arXiv subjects

Alexandre Pannier

Publications and source records attributed to Alexandre Pannier.

11 recordsLinked to original sources

arXiv2026

Functional integration by parts formulae for stochastic Volterra processes

We investigate integration by parts (IBP) formulae for stochastic Volterra equations and we establish the smoothing effect of the expectation. Due to the inherent path-dependent dynamics of this class of processes, standard Bismut--Elworthy--Li (BEL) formulae and lifting procedures fail to produce representations for directional derivatives with respect to the initial curve. We exhibit a new type of fractional IBP for these derivatives which, by means of the Riemann--Liouville fractional derivative, interpolates between the standard chain rule and a pure BEL formula with Cameron--Martin path directions. Our assumptions describe precisely the trade-off between the direction's and the test function's regularities. Crucially, we reveal that more roughness leads to more smoothing: for a power-law kernel with Hurst parameter $H\in(0,1/2)$, we show that the expectation is differentiable along constant directions provided that the test function has H\"older continuity $\beta>2H$. The proof of the formula relies on a careful analysis of the conditional expectation's temporal regularity and on the well-posedness of its Riemann--Liouville derivative. We complement these results with a BEL formula along all square integrable directions whenever the noise is additive, a second order BEL formula and an application to forward and rough volatility models. In the latter case, the derivative is interpreted as the sensitivity with respect to the whole initial forward variance curve.

math.PR↗

arXiv2025

Kolmogorov equations for stochastic Volterra processes with singular kernels

We associate backward and forward Kolmogorov equations to a class of fully nonlinear Stochastic Volterra Equations (SVEs) with convolution kernels $K$ that are singular at the origin. Working on a carefully chosen Hilbert space $\mathcal{H}_1$, we rigorously establish a link between solutions of SVEs and Markovian mild solutions of a Stochastic Partial Differential Equation (SPDE) of transport-type. Then, we obtain two novel It\^o formulae for functionals of mild solutions and, as a byproduct, show that their laws solve corresponding Fokker-Planck equations. Finally, we introduce a natural notion of "singular" directional derivatives along $K$ and prove that (conditional) expectations of SVE solutions can be expressed in terms of the unique solution to a backward Kolmogorov equation on $\mathcal{H}_1$. Our analysis relies on stochastic calculus in Hilbert spaces, the reproducing kernel property of the state space $\mathcal{H}_1,$ as well as crucial invariance and smoothing properties that are specific to the SPDEs of interest. In the special case of singular power-law kernels, our conditions guarantee well-posedness of the backward equation either for all values of the Hurst parameter $H,$ when the noise is additive, or for all $H>1/4$ when the noise is multiplicative.

math.PR↗

arXiv2025

A BDG inequality for stochastic Volterra integrals

We establish Burkholder-Davis-Gundy-type inequalities for stochastic Volterra integrals with a completely monotone convolution kernel, which may exhibit singular behaviour at the origin. When the supremum is taken over a finite interval, the upper bound depends linearly on the $L^\gamma$-norm of the kernel, for any $\gamma>2$. We demonstrate the utility of this inequality in quantifying the pathwise distance between two stochastic Volterra equations with distinct kernels, with a particular emphasis on the multifactor Markovian approximation. For kernels that decay sufficiently fast, we derive an alternative inequality valid over an infinite time interval, providing uniform-in-time bounds for mean-reverting stochastic Volterra equations. Finally, we compare our findings with existing results in the literature.

math.PR↗

arXiv2024

A path-dependent PDE solver based on signature kernels

We develop a kernel-based solver for path-dependent PDEs (PPDEs) along with a convergence theory. Our numerical scheme leverages signature kernels, a recently introduced class of kernels on path-space. Specifically, we solve an optimal recovery problem by approximating the solution of a PPDE with an element of minimal norm in the signature reproducing kernel Hilbert space constrained to satisfy the PPDE at a finite collection of collocation paths. In the linear case, we show that the optimisation has a unique closed-form solution expressed in terms of signature kernel evaluations at the collocation paths. Under strict assumptions, we prove consistency of the proposed scheme, guaranteeing convergence to the PPDE solution as the number of collocation points increases. Finally, several numerical examples are presented, in particular in the context of option pricing under rough volatility. Our numerical scheme constitutes a valid alternative to the ubiquitous Monte Carlo methods.

