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Register a new userInfinite Dimensional Pathwise Volterra Processes Driven by Gaussian Noise -- Probabilistic Properties and Applications
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We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the Holder continuous trajectories of Hilbert-valued Gaussian processes. To this end, we extend the Volterra sewing lemma from cite{HarangTindel} to the two dimensional case, in order to construct two dimensional operator-valued Volterra integrals of Young type. We prove that the covariance operator associated to infinite dimensional Volterra processes can be represented by such a two dimensional integral, which extends the current notion of representation for such covariance operators. We then discuss a series of applications of these results, including the construction of a rough path associated to a Volterra process driven by Gaussian noise with possibly irregular covariance structures, as well as a description of the irregular covariance structure arising from Gaussian processes time-shifted along irregular trajectories. Furthermore, we consider an infinite dimensional fractional Ornstein-Uhlenbeck process driven by Gaussian noise, which can be seen as an extension of the volatility model proposed by Rosenbaum et al. in cite{ElEuchRosenbaum}.
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As Gaussian processes are used to answer increasingly complex questions, analytic solutions become scarcer and scarcer. Monte Carlo methods act as a convenient bridge for connecting intractable mathematical expressions with actionable estimates via sampling. Conventional approaches for simulating Gaussian process posteriors view samples as draws from marginal distributions of process values at finite sets of input locations. This distribution-centric characterization leads to generative strategies that scale cubically in the size of the desired random vector. These methods are prohibitively expensive in cases where we would, ideally, like to draw high-dimensional vectors or even continuous sample paths. In this work, we investigate a different line of reasoning: rather than focusing on distributions, we articulate Gaussian conditionals at the level of random variables. We show how this pathwise interpretation of conditioning gives rise to a general family of approximations that lend themselves to efficiently sampling Gaussian process posteriors. Starting from first principles, we derive these methods and analyze the approximation errors they introduce. We, then, ground these results by exploring the practical implications of pathwise conditioning in various applied settings, such as global optimization and reinforcement learning.
We use Yosida approximation to find an It^o formula for mild solutions $left{X^x(t), tgeq 0right}$ of SPDEs with Gaussian and non-Gaussian coloured noise, the non Gaussian noise being defined through compensated Poisson random measure associated to a Levy process. The functions to which we apply such It^o formula are in $C^{1,2}([0,T]times H)$, as in the case considered for SDEs in [9]. Using this It^o formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such It^o formula to an It^o formula for mild solutions introduced by Ichikawa in [8], and an It^o formula written in terms of the semigroup of the drift operator [11] which we extend before to the non Gaussian case.
We extend recent results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel $K(t,s)$ and inhomogeneous drift and diffusion coefficients $b(s,X_s)$ and $sigma(s,X_s)$. In the case of affine $b$ and $sigma sigma^T$ we show how the conditional Fourier-Laplace functional can be represented by a solution of an inhomogeneous Riccati-Volterra integral equation. For a kernel of convolution type $K(t,s)=overline{K}(t-s)$ we establish existence of a solution to the stochastic inhomogeneous Volterra equation. If in addition $b$ and $sigma sigma^T$ are affine, we prove that the conditional Fourier-Laplace functional is exponential-affine in the past path. Finally, we apply these results to an inhomogeneous extension of the rough Heston model used in mathematical finance.
The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter greater than 1/4 and less than 1/2 in the space variable.
We extend the recently developed rough path theory for Volterra equations from (Harang and Tindel, 2019) to the case of more rough noise and/or more singular Volterra kernels. It was already observed in (Harang and Tindel, 2019) that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of non-geometric rough paths developed in (Gubinelli, 2010) and (Hairer and Kelly, 2015) we provide a simple description of the Volterra rough path and the controlled Volterra process in terms of rooted trees, and with this description we are able to solve rough volterra equations in driven by more irregular signals.
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