arXiv subjects

Paolo Pigato

Publications and source records attributed to Paolo Pigato.

22 records · Page 2Linked to original sources

arXiv2016

Tube estimates for diffusion processes under a weak Hörmander condition

We consider a diffusion process under a local weak Hörmander condition on the coefficients. We find Gaussian estimates for the density in short time and exponential lower and upper bounds for the probability that the diffusion remains in a small tube around a deterministic trajectory (skeleton path), explicitly depending on the radius of the tube and on the energy of the skeleton path. We use a norm which reflects the non-isotropic structure of the problem, meaning that the diffusion propagates in $\mathbb{R}^2$ with different speeds in the directions $σ$ and $[σ,b]$. We establish a connection between this norm and the standard control distance.

math.PR↗

arXiv2016

Diffusions under a local strong Hörmander condition. Part II: tube estimates

We study lower and upper bounds for the probability that a diffusion process in $\mathbb{R}^n$ remains in a tube around a skeleton path up to a fixed time. We assume that the diffusion coefficients $σ_1,\ldots,σ_d$ may degenerate but they satisfy a strong Hörmander condition involving the first order Lie brackets around the skeleton of interest. The tube is written in terms of a norm which accounts for the non-isotropic structure of the problem: in a small time $δ$, the diffusion process propagates with speed $\sqrtδ$ in the direction of the diffusion vector fields $σ_{j}$ and with speed $δ=\sqrtδ\times \sqrtδ$ in the direction of $[σ_{i},σ_{j}]$. The proof consists in a concatenation technique which strongly uses the lower and upper bounds for the density proved in the part I.

math.PR↗

arXiv2016

Diffusions under a local strong Hörmander condition. Part I: density estimates

We study lower and upper bounds for the density of a diffusion process in ${\mathbb{R}}^n$ in a small (but not asymptotic) time, say $δ$. We assume that the diffusion coefficients $σ_1,\ldots,σ_d$ may degenerate at the starting time $0$ and point $x_0$ but they satisfy a strong Hörmander condition involving the first order Lie brackets. The density estimates are written in terms of a norm which accounts for the non-isotropic structure of the problem: in a small time $δ$, the diffusion process propagates with speed $\sqrtδ$ in the direction of the diffusion vector fields $σ_{j}$ and with speed $δ=\sqrtδ\times \sqrtδ$ in the direction of $[σ_{i},σ_{j}]$. In the second part of this paper, such estimates will be used in order to study lower and upper bounds for the probability that the diffusion process remains in a tube around a skeleton path up to a fixed time.

math.PR↗

arXiv2014

Multi-scaling of moments in stochastic volatility models

We introduce a class of stochastic volatility models $(X_t)_{t \geq 0}$ for which the absolute moments of the increments exhibit anomalous scaling: $\E\left(|X_{t+h} - X_t|^q \right)$ scales as $h^{q/2}$ for $q < q^*$, but as $h^{A(q)}$ with $A(q) < q/2$ for $q > q^*$, for some threshold $q^*$. This multi-scaling phenomenon is observed in time series of financial assets. If the dynamics of the volatility is given by a mean-reverting equation driven by a Levy subordinator and the characteristic measure of the Levy process has power law tails, then multi-scaling occurs if and only if the mean reversion is superlinear.

math.PR↗