Time Consistent and Currency Invariant Risk Adjusted Valuations

In order to study time-consistent monetary risk measures in a continuous time, finite horizon setting, there arises a need to define risk-adjusted valuations at all dates in the time continuum. One approach of tackling this issue is to define risk-adjusted valuations at all dates and call this sequence of risk valuations as the risk valuation process. However, monetary risk measures enjoy the translation invariance property which enables us to define monetary risk-adjusted valuations at intermediate stopping times in terms of the date-0 risk measure.We characterize these intermediate risk-adjusted valuations in terms of the date-0 risk measure for strongly relevant time-consistent monetary risk measures. We develop a representation for strongly relevant monetary risk measures which are time consistent at certain specified dates. We develop two methods for constructing time-consistent monetary convex risk measures in a discrete time setting.We seek to characterize monetary convex risk measures which are both time consistent and currency invariant. We define currency invariance for monetary convex risk measures. We show that only trivial monetary convex risk measures satisfy both time consistency and currency invariance for all currency exchange rate processes.We define weak time consistency and currency invariance for concave risk-adjusted valuation processes. We show that all time-consistent concave risk-adjusted valuation processes are trivially currency invariant. We propose a class of concave risk-adjusted valuation processes, which we call Psuedo-Monetary Concave Risk Adjusted Valuation Processes(PCRVPs), which are not necessarily monetary but possess some of the appeal of monetary concave risk-adjusted valuations. We develop a representation for PCRVPs which are weakly time consistent at certain specified dates. This representation is similar to the one we develop for monetary convex risk measures. We show that the only a very specific class of Psuedo-Monetary Coherent Risk Adjusted Valuation Processes are currency invariant for all exchange rate processes