In this paper, we prove a kind of Abelian theorem for a class of stochastic volatility models \((X,V),\) where both the state process \(X\) and the volatility process \(V\) may have jumps. Our results relate the asymptotic behavior of the characteristic function of \(X_{\Delta}\) for some \(\Delta>0\) in a stationary regime to the Blumenthal-Getoor indexes of the L\'evy processes driving the jumps in \(X\) and \(V\). The results obtained are used to construct consistent estimators for the above Blumenthal-Getoor indexes based on low-frequency observations of the state process \(X\). We derive the convergence rates for the corresponding estimator and show that these rates can not be improved in general.

We consider the extinction events of Galton-Watson processes with countably infinitely many types. In particular, we construct truncated and augmented Galton-Watson processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the global extinction probability vector of the Galton-Watson process with countably infinitely many types. Besides giving rise to a family of new iterative methods for computing the global extinction probability vector, our approach paves the way to new global extinction criteria for branching processes with countably infinitely many types.

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In this note, we consider the construction of a one-dimensional stable Langevin type process confined in the upper half-plane and submitted to diffusive-reflective boundary conditions whenever the particle position hits 0. We show that two main different regimes appear according to the values of the chosen parameters. We then use this study to construct the law of a (free) stable Langevin process conditioned to stay positive, thus extending earlier works on integrated Brownian motion. This construction further allows to obtain the exact asymptotics of the persistence probability of the integrated stable Lévy process. In addition, the paper is concluded by solving the associated trace problem in the symmetric case.

Asymptotic behavior of large excursions probabilities are evaluated for Euclidean norm of a wide class of Gaussian non-stationary vector processes with independent identically distributed components. It is assumed that the components have means zero and variances reaching its absolute maximum at only one point of the considered time interval. The Bessel process is an important example of such processes.

We investigate large deviations for the empirical measure of the forward and backward recurrence time processes associated with a classical renewal process with arbitrary waiting-time distribution. The Donsker-Varadhan theory cannot be applied in this case, and indeed it turns out that the large deviations rate functional differs from the one suggested by such a theory. In particular, a non-strictly convex and non-analytic rate functional is obtained.

We obtain Calderón–Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous nonlinear coefficients. We then derive the well-posedness of the martingale problem associated with related degenerate operators, and therefore uniqueness in law for the corresponding stochastic differential equations. Some density estimates are established as well.

Let a be a finite signed measure on [-r,0], Z a Lévy process (that is a real process with independent stationary increments and càdlàg paths). A linear stochastic delay differential equation

X(t)=X(0)+∫ 0 t ∫ [-r,0] X(s+u)da(u)ds+Z(t),t≥0,(1)driven by Z is studied, only càdlàg solutions to (1) such that Z and (X(t),-r≤t≤0) are independent being considered. Set h(λ)=λ-∫ [-r,0] exp(λu)da(u) and v 0 =sup{Reλ∣λ∈ℂ,h(λ)=0}. Let the Lévy measure of jumps of the process Z be denoted by F. It is shown that there exists a stationary solution to (1) if and only if v 0 <0 and ∫ |y|>1 log|y|dF(y)<∞. If X is a stationary solution to (1), then X(t) equals in law to ∫ 0 ∞ x 0 (t)dZ(t), where x 0 is the fundamental solution of the deterministic counterpart (Z≡0) to (1).

Given a Lévy process (Lt)t≥0 and an independent nondecreasing process (time change) (T(t))t≥0, we consider the problem of statistical inference on T based on low-frequency observations of the time-changed Lévy process LT(t). Our approach is based on the genuine use of Mellin and Laplace transforms. We propose a consistent estimator for the density of the increments of T in a stationary regime, derive its convergence rates and prove the optimality of the rates. It turns out that the convergence rates heavily depend on the decay of the Mellin transform of T. Finally, the performance of the estimator is analysed via a Monte Carlo simulation study.

The Multifractal Embedded Branching Process (MEBP) process and Canonical Embedded Branching Process (CEBP) process were introduced by Decrouez and Jones (2012). The CEBP is a process in which the crossings of dyadic intervals constitute a branching process. An MEBP process is defined as a mul- tifractal time-change of a CEBP process, where the time-change is such that both it and the CEBP can be simulated on-line. In this paper, under various moment conditions, we show that CEBP processes have a constant modulus of continuity, obtain the Hausdorff spectrum of the time-change, and thus obtain the Hausdorff spectrum of an MEBP process.

In this article, we consider the problem of sampling from a probability measure π having a density on R d proportional to x↦ e− U (x). The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential U is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in V-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and π, as well as weak error bounds. Numerical experiments are presented which support our findings.

We provide sharp error bounds for the difference between the transition densities of some multidimensional Continuous Time Markov Chains (CTMC) and the fundamental solutions of some fractional in time Partial (Integro) Differential Equations (P(I)DEs). Namely, we consider equations involving a time fractional derivative of Caputo type and a spatial operator corresponding to the generator of a non degenerate Brownian or stable driven Stochastic Differential Equation (SDE).