Consistent Risk Measures and a non-linear Extension of Backwards Martingale Convergence.

in Z.-Q. Chen; N. Jacob; M. Takeda (Eds.) Festschrift Masatoshi Fukushima (2015)

We study the behavior of conditional risk measures along decreasing σ-fields. Under a condition of consistency, we prove a non-linear extension of backwards martingale convergence. In particular we show ... [more ▼]

We study the behavior of conditional risk measures along decreasing σ-fields. Under a condition of consistency, we prove a non-linear extension of backwards martingale convergence. In particular we show the existence of a limiting conditional risk measure with respect to the tail field, we describe its dual representation in terms of a limiting penalty function, and we show that consistency extends to the tail field. Moreover, we clarify the structure of global risk measures which are consistent with the given sequence of conditional risk measures. [less ▲]

Risk assessment for uncertain cash flows: model ambiguity, discounting ambiguity, and the role of bubbles

in Finance and Stochastics (2012), 16(4), 669-709

We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito et al. (Electron. J. Probab. 11(3):57–106, 2006). These risk measures ... [more ▼]

We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito et al. (Electron. J. Probab. 11(3):57–106, 2006). These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalised probability measures on the optional σ-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for time consistency. In particular, we show how “bubbles” may appear in the dynamic penalisation, and how they cause a breakdown of asymptotic safety of the risk assessment procedure. [less ▲]