Processes Governed by Stochastic Differential Equations

In industrial operations, there often exist random fluctuations in the operating conditions from a variety of sources. Aris and Amundson (1958) show how fluctuations in the outputs of a continuous reactor caused by specified fluctuations in the input variables can be estimated by linearization. 

For dynamic effects, stochastic differential equations pose a reasonable alternative to model such situations. A stochastic differential equation is written in the following form  

The fluctuations are often modeled as a Wiener process, i.e., i (t)dt = dW t) (Gardiner, 2003) the function is known as white noise. For its precise mathematical definition and that of the Wiener process, the reader is referred to Gardiner (2003). Then Equation (2.44) is replaced by 

For information on how stochastic differential equations can be used to analyze the effect of the fluctuations in operating conditions, see Rao et al. (1974a). The actual method of solving nonlinear stochastic equations can be found in Rao et al. (1974b). Algorithms simpler than Rao et al. (1974b) are reported elsewhere (e.g., Pardoux and Talay, 1985). 

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