As for the Riemann and Lebesgue integral, the definition of stochastic integral is theoretical and it is not possible to use it directly for practical purposes, apart from some particular cases. Classical results reduce the problem of the computation of a Riemann integral to the determination of a primitive of the integrand function; in stochastic integration theory, the concept of primitive is translated into “integral terms” by the Itô-Doeblin formula1. This formula extends Theorem 3.70 in a probabilistic framework and lays the grounds for differential calculus for Brownian motion: as we have already seen the Brownian motion paths are generally irregular and so an integral interpretation of differential calculus for stochastic processes is natural.
KeywordsBrownian Motion Call Option Quadratic Variation American Option Underlying Asset
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