Stochastic Differential Equations With Applications to Biomedical Signal Processing

Dynamic behavior of biological systems is often governed by complex physiological processes that are inherently stochastic.Therefore most physiological signals belong to the group of stochastic signals for which it is impossible to predict an exact future value even if we know its entire past history.That is there is always an aspect of a signal that is inherently random i.e. unknown.Commonly used biomedical signal processing techniques often assume that observed parameters and variables are deterministic in nature and model randomness through so called observation errors which do not influence the stochastic nature of underlying processes (e.g., metabolism, molecular kinetics, etc.).An alternative approach would be based on the assumption that the governing mechanisms are subject to instantaneous changes on a certain time scale.As an example fluctuations in the respiratory rate and/or concentration of oxygen (or equivalently partial pressures) in various compartments is strongly affected by a metabolic rate, which is inherently stochastic and therefore is not a smooth process.As a consequence one of the mathematical techniques that is quickly assuming an important role in modeling of biological signals is stochastic differential equations (SDE) modeling.These models are natural extensions of classic deterministic models and corresponding ordinary differential equations.In this chapter we will present computational framework necessary for successful application of SDE models to actual biomedical signals.To accomplish this task we will first start with mathematical theory behind SDE models.These models are used extensively in various fields such as financial engineering, population dynamics, hydrology, etc.Unfortunately, most of the literature about stochastic differential equations seems to place a large emphasis on rigor and completeness using strict mathematical formalism that may look intimidating to non-experts.In this chapter we will attempt to present answer to the following questions: in what situations the stochastic differential models may be applicable, what are the essential characteristics of these models, and what are some possible tools that can be used in solving them.We will first introduce mathematical theory necessary for understanding SDEs.Next, we will discuss both univariate and multivariate SDEs and discuss the corresponding computational issues.We will start with introducing the concept of stochastic integrals and illustrate the solution process using one univariate and one multivariate example.To address the computational complexity in realistic biomedical signal models we will further discuss the aforementioned biochemical transport model and derive the stochastic integral solution www.intechopen.

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