Probability

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By Dynkin E.B.

The Annals of Probability1993, Vol. 21, No. three, pp 1185-1262

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Additional resources for Branching processes and PDEs

Sample text

The martingale characterization can be extended in the context of imperfect financial models, mainly financial models with proportional transaction costs, short sale constraints, convex cone constraints, etc. We can observe three main lines of research generalizing these initial results. The first one applies in the context of imperfect financial markets for a model with transaction costs. The second line of research expands the restricted feasible portfolio case, usually cone constraints. The third research direction and the most recent one is based on the assumption that the price is non-linear with respect to the portfolio.

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A typical example where hedging effects occur and X and Y are not comonotone (comonotonicity := for all is the following one from Chateneauf et al. (1996). Suppose that X offers 1000 if even B occurs, 5000 otherwise, Y offers 5000 if B occurs, 1000 otherwise. Clearly X and Y are not comonotone and X (resp Y) is a hedge against Y (resp. X) since X + Y is riskless: it offers 6000 with certainty. So, subadditivity for is required. , if X and Y are comonotone. Then the third axiom (Comonotonicity Premium) expresses for all equality holds if X and Y are comonotone.