XIII Symposium on Probability and Stochastic Processes

These notes were written with the occasion of the XIII Symposium on Probability and Stochastic Processes at UNAM. We will introduce general reflected diffusions with instantaneous reflection when hitting the boundary. Two main tools for studying these processes are presented: the submartingale problem, and stochastic differential equations. We will see how these two complement each other. In the last sections, we will see in detail two processes to which this theory applies nicely, and uniqueness of a stationary distribution holds for them, despite the fact they are degenerate.

Random walks conditioned to stay positive are a prominent topic in fluctuation theory. One way to construct them is as a random walk conditioned to stay positive up to time n, and let n tend to infinity. A second method is conditioning instead to stay positive up to an independent geometric time, and send its parameter to zero. The multidimensional case (condition the components of a d-dimensional random walk to be ordered) was solved by Eichelsbacher and König (Ordered random walks, Electron J Probab 13(46):1307–1336, 2008. MR 2430709) using the first approach, but some moment conditions need to be imposed. Our approach is based on the second method, which has the advantage to require a minimal restriction, needed only for the finiteness of the h-function in certain cases. We also characterize when the limit is Markovian or sub-Markovian, and give several reexpresions of the h-function. Under some conditions given by Ignatiouk-Robert (Harmonic functions of random walks in a semigroup via ladder heights. ArXiv e-prints, 2018), it can be proved that our h-function is the only harmonic function which is zero outside the Weyl chamber \(\{x=(x_1,\ldots , x_d)\in {\mathbb {R}}^d: x_1<\cdots < x_d\}\).

We prove that the rate of convergence for the central limit theorem in finite free convolution is of order n ^−1∕2.

Last passage times arise in a number of areas of applied probability, including risk theory and degradation models. Such times are obviously not stopping times since they depend on the whole path of the underlying process. We consider the problem of finding a stopping time that minimises the L ¹-distance to the last time a spectrally negative Lévy process X is below zero. Examples of related problems in a finite horizon setting for processes with continuous paths are by Du Toit et al. (Stochastics Int J Probab Stochastics Process 80(2–3):229–245, 2008) and Glover and Hulley (SIAM J Control Optim 52(6):3833–3853, 2014), where the last zero is predicted for a Brownian motion with drift, and for a transient diffusion, respectively.

As we consider the infinite horizon setting, the problem is interesting only when the Lévy process drifts to ∞ which we will assume throughout. Existing results allow us to rewrite the problem as a classic optimal stopping problem, i.e. with an adapted payoff process. We use a direct method to show that an optimal stopping time is given by the first passage time above a level defined in terms of the median of the convolution with itself of the distribution function of −inf_t≥0X _t. We also characterise when continuous and/or smooth fit holds

We review combinatorial properties of solitons of the Box-Ball system introduced by Takahashi and Satsuma (J Phys Soc Jpn 59(10):3514–3519, 1990). Starting with several definitions of the system, we describe ways to identify solitons and review a proof of the conservation of the solitons under the dynamics. Ferrari et al. (Soliton decomposition of the box-ball system (2018). arXiv:1806.02798) proposed a soliton decomposition of a configuration into a family of vectors, one for each soliton size. Based on this decompositions, the authors (Ferrari and Gabrielli, Electron. J. Probab. 25, Paper No. 78–1, 2020) propose a family of measures on the set of excursions which induces invariant distributions for the Box-Ball System. In the present paper, we propose a new soliton decomposition which is equivalent to a branch decomposition of the tree associated to the excursion, see Le Gall (Une approche élémentaire des théorèmes de décomposition de Williams. In: Séminaire de Probabilités, XX, 1984/85, vol. 1204, pp. 447–464. Lecture Notes in Mathematics. Springer, Berlin (1986)). A ball configuration distributed as independent Bernoulli variables of parameter λ < 1∕2 is in correspondence with a simple random walk with negative drift 2λ − 1 and having infinitely many excursions over the local minima. In this case the soliton decomposition of the walk consists on independent double-infinite vectors of iid geometric random variables (Ferrari and Gabrielli, Electron. J. Probab. 25, Paper No. 78–1, 2020). We show that this property is shared by the branch decomposition of the excursion trees of the random walk and discuss a corresponding construction of a Geometric branching process with independent but not identically distributed Geometric random variables.

This paper studies the invertibility property of continuous time moving average processes driven by a Lévy process. We provide of sufficient conditions for the recovery of the driving noise. Our assumptions are specified via the kernel involved and the characteristic triplet of the background driving Lévy process.