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Pathwise large deviations for the rough Bergomi model: Corrigendum

Published online by Cambridge University Press:  16 September 2021

Stefan Gerhold*
Affiliation:
TU Wien
Antoine Jacquier*
Affiliation:
TU Wien
Mikko Pakkanen*
Affiliation:
TU Wien
Henry Stone*
Affiliation:
Imperial College
Thomas Wagenhofer*
Affiliation:
TU Wien
*
*Postal address: Wiedner Hauptstrasse 8–10, 1040 Vienna, Austria
***Postal address: Imperial College, South Kensington Campus, London SW7 2AZ, UK
***Postal address: Imperial College, South Kensington Campus, London SW7 2AZ, UK
***Postal address: Imperial College, South Kensington Campus, London SW7 2AZ, UK
*Postal address: Wiedner Hauptstrasse 8–10, 1040 Vienna, Austria
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Abstract

This note corrects an error in the definition of the rate function in Jacquier, Pakkanen, and Stone (2018) and slightly simplifies some proofs.

Type
Corrigendum
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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 $\mu,\nu \in \mathcal{C}^*$ ,

\begin{align*} \mu\big(\mathcal{I}^\varphi(\mathcal{I}^\varphi)^* \nu\big) &= \int_0^1 \int_0^t \varphi(u,t) \int_u^1 \varphi(u,s)\, \nu(\mathrm{d} s)\, \mathrm{d} u\, \mu(\mathrm{d} t) \\[2pt] &= \int_0^1 \int_0^1 \int_0^{s \wedge t} \varphi(u,t) \varphi(u,s)\, \mathrm{d} u\, \nu(\mathrm{d} s)\, \mu(\mathrm{d} t) \\[2pt] &= \int_0^1 \int_0^1 \mathbb{E}[Y_t Y_s]\, \nu(\mathrm{d} s)\, \mu(\mathrm{d} t) = \mathbb{E}[ \mu(Y) \nu (Y)],\end{align*}

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

\begin{equation*} \mathcal{I}^{\Phi} f := \int_0^\cdot \Phi(u,\cdot)f(u) \, \mathrm{d} u. \end{equation*}

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

\begin{equation*} \Phi = \begin{pmatrix} \varphi_1 &\quad 0 \\[3pt] \rho \varphi_2 &\quad \sqrt{1-\rho^2}\varphi_2 \end{pmatrix}. \end{equation*}

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

\begin{equation*} \Psi = \begin{pmatrix} K_\alpha &\quad 0 \\[3pt] \rho &\quad \sqrt{1-\rho^2} \end{pmatrix}. \end{equation*}

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

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

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

  4. 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. 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.

Footnotes

The online version of this article has been updated since original publication. A notice detailing the changes has also been published at https://doi.org/10.1017/jpr.2018.72

References

Jacquier, A., Pakkanen, M. S. and Stone, H. (2018). Pathwise large deviations for the rough Bergomi model. J. Appl. Prob. 55, 10781092.CrossRefGoogle Scholar
Lifshits, M. (2012). Lectures on Gaussian Processes. Springer, Heidelberg.CrossRefGoogle Scholar
Stone, H. (2019). Rough volatility models: small-time asymptotics and calibration. PhD thesis. Imperial College.Google Scholar