Directed continuous-time random walk with memory

In this paper, we are addressing the old problem of long-term nonlinear autocorrelation function versus short-term linear autocorrelation function. As continuous-time random walk (CTRW) can describe almost all possible kinds of diffusion, it seems to be an excellent tool to use. To be more precise, for instance, CTRW can successfully describe the short-term negative autocorrelation of returns in high-frequency financial data (caused by the bid-ask bounce phenomena). We observe long-term autocorrelation of absolute values of returns. Can it also be described by the CTRW model? And maybe more importantly, to what extent can it be explained by the same phenomena? To refer to these questions, we propose a new directed CTRW model with memory. The canonical CTRW trajectory consists of spatial jumps preceded by waiting times. In directed CTRW, we consider the case with positive spatial jumps only. We take into account the memory in the model as each spatial jump depends on the previous one. This model, based on simple assumptions, allowed us to obtain the general formula covering most popular types of nonlinear autocorrelation functions.