Multidimensional coherent risk measures and their application to the solution of problems of financial mathematics

We deal with multidimensional coherent and convex risk measures. The approach described takes into account risks of changing currency exchange rates and transaction costs