1. Corrected rate function
Note that the correct rate function also appears in the PhD thesis [Reference Stone3] (see Proposition 1.4.18), but with a different proof. We first give a slightly simplified proof of [Reference Jacquier, Pakkanen and Stone1, Theorem 3.1]. Any unexplained notation is as in [Reference Jacquier, Pakkanen and Stone1].
Let $Y:= \int_0^\cdot \varphi(u,\cdot)\,\mathrm{d} W_u$ be the Gaussian process from that theorem, and $K_Y:\mathcal{C}^* \to \mathcal{C}$ its covariance operator (definition in [Reference Lifshits2, p. 5]). As noted in [Reference Jacquier, Pakkanen and Stone1], $\mathcal{I}^\varphi$ is injective by Titchmarsh’s convolution theorem. By the factorization theorem [Reference Lifshits2, Theorem 4.1] and the discussion in [Reference Lifshits2, pp. 32–33], it suffices to verify the factorization identity $\mathcal{I}^\varphi(\mathcal{I}^\varphi)^*=K_Y$ to conclude that the reproducing kernel Hilbert space (RKHS) is the image $\mathcal{I}^\varphi\big(L^2([0,1])\big)$ . By Fubini’s theorem, we have $(\mathcal{I}^\varphi)^* \mu = \int_\cdot^1 \varphi(\cdot,t)\mu(\mathrm{d} t)$ for any measure $\mu \in \mathcal{C}^*$ . We then compute, for $\mu,\nu \in \mathcal{C}^*$ ,
which proves the theorem.
The second definition in [Reference Jacquier, Pakkanen and Stone1, (2.3)] should be replaced by the following one.
Definition 1. For $\Phi:\mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{R}^{2\times2},$ define $\mathcal{I}^{\Phi}:L^2([0,1],\mathbb{R}^2) \to L^2([0,1],\mathbb{R}^2)$ by
The following theorem replaces [Reference Jacquier, Pakkanen and Stone1, Theorem 3.2].
Theorem 1. Let $\varphi_1,\varphi_2$ satisfy [Reference Jacquier, Pakkanen and Stone1, Assumption 3.1], and define $Y_i:= \int_0^\cdot \varphi_i(u,\cdot) \, \mathrm{d} W_u^i$ , $i=1,2$ , where $W^1$ and $W^2$ are standard Brownian motions with correlation parameter $\rho \in(1,1)$ . Then, the RKHS of $(Y_1,Y_2)$ is $\mathcal{H}^\Phi := \{ \mathcal{I}^\Phi f : f \in L^2([0,1],\mathbb{R}^2) \}$ , with inner product $\langle \mathcal{I}^\Phi f, \mathcal{I}^\Phi g \rangle = \langle f,g \rangle$ , where
Proof. Analogous to the proof above. Injectiveness of $\mathcal{I}^\Phi$ follows from the Titchmarsh convolution theorem. We have $(\mathcal{I}^\Phi)^*\mu = \int_\cdot^1 \Phi^{\top}(\cdot,t) \mu(\mathrm{d} t)$ for any measure $\mu\in(\mathcal{C}^2)^*$ . The factorization identity $\mathcal{I}^\Phi(\mathcal{I}^\Phi)^*=K_{Y_1,Y_2}$ is verified as above.
Theorem 1 implies the following corollary, which replaces [Reference Jacquier, Pakkanen and Stone1, Corollary 3.2].
Corollary 1. The RKHS of the measure induced on $\mathcal{C}^2$ by the process (Z,B) is $\mathcal{H}^{\Psi}$ , where
Consequently, $\ \cdot \_{\mathcal{H}^{\Psi}}$ should replace $\ \cdot \_{\mathcal{H}_\rho^{K_\alpha}}$ in line 4 of p. 1083 and in the proof of [Reference Jacquier, Pakkanen and Stone1, Theorem 2.1] on p. 1088. The special case $\rho=0$ requires no separate treatment, and the result agrees with [Reference Jacquier, Pakkanen and Stone1, Section 5].
2. Minor corrections

1. On p. 1079, last line of the introduction: replace $\int_0^1$ by $\int_0^\cdot$ .

2. On p. 1084, definition of topological dual: add ‘continuous’ before ‘linear functionals’.

3. On p. 1085, second displayed formula: after the second $=$ , replace f by $\Gamma(f^*)$ .

4. In the statement of Theorem 3.4, $\varepsilon \mu$ should be replaced by $\mu(\varepsilon^{1/2}\,\cdot)$ . The speed $\varepsilon^{\beta}$ resulting from the application of the theorem on p. 1088 is correct, though.

5. First line of p. 1089: Replace $v_0^{1+\beta}$ by $v_0 \varepsilon^{1+\beta}$ . To make the estimate work for $t=0$ , confine $\varepsilon$ to the finite interval [0,1] instead of $\mathbb{R}^+$ in line ${4}$ of p. 1088.