Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

7Stochastic differential equations

7.1 Existence and uniqueness theorem and main proprieties of the solution

Let be a complete probability space and a standard Wiener process defined on it. Let be a non‐anticipative filtration. Let be a ‐measurable r.v. (therefore independent of the Wiener process); in particular, it can be a deterministic constant. Having defined in Chapter  the Itô stochastic integrals, the stochastic integral equation

(7.1)

does have a meaning for if, as in Definition 6.4 of an Itô process,

Of course we need to impose appropriate conditions on and to insure that