Semimartingale approach

Abstract

In this paper, we present the definition of semimartingale and give the treatment of stochastic integration as a Riemann-type limit of sums. This approach is not like the classical one, which defines a semimartingale to be the sum of a local martingale and a finite variation process. We also apply the theory of stochastic integration to obtain the integro-differential equation for the value of an Asian option when the stock price is driven by a $L\grave{e} vy$ process. Here we use the technique of change of $num\grave{e} raire$ used frequently in the literature to remove the path dependency in option pricing problem.