This paper presents an investigation into a random walk on a cycle graph with restricted forward movement at most $m$ steps, known as the forward jump random walk. The study derives precise formulas for the probability mass function of the arriving state, the hitting time, and its expected value...

High-frequency financial data are characterized by a set of ubiquitous statistical properties that prevail with surprising uniformity. While these 'stylized facts' have been well-known for decades, attempts at their behavioral explanation have remained scarce. However, recently a new branch of...

In this paper, we draw upon the close relationship between statistical physics and mathematical finance to develop a suite of models for financial bubbles and crashes. By modifying previous approaches, we are able to derive novel analytical formulae for evaluation problems and for the expected...