A scalar stochastic differential equation (SDE)
involves a the Wiener process W t , t ≥ 0, which is one of the most fundamental stochastic processes and is often called a Brownian motion (see Brownian Motion and Diffusions). A Wiener process is a Gaussian process with W 0 = 0 with probability 1 and \(\mathcal{N}(0,t - s)\)-distributed increments W t − W s for 0 ≤ s < t where the increments \({W}_{{t}_{2}} - {W}_{{t}_{1}}\) and \({W}_{{t}_{4}} - {W}_{{t}_{3}}\) on non-overlapping intervals, (i.e., with 0 ≤ t 1 < t 2 ≤ t 3 < t 4) are independent random variables. It follows from the Kolmogorov criterion that the sample paths of a Wiener process are continuous. However, they are nowhere differentiable.
Consequently, an SDE is not a differential equation at all, but only a symbolic representation for the stochastic integral equation
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References and Further Reading
Kloeden PE, Platen E (1992) The numerical solution of stochastic differential equations. Springer, Berlin (3rd revised edition, 1999)
Øksendal B (2003) Stochastic differential equations. an introduction with applications. Springer, Berlin (6th edition, Corr. 4th printing, 2007)
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Kloeden, P.E. (2011). Stochastic Differential Equations. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_569
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DOI: https://doi.org/10.1007/978-3-642-04898-2_569
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