Brownian integration

Abstract

In this chapter we introduce the elements of stochastic integration theory that are necessary to treat some financial models in continuous time. In Paragraph 3.4 we gave grounds for the interest in the study of the limit of a Riemann-Stieltjes sum of the form

$$ \sum\limits_{k = 1}^N {u_{t_{k - 1} } \left( {W_{t_k } - W_{t_{k - 1} } } \right)} $$

(4.1)

as the refinement parameter of the partition {t₀, . . . , t _N } tends to zero. In (4.1) W is a real Brownian motion that represents a risky asset and u is an adapted process that represents an investment strategy: if the strategy is self-financing, the limit of the sum in (4.1) is equal to the value of the investment.