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Statistics for Finance by Jan Nygaard Nielsen, Henrik Madsen, Erik Lindström

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Chapter 8

Stochastic differential equations

Having established stochastic calculus in the Itō sense in the last chapter, we are now prepared to consider stochastic differential equations. For ease of notation, we shall in general only state the important results for univariate SDEs, but a few results will be generalized to multivariate SDEs.

We repeat that the notion of stochastic differential equations (SDEs) is merely a shorthand notation for stochastic integral equations. The latter may be defined in several ways, but we restrict our discussion to stochastic integrals in the Itō sense. Unfortunately, this implies that the well-known chain rule for variable transformations must be replaced by the so-called Itō formula, which will be introduced ...

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