We derive the asymptotic laws of winding numbers for planar isotropic stable L evy processes and walks of index 2 (0; 2).

Throughout this work, we shall consider a real-valued Lévy process ξ = (ξt, t ≥ 0), refering to [5, 60] for background. This means that the process ξ starts from ξ0 = 0, has right-continuous paths with left-limits, and if (Ft)t≥0 denotes the natural filtration generated by ξ, then the increment ξt+s − ξt is independent of Ft and has the same law as ξs for every s, t ≥ 0. It is well-known (see e...

The paper establishes strong convergence results for the joint convergence of sequential order statistics. There exists an explicit construction such that almost sure convergence to extremal processes follows. If a partial sum of rowwise i.i.d. random variables is attracted by a non-Gaussian limit law then the results apply to invariance principles for sums of extreme sequential order statistic...

Solution to the optimal stopping problem V (x) = sup E(x + X) + is given, where X = fXtg t0 is a L evy process, and the supremum is taken over the class of stopping times. Results are expressed in terms of the distribution of the random variable M = sup t Xt, under the hypothesis E(M) < +1, and simple conditions for this hypothesis to hold are given. Based on this, the prophet inequality V (x) ...

We consider different generalizations of the Fokker–Planck equation (FPE) devised to describe L evy processes in potential force fields. We show that such generalizations can proceed along different lines. On one hand, L evy statistics can emerge from the fractal temporal nature of the underlying process, i.e., a high variability in the rate of microscopic events. On the other hand, they may be...

By space-fractional (or L evy-Feller) diiusion processes we mean the processes governed by a generalized diiusion equation which generates all L evy stable probability distributions with index (0 < 2), including the two symmetric most popular laws, Cauchy (= 1) and Gauss (= 2). This generalized equation is obtained from the standard linear diiusion equation by replacing the second-order space d...

Solution to the optimal stopping problem for a L evy process and reward functions (e x ?K) + and (K ?e x) + , discounted at a constant rate is given in terms of the distribution of the overall supremum and innmum of the process killed at this rate. Closed forms of this solutions are obtained under the condition of positive jumps mixed-exponentially distributed. Results are interpreted as admiss...

By a classical result of P. L evy, the Brownian motion (B t) t0 on R may be characterized as a continuous process on R such that (B t) t0 and (B 2 t ? t) t0 are martingales. Generalizations of this result are usually obtained in the setting of the so-called martingale problem. This paper contains a variant of the martingale problem for stochastic processes on locally compact groups with indepen...

semilinear stochastic evolution equations with multiplicative l&apos;evy noise are considered‎. ‎the drift term is assumed to be monotone nonlinear and with linear growth‎. ‎unlike other similar works‎, ‎we do not impose coercivity conditions on coefficients‎. ‎we establish the continuous dependence of the mild solution with respect to initial conditions and also on coefficients. ‎as corollarie...

We consider some classes of stationary, counting{measure{valued Markov processes and their companions under time{reversal. Examples arise in the L evy{It^ o decomposition of stable Ornstein{Uhlenbeck processes, the large{time asymptotics of the standard additive coalescent, and extreme value theory. These processes share the common feature that points in the support of the evolving counting{ me...