Hausdorff dimension of the range and the graph of stable-like processes

We determine the Hausdorff dimension for the range of a class of pure jump Markov processes in $\mathbb{R}^d$, which turns out to be random and depends on the trajectories of these processes. The key argument is carried out through the SDE representation of these processes. The method developed here also allows to compute the Hausdorff dimension for the graph.Comment: 16 pages, accepted by Journal of Theoretical Probabilit

On flushed partitions and concave compositions

In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews' partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order $f(q)$, $\phi(q)$ and $\psi(q)$. An identity of Ramanujan is proved combinatorially. Several new identities are also established.Comment: 19 page

On savings ratio.

This paper explores the factors that affect saving and consumption behaviours, in a context where some believe that the high savings ratio of the East Asia and oil-producing countries is one major cause for the global imbalances and the crisis. The paper elaborates on the factors behind the high savings ratios in East Asia and oil producing countries and low savings ratios in the United States. It argues that the high savings in East Asia can mainly be explained by cultural and structural factors in nature. Simple adjustments in nominal exchange rate cannot influence savings behaviour. The paper also argues that savings increased in East Asia as a result of rescue plans designed by international organisations after the 1997 crisis. Encouraging a reduction in savings ratio has been an important component of the national economic policies in China since 2005. The low savings ratios in the United States over the recent period are mainly due to euphoria sentiments and an analysis of time series suggests that no significant causal relationship exists between savings and consumption behaviours in the United States and those in East Asia. The paper discusses a set of options for adjusting savings ratio over the time. They include a switch to a more balanced growth pattern, enhanced regulation of the international speculative capital flows and the channeling of more savings into developing countries and emerging markets. In addition, a reform of the international monetary system is in order so as to prevent the over-concentration of foreign assets in one particular currency.

Multifractal properties of sample paths of ground state-transformed jump processes

We consider a class of L\'evy-type processes with unbounded coefficients, arising as Doob $h$-transforms of Feynman-Kac type representations of non-local Schr\"odinger operators, where the function $h$ is chosen to be the ground state of such an operator. First, we show the existence of a c\`adl\`ag version of the so-obtained ground state-transformed processes. Next, we prove that they satisfy a related stochastic differential equation with jumps. Making use of this SDE, we then derive and prove the multifractal spectrum of local H\"older exponents of sample paths of ground state-transformed processes.Comment: 23 page