Abstract
We prove that an optional process of non-exploding realized power variation along stopping times possesses almost surely làglàd paths. This result is useful for the analysis of some imperfect market models in mathematical finance. In the finance applications variation naturally appears along stopping times and not pathwise. On the other hand, if the power variation were only taken along deterministic points in time, the assertion would obviously be wrong.
Highlights
In financial market models with proportional transaction costs and effective friction trading strategies have to be almost surely of finite variation in order to avoid infinite losses
We prove that an optional process of non-exploding realized power variation along stopping times possesses almost surely làglàd paths
In models with a “large” trader having a smooth impact on the price process of an illiquid stock, as introduced by Bank and Baum [1] and Çetin, Jarrow, and Protter [3], a trading strategy should be of non-exploding quadratic variation
Summary
In financial market models with proportional transaction costs and effective friction trading strategies have to be almost surely of finite variation in order to avoid infinite losses (see Campi and Schachermayer [2]). Freedman [5], Proposition 70 and the arguments given on pages 48 and 49) This means that in a pathwise sense the quadratic variation does not exist and is exploding, but if grid points are restricted to stopping times the realized quadratic variation converges to t in probability. If the power variation is only taken along deterministic points in time, a non-exploding variation does obviously not imply that paths possess left and right limits, see Example 3.1 for an easy counterexample The reason for this is that for processes having neither left- nor right-continuous paths arbitrary sequences of grids with vanishing mesh do not always capture the entire variation
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