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On Esscher Transforms In Discrete Finance Models

H. Bühlmann, F. Delbaen, P. Embrechts, A. Shiryaev
Published 1998 · Economics

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The object of our study iswhere each Sn is a m-dimensional stochastic (real valued) vector, i.e.denned on a probability space (I©, , P) and adapted to a filtration (n)0≤n≤N with 0 being the Iƒ-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.Definition 1(a) A sequence of random vectorswhereis called a trading strategy. Since our time horizon ends at time N we must always have I‘N â‰i 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 I‘n at time n.(b) The sequence of random variableswhere Sn stands for the payment stream generated by I‘ (set I‘−1 â‰i 0).
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