This paper introduces the multiple stochastic integrals with respect to martingales which are generalizations of the well-known multiple Wiener integrals constructed by K. Ito. For deterministic integrands, the space $&^p(p=2,3, ...)$ of real-valued functions defined on $[O,T]^p$ with some conditions constitutes an inner product space by virtue of the quadratic variation process of a given $L^{2p}$-martingale. We define a bounded linear operator $I_p$ from the completion of $&^p$ into $L^2(Ω,\zeta,P)$, which we shall call the multiple stochastic integral. For random integrands, we also introduce the inner product space $&^p(p=2,3,...)$ of real-valued functions defined on $[O,T]^p × Ω$ with some conditions, and we define the corresponding multiple stochastic integral as a bounded linear operator $I_p$ from the completion of $&^p$ into $L^2(Ω,\zeta,P)$. In both cases, some fundamental properties are obtained.
