Financial Modelling and Memory: Mathematical System

Data based search for patterns are of concern to almost all fields of studies and research. To do so, parametric mathematical models are constructed to explain observed data sets, forecast future prices, etc. The models we use are extremely varied, seeking to be reliable, robust and explanatory. Financial models for example have for the most part assumed models that are based on drift and randomness to construct models of asset prices. Typical examples include random walks and stochastic (Brownian motion) differential equations (such as a lognormal process) as well as Poisson and Jump stochastic processes. These latter models are based on events that occur at random and independent times. They assume also a number of simplifying assumptions, all of which are either explicit or implicit. For example, stochastic differential equations assume that data can be expressed in terms of two statistical properties: the data drift and its volatility, both of which are defined in terms of underlying (Brownian motion) normally and statistically distributed events. By the same token, Poisson-like jump processes are defined in terms of a memory-less process (with independent inter-events time distributions). To circumvent some of their assumptions, more complex and multi-variable models are used to account for observations that such models fail to explain. For the most part these models are based on both a rationality and an experience acquired based on theoretical constructs and on data analyses. These models are extremely useful, and provide an ex-ante interpretation for the behavior of data sets as well define the statistical properties of observed financial variables. Ex-post, however, these models may default since all models are merely an educated hypothesis of underlying processes. Modeling financial processes is therefore a work in process, in search for coherent and complete mathematical systems that can on the one hand be justified theoretically and on the other account far more precisely for observed data sets.