The canonical Continuous-Time Random Walk (CTRW) formalism was originally introduced by physicists, Montroll and Weiss, in 1965 as a way to render time continuous in the classical random walk. In both processes mentioned jumps of the  process value are random and independent. The difference is in the time dimension. In the classical random walk time intervals between jumps are constant while in Continuous-Time Random Walk intervals are random and independent, just like the jumps of the process value.

Continuous-Time Random Walk models with memory, developed in my thesis, are more general class of the stochastic processes. In those models jumps of the process value are still random but not necessarily independent. In my thesis I analyzed two versions of the CTRW model:
•    model with one-step memory - jump of the process value depends on one preceding jump
•    model with two-step memory - jump of the process value depends on two preceding jumps
In both models (one and two step memory) dependences of the consecutive jumps were motivated by the negative feedback. The negative feedback is encountered both in nature and in socio-economical systems as a counteraction against some exogenous factors which aim at restoring of the initial conditions of these systems. This effect is well defined for systems in equilibrium or, approximately, in partial equilibrium by the commonly known Le Chatelier-Braun principle of contrariness.

The strong mutual dependence between consecutive jumps of a share price has been observed on financial markets in contrast to its weak statistical dependence on time intervals between consecutive trades. Importantly, this strong dependence originating in the market microstructure can deeper be understood by analysis of  the order book. The order book is a deterministic system developed to organize the double auction market, such as stock market. This book contains different kinds of buy and sell orders. The most prominent feature of this auction is the so-called bid-ask spread. This spread is a positive and decisive difference between the lowest available sale offer (ask) price that sellers are willing to accept and the highest purchase (bid) price of an asset that buyers are willing to pay. The existence of the bid-ask spread and intraday dynamics of transaction prices in markets with a moderate liquidity, as is apparently the case of emerging markets (e.g. the Polish market), leads to the phenomenon called bid-ask bounce. The presence of this bounce results in a strong anti-correlation of successive price changes.

The principal aim of my thesis was to describe stochastic evolution of a typical share price on  a financial market with a moderate liquidity, on a high-frequency time scale. This evolution is a short-term anti-correlated stochastic process, which I described in the frame of the CTRW model with memory. The  model was mainly validated by satisfactory agreement of theoretical velocity autocorrelation function with its empirical counterpart obtained for the continuous quotation or tick-by-tick data. 

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