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On Azéma-Yor processes, their optimal properties and the Bachelier-drawdown equation. (English) Zbl 1239.60031

The Azéma-Yor process associated with a primitive \(U\) of a locally bounded Borel function \(u\) on \([a,+\infty)\) and a semimartingale \(X\) with continuous running maximum \(\overline{X}\) is defined as \[ M^U_t(X):=U(\overline{X}_t)-u(\overline{X}_t)(\overline{X}_t)-X_t). \] The set of Azéma-Yor processes associated with the same max-continuous semimartingale \(X\) has a group structure that is indexed by functions and this property happens to be very useful. The authors first consider the Bachelier equation \[ dY_t=\phi(\overline{Y}_t)dX_t \] associated with \(X\) and a positive locally bounded function \(\phi\). They prove that \(Y\) is the Azéma-Yor process associated with \(U\) and some function \(U\) related to \(\phi\). They also prove that the Bachelier equation is equivalent to the so-called drawdown equation. Solutions of the latter have the drawdown property: they always stay above a given function of their past maximum.
They investigate in detail Azéma-Yor martingales defined from a nonnegative local martingale converging to zero at infinity. They establish relations between average value at risk, drawdown function, Hardy-Littelwood transform and its inverse. The construction of Azéma-Yor martingales with given terminal law allows them to retrieve the Azéma-Yor solution to the Skorokhod embedding problem. Finally, they characterize Azéma-Yor martingales showing that they are optimal relative to the concave ordering of terminal variables among martingales whose maximum stochastically dominates a given benchmark.

MSC:

60G44 Martingales with continuous parameter
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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