math.NA↗

arXiv2023

Path-dependent PDEs for volatility derivatives

We regard options on VIX and Realised Variance as solutions to path-dependent partial differential equations (PDEs) in a continuous stochastic volatility model. The modeling assumption specifies that the instantaneous variance is a $C^3$ function of a multidimensional Gaussian Volterra process; this includes a large class of models suggested for the purpose of VIX option pricing, either rough, or not, or mixed. We unveil the path-dependence of those volatility derivatives and, under a regularity hypothesis on the payoff function, we prove the well-posedness of the associated PDE. The latter is of heat type, because of the Gaussian assumption, and the terminal condition is also path-dependent. Furthermore, formulae for the greeks are provided, the implied volatility is shown to satisfy a quasi-linear path-dependent PDE and, in Markovian models, finite-dimensional pricing PDEs are obtained for VIX options.

math.PR↗

arXiv2023

Rough volatility, path-dependent PDEs and weak rates of convergence

In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional It\^o formula developed by [Viens, F., & Zhang, J. (2019). A martingale approach for fractional Brownian motions and related path dependent PDEs. Ann. Appl. Probab.]. We then leverage these tools to study weak rates of convergence for discretised stochastic integrals of smooth functions of a Riemann-Liouville fractional Brownian motion with Hurst parameter $H \in (0,\frac{1}{2})$. These integrals approximate log-stock prices in rough volatility models. We obtain the optimal weak error rates of order $1$ if the test function is quadratic and of order $(3H+\frac{1}{2})\wedge1$ if the test function is five times differentiable; in particular these conditions are independent of the value of $H$.

math.PR↗

arXiv2022

On the large-time behaviour of affine Volterra processes

We show the existence of a stationary measure for a class of multidimensional stochastic Volterra systems of affine type. These processes are in general not Markovian, a shortcoming which hinders their large-time analysis. We circumvent this issue by lifting the system to a measure-valued stochastic evolution equation introduced by Cuchiero and Teichmann~\cite{CT18}, whence we retrieve the Markov property. Leveraging on the associated generalised Feller property, we extend the Krylov-Bogoliubov theorem to this infinite-dimensional setting and thus establish an approach to the existence of invariant measures. We present concrete examples, including the rough Heston model from Mathematical Finance.

math.PR↗

arXiv2021

Rough multifactor volatility for SPX and VIX options

We provide explicit small-time formulae for the at-the-money implied volatility, skew and curvature in a large class of models, including rough volatility models and their multi-factor versions. Our general setup encompasses both European options on a stock and VIX options, thereby providing new insights on their joint calibration. The tools used are essentially based on Malliavin calculus for Gaussian processes. We develop a detailed theoretical and numerical analysis of the two-factor rough Bergomi model and provide insights on the interplay between the different parameters for joint SPX-VIX smile calibration.

q-fin.MF↗

arXiv2020

Pathwise large deviations for white noise chaos expansions

We consider a family of continuous processes $\{X^\varepsilon\}_{\varepsilon>0}$ which are measurable with respect to a white noise measure, take values in the space of continuous functions $C([0,1]^d:\mathbb{R})$, and have the Wiener chaos expansion \[ X^\varepsilon = \sum_{n=0}^{\infty} \varepsilon^n I_n \big(f_n^{\varepsilon} \big). \] We provide sufficient conditions for the large deviations principle of $\{X^\varepsilon\}_{\varepsilon>0}$ to hold in $C([0,1]^d:\mathbb{R})$, thereby refreshing a problem left open by P\'erez-Abreu (1993) in the Brownian motion case. The proof is based on the weak convergence approach to large deviations: it involves demonstrating the convergence in distribution of certain perturbations of the original process, and thus the main difficulties lie in analysing and controlling the perturbed multiple stochastic integrals. Moreover, adopting this representation offers a new perspective on pathwise large deviations and induces a variety of applications thereof.

math.PR↗

arXiv2020

Large and moderate deviations for stochastic Volterra systems

We provide a unified treatment of pathwise Large and Moderate deviations principles for a general class of multidimensional stochastic Volterra equations with singular kernels, not necessarily of convolution form. Our methodology is based on the weak convergence approach by Budhijara, Dupuis and Ellis. We show in particular how this framework encompasses most rough volatility models used in mathematical finance and generalises many recent results in the literature.

math.PR↗

arXiv2019

On the uniqueness of solutions of stochastic Volterra equations

We prove strong existence and uniqueness, and H\"older regularity, of a large class of stochastic Volterra equations, with singular kernels and non-Lipschitz diffusion coefficient. Extending Yamada-Watanabe's theorem, our proof relies on an approximation of the process by a sequence of semimartingales with regularised kernels. We apply these results to the rough Heston model, with square-root diffusion coefficient, recently proposed in Mathematical Finance to model the volatility of asset prices.

math.PR↗