The object of our study is
where each Sn is a m-dimensional stochastic (real valued) vector, i.e.
denned on a probability space (Ω, , P) and adapted to a filtration (n)0≤n≤N with 0 being the σ-algebra consisting of all null sets and their complements. In this paper we interpret as the value of some financial asset k at time n.
Remark: If the asset generates dividends or coupon payments, think of as to include these payments (cum dividend process). Think of dividends as being reinvested immediately at the ex-dividend price.
(a) A sequence of random vectors
is called a trading strategy. Since our time horizon ends at time N we must always have ϑN ≡ 0.
The interpretation is obvious: stands for the number of shares of asset k you hold in the time interval [n,n + 1). You must choose ϑn at time n.
(b) The sequence of random variables
where Sn stands for the payment stream generated by ϑ (set ϑ−1 ≡ 0).