Vector Stochastic Integrals and the Fundamental Theorems of Asset Pricing

This paper deals with the foundations of stochastic mathematical finance and has three main purposes: (1) To present a self-contained construction of the vector stochastic integral $H\bullet X$ with respect to a $d$-dimensional semimartingale $X=(X_t^1,\dots ,X_t^d)$. This notion is more general than the componentwise stochastic integral $\sum _{i=1}^d H^i\bullet X^i$. (2) To show that vector stochastic integrals are important in mathematical finance. To be more precise, the notion of componentwise stochastic integral is insufficient in the First and Second Fundamental Theorems of Asset Pricing. (3) To prove the Second Fundamental Theorem of Asset Pricing in the general setting, i.e. in the continuous-time case for a general multidimensional semimartingale. The proof is based on the martingale techniques and, in particular, on the properties of the vector stochastic integral.