Sieving random iterative function systems

It is known that backward iterations of independent copies of a contractive random Lipschitz function converge almost surely under mild assumptions. By a sieving (or thinning) procedure based on adding to the functions time and space components, it is possible to construct a scale invariant stochastic process. We study its distribution and paths properties. In particular, we show that it is c\`adl\`ag and has finite total variation. We also provide examples and analyse various properties of particular sieved iterative function systems including perpetuities and infinite Bernoulli convolutions, iterations of maximum, and random continued fractions.Comment: 36 pages, 2 figures; accepted for publication in Bernoull

Politics and Volatility

We investigate how politics (party orientation, national elections, and strength of democratic institutions) affect stock market volatility. We hypothesize that labor-intensive industries, industries with larger exposure to foreign trade, industries whose operations require efficient contracts, and industries susceptible to government expropriation are more sensitive to changes in political environment. Using a large panel of industry-country-year observations, we show that politically-sensitive industries exhibit higher volatilities during national elections. Volatility is also higher for labor-intensive industries under leftist governments. Moreover, governance-sensitive industries and industries under a higher risk of expropriation are more volatile when democratic institutions are weak. The rise in volatility is driven largely by systematic risk rather than firm-specific risk. The results are consistent with the 'peso problem' hypothesis that uncertainty about future government policies can increase stock market volatility.

Facial structure of strongly convex sets generated by random samples

The $K$-hull of a compact set $A\subset\mathbb{R}^d$, where $K\subset \mathbb{R}^d$ is a fixed compact convex body, is the intersection of all translates of $K$ that contain $A$. A set is called $K$-strongly convex if it coincides with its $K$-hull. We propose a general approach to the analysis of facial structure of $K$-strongly convex sets, similar to the well developed theory for polytopes, by introducing the notion of $k$-dimensional faces, for all $k=0,\dots,d-1$. We then apply our theory in the case when $A=\Xi_n$ is a sample of $n$ points picked uniformly at random from $K$. We show that in this case the set of $x\in\mathbb{R}^d$ such that $x+K$ contains the sample $\Xi_n$, upon multiplying by $n$, converges in distribution to the zero cell of a certain Poisson hyperplane tessellation. From this results we deduce convergence in distribution of the corresponding $f$-vector of the $K$-hull of $\Xi_n$ to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the $f$-vector.Comment: 40 pages, 3 figures; Corollary 6.6 has been corrected and new Theorem 7.3 has been adde

Small mass asymptotic for the motion with vanishing friction

We consider the small mass asymptotic (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and boundary condition of this limiting Markov process and prove the convergence.Comment: final version for publication, accepted by Stochastic Processes and their Application