The paper uses Lévy processes and bivariate Lévy copulae in order to model the behavior of intraday log-returns. Based on assumptions about the form of marginal tail integrals and a Clayton Lévy copula, the model allows for capturing intraday cross-dependency. The model is applied to VaR of the portfolios constructed on stock returns as well as on cryptocurrencies. The proposed method shows fair performance compared to classical time series models.

In the present work, we consider spectrally positive Lévy processes $(X_t,t\ge 0)$ not drifting to $+\infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting $0$. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law ${\mathbb{P}}_x^{☆}$ of this conditioned process (starting at $xgt;0$) is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths, this martingale...

It is well established that resource variability generated by spatial patchiness and turbulence is an important influence on the growth and recruitment of planktonic fish larvae. Empirical data show fractal-like prey distributions, and simulations indicate that scale-invariant foraging strategies may be optimal. Here we show how larval growth and recruitment in a turbulent environment can be formulated as a hitting time problem for a jump-diffusion process. We present two theoretical results. Firstly,...

Let ${\tau }_D\left(Z\right)$ be the first exit time of iterated Brownian motion from a domain $D\subset {ℝ}^n$ started at $z\in D$ and let $P_z[{\tau }_D\left(Z\right)>t]$ be its distribution. In this paper we establish the exact asymptotics of $P_z[{\tau }_D\left(Z\right)>t]$ over bounded domains as an improvement of the results in DeBlassie (2004) [DeBlassie, Ann. Appl. Prob.14 (2004) 1529–1558] and Nane (2006) [Nane, Stochastic Processes Appl.116 (2006) 905–916], for $z\in D$
 ${\displaystyle \underset{t\to \infty }{lim}{t}^{-1/2}exp\left(\frac{3}{2}{\pi }^{2/3}{\lambda }_D^{2/3}{t}^{1/3}\right)P_z[{\tau }_D\left(Z\right)>t]=C\left(z\right),}$
where $C\left(z\right)=\left({\lambda }_D{2}^{7/2}\right)/\sqrt{3\pi }{\left(\psi \left(z\right){\int }_D\psi \left(y\right)\mathrm{d}y\right)}^2$. Here λD is the first eigenvalue of the Dirichlet Laplacian $\frac{1}{2}\Delta$ in D, and ψ is the eigenfunction corresponding...