2r}]$  exist and are uniformly bounded with respect to  $M$  and  $n = 0, \ldots, M$
2. for all  $G \in \mathcal{C}_P^{2(p+1)}(\mathbb{R}^d, \mathbb{R})$ , if  $\widehat{\mathbf{x}}_n = \mathbf{x}(t_n)$ , then for some  $K \in \mathcal{C}_P^0(\mathbb{R}^d, \mathbb{R})$ , we have
$$
\left| \mathbb {E} \left[ G \left(\mathbf {x} \left(t _ {n + 1}\right)\right) \right] - \mathbb {E} \left[ G \left(\widehat {\mathbf {x}} _ {n + 1}\right) \right] \right| \leq K \left(\widehat {\mathbf {x}} _ {n}\right) h ^ {p + 1}.
$$
Then  $\widehat{\mathbf{x}}$  is of weak global order  $p$ .
Lemma C.7. Suppose that  $\widehat{\mathbf{x}}_0$  has moments of all orders and that, for  $h < 1$ ,
$$
\left| \mathbb {E} \left[ \Phi \left(\widehat {\mathbf {x}} _ {n}, h _ {n}, I _ {n}\right) - \widehat {\mathbf {x}} _ {n} \right] \right| \leqslant K (1 + | \widehat {\mathbf {x}} _ {n} |) h
$$
$$
\left| \Phi (\widehat {\mathbf {x}} _ {n}, h _ {n}, I _ {n}) - \widehat {\mathbf {x}} _ {n} \right| \leqslant X _ {n} (1 + | \widehat {\mathbf {x}} _ {n} |) h ^ {1 / 2}
$$
where  $X_{n}$  has moments of all orders. Then Condition 1 in Theorem C.6 is fulfilled.
C.2 Convergence of SEEDS-1
In this section we will prove that that SEEDS-1 as described above is of global strong order 1.0.
C.2.1 Strong Itô-Taylor approximation
Then the truncated Itô-Taylor expansion of the analytic solution  $\mathbf{x}_t$  of the VP NP NRSDE starting from  $\mathbf{x}_s$  is given, for  $\epsilon \sim \mathcal{N}(0,1)$ , by
$$
\begin{array}{l} \mathbf {x} _ {t} = \frac {\alpha_ {t}}{\alpha_ {s}} \mathbf {x} _ {s} - 2 \alpha_ {t} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} e ^ {- \lambda} \widehat {\epsilon} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) d \lambda - \sqrt {2} \alpha_ {t} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} e ^ {- \lambda} d \boldsymbol {\omega} _ {\lambda} \\ { = } { \frac { \alpha _ { t } } { \alpha _ { s } } \mathbf { x } _ { s } - 2 \bar { \sigma } _ { t } ( e ^ { h } - 1 ) \boldsymbol { \epsilon } _ { \theta } ( \mathbf { x } _ { s } , s ) - \bar { \sigma } _ { t } \sqrt { e ^ { 2 h } - 1 } \boldsymbol { \epsilon } + \mathcal { R } _ { 1 } } \\ { = } { \frac { \alpha _ { t } } { \alpha _ { s } } \mathbf { x } _ { s } - 2 \bar { \sigma } _ { t } ( e ^ { h } - 1 ) \boldsymbol { \epsilon } _ { \theta } ( \mathbf { x } _ { s } , s ) - \bar { \sigma } _ { t } \sqrt { e ^ { 2 h } - 1 } \boldsymbol { \epsilon } + \mathcal { O } ( h ) , } \\ \end{array}
$$
where the symbol  $\mathcal{O}(h^p)$  represent terms  $\mathbf{u}$  such that  $\| \mathbf{u}\| \leqslant |K(\mathbf{x}_s)|h^p$ , for  $K\in \mathcal{C}_P^0 (\mathbb{R}^d,\mathbb{R})$  and small  $h > 0$ . The SEEDS-1 scheme corresponds to such truncated Ito-Taylor expansion containing only the time and Wiener integrals of multiplicity one. As such, since  $G_{t}g(t) = 0$  and assuming Lipschitz and linear growth conditions on  $\epsilon_{\theta}$  as in Assumption C.1,  $\mathbf{x}_t$  can be interpreted as an order 1.0 strong Ito-Taylor approximation [17, Theorem 10.6.3] of the solution to (42).
C.2.2 Continuous approximation of SEEDS-1
Let  $\tilde{\alpha}_t\coloneqq \frac{1}{2}\beta_dt^2 +\beta_m t$  , where  $\beta_{d} = \beta_{\mathrm{max}} - \beta_{\mathrm{min}} = 19.9,\beta_{m} = \beta_{\mathrm{min}} = 0.1$  . We have
$$
\sigma_ {t} = \sqrt {e ^ {\tilde {\alpha} _ {t}} - 1}, \qquad \alpha_ {t} = e ^ {- \frac {1}{2} \tilde {\alpha} _ {t}} = \frac {1}{\sqrt {\sigma_ {t} ^ {2} + 1}}
$$
so, in particular, as  $T = 1$ , we have  $\tilde{\alpha}_{t_0} = \tilde{\alpha}_1 = \frac{1}{2} (\beta_{\mathrm{max}} - \beta_{\mathrm{min}}) + \beta_{\mathrm{min}} = \frac{1}{2} (\beta_{\mathrm{max}} + \beta_{\mathrm{min}}) \approx 10.05$  and  $\tilde{\alpha}_{t_{M+1}} = \tilde{\alpha}_0 = 0$ . We deduce  $\alpha_{t_0} = \alpha_1 \approx e^{-\frac{10.05}{2}} < 1$ ,  $\alpha_{t_{M+1}} = \alpha_0 = 1$ . Next,  $\sigma_{t_0} = \sigma_1 \approx \sqrt{e^{10.05} - 1} > 1$  and  $\sigma_{t_{M+1}} = \sigma_0 = 0$ . As such,  $\lambda_{t_0} = -\log(\sigma_1) := -L_0 < 0$  and  $\lambda_t \underset{t \to t_{M+1}}{\longrightarrow} +\infty$ .
As such, we will set  $t_M = t_{M+1} + \varepsilon$  the end time so that  $\lambda_{t_M} = L_0$  is finite. This implies that, for  $\lambda \in [-L_0, +\infty[$ ,  $0 < e^{-L_0} \leqslant e^{\lambda} < 1 < e^{\lambda_{t_M}}$  and, for  $\hat{T} = \lambda_{t_M} - \lambda_{t_0}$ ,  $e^h = e^{\lambda_t - \lambda_s} \leqslant e^{\hat{T}}$ . Now set, for  $\lambda \in [-L_0, +\infty[$ ,
$$
\hat {\alpha} _ {\lambda} := \sqrt {\frac {1}{1 + e ^ {- 2 \lambda}}}, \qquad \hat {\sigma} _ {\lambda} := \sqrt {\frac {1}{1 + e ^ {+ 2 \lambda}}}.
$$
Then, as  $\lambda$  increases,  $\hat{\alpha}_{\lambda}$  increases starting from  $0 < \hat{\alpha}_{\lambda_{t_0}} < 1$  and  $\hat{\alpha}_{\lambda} \underset{\lambda \to +\infty}{\longrightarrow} 1$  while at the same time  $\hat{\sigma}_{\lambda}$  decreases starting from  $0 < \hat{\sigma}_{\lambda_{t_0}} < 1$  and  $\hat{\sigma}_{\lambda} \underset{\lambda \to +\infty}{\longrightarrow} 0$ . As such, we can rewrite the exact solution (43) as
$$
\widehat {\mathbf {x}} _ {\lambda_ {t}} = \frac {\hat {\alpha} _ {\lambda_ {t}}}{\hat {\alpha} _ {\lambda_ {s}}} \widehat {\mathbf {x}} _ {\lambda_ {s}} - 2 \hat {\alpha} _ {\lambda_ {t}} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} e ^ {- \lambda} \hat {\epsilon} _ {\theta} (\widehat {\mathbf {x}} _ {\lambda}, \lambda) \mathrm {d} \lambda - \sqrt {2} \hat {\alpha} _ {\lambda_ {t}} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} e ^ {- \lambda} \mathrm {d} \bar {\omega} (\lambda). \tag {48}
$$
Notice that  $\frac{\hat{\alpha}_{\lambda_t}}{\hat{\alpha}_{\lambda_s}}$  is bounded for all  $t, s$  by
$$
\frac {\hat {\alpha} _ {\lambda_ {t}}}{\hat {\alpha} _ {\lambda_ {s}}} \leqslant \frac {1}{\hat {\alpha} _ {\lambda_ {t _ {0}}}}
$$
and  $0 < \hat{\alpha}_{\lambda} < 1$  for all  $\lambda \in [-L_0, + \infty[$ .
Recall that the SEEDS-1 is defined recursively as
$$
\mathbf {y} _ {t _ {0}} \leftarrow \mathbf {x} _ {T}, \mathbf {y} _ {t _ {i}} \leftarrow \frac {\alpha_ {t _ {i}}}{\alpha_ {t _ {i - 1}}} \mathbf {y} _ {t _ {i - 1}} - 2 \bar {\sigma} _ {t _ {i}} (e ^ {h _ {i}} - 1) \epsilon_ {\theta} (\mathbf {y} _ {t _ {i - 1}}, t _ {i - 1}) - \bar {\sigma} _ {t _ {i}} \sqrt {e ^ {2 h _ {i}} - 1} \epsilon_ {i}
$$
and, for simplicity, we will denote  $\mathbf{y}_{\lambda_{t_i}}$  for the iterates (48).
Define a continuous approximation of SEEDS-1 as follows. For  $\hat{h} = t - s$ , we write  $\hat{s} = [s / \hat{h}]\hat{h}$  where  $[x]$  denotes the largest integer lesser or equal to  $x$  and  $I_{[A]}$  is the indicator function associated to a set  $A$ . We define the step function:
$$
\widehat {\mathbf {y}} (\lambda) := \sum_ {k \geqslant 0} I _ {[ \lambda_ {t _ {k}}, \lambda_ {t _ {k + 1}} ]} \mathbf {y} _ {\lambda_ {t _ {k}}}
$$
and the continuous approximation
$$
\mathbf {y} (t) := \frac {\alpha_ {t}}{\alpha_ {t _ {0}}} \mathbf {y} (t _ {0}) - 2 \alpha_ {t} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} e ^ {- \hat {\lambda}} \hat {\epsilon} _ {\theta} (\hat {\mathbf {y}} (\lambda), \hat {\lambda}) d \lambda - \sqrt {2} \alpha_ {t} \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} e ^ {- \lambda} d \boldsymbol {\omega} _ {\lambda}.
$$
Proposition C.8. There are two constants  $C_1, C_2$  independent of  $h$  such that, for all  $t \in [0, T]$ , we have
$$
\begin{array}{l} \mathbb {E} \left[ \sup  _ {t _ {0} \leqslant t \leqslant t _ {M}} | \mathbf {y} (t) | ^ {2} \right] \leqslant C _ {1} \\ \mathbb {E} \left[ | \mathbf {y} (t) - \widehat {\mathbf {y}} (t) | ^ {2} \right] \leqslant C _ {2} h ^ {2}. \\ \end{array}
$$
Proof. Recall the standard inequality  $(a + b + c)^2 \leqslant 3(a^2 + b^2 + c^2)$  for  $a, b, c \in \mathbb{R}$ . Then
$$
\left| \mathbf {y} (t) \right| ^ {2} \leqslant 3 \left[ \left| \frac {\alpha_ {t}}{\alpha_ {t _ {0}}} \mathbf {y} _ {t _ {0}} \right| ^ {2} + 4 \alpha_ {t} ^ {2} \left| \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} e ^ {- \hat {\lambda}} \hat {\epsilon} _ {\theta} (\hat {\mathbf {y}} (\lambda), \hat {\lambda}) d \lambda \right| ^ {2} + 2 \alpha_ {t} ^ {2} \left| \int_ {\lambda_ {t _ {0}}} ^ {\lambda_ {t}} e ^ {- \lambda} d \omega_ {\lambda} \right| ^ {2} \right].
$$
Using the fact that  $\alpha_{t} \leqslant 1$ , we have, by writing  $\hat{T} = \lambda_{t_M} - \lambda_{t_0}$  and taking the expectation: