7Stochastic differential equations

7.1 Existence and uniqueness theorem and main proprieties of the solution

Let images be a complete probability space and images a standard Wiener process defined on it. Let images be a non‐anticipative filtration. Let images be a images‐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)equation

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

equation

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

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