The Processes with Dependent Increments as Mathematical Models of the Interest Rate Processes

Abstract

Processes of the interest rates and other financial indexes in continuous time are usually modeled in the literature by stochastic processes with independent increments. Such processes are described by the stochastic differential equations and are the Markov processes. As it follows from the theory the stationary stochastic process is the Markov process (in the wide sense) if and only if the normalized correlation function is exponential. In other words the stochastic processes with independent increments generate the data series with the exponential correlation functions. At the same time the correlation functions of real data series have often non-exponential correlation functions. For example such functions are typical for the US Treasury Security Yield Rate, Internal Rate of Yield on UK 2.5 % Consols, UK Dividend Yield Rate for Shares and other financial data series. Therefore in order that to fit a mathematical model to some real financial data it should be used a stochastic processes with dependent increments. Such processes have more flexible structure that allows obtain the necessary properties. In present paper it is proposed a way for the construction of the process with dependent increments. For that it is supposed that the stochastic process of the interest rate (or other financial index) has a derivative of the some order and this derivative is the process with independent increments. In other words the stochastic process of the interest rate is described by the stochastic differential equation of some order more than first. It results in the more relevant mathematical models. If the coefficients of stochastic differential equations are constant then the solutions in the explicit form are derived. On practice the derivatives of the interest rate processes are non-observed therefore the practical forms of solutions can not include the values of derivatives. Therefore it is discussed a problem of exclusion of these values from solutions. It is shown that these solutions exist and they are determined on discrete set of time instants. The case when the first derivative of process of interest rate has independent increments is described in details. The offered approach is illustrated by the analysis of actual time series of the yield rates of the US Treasury Securities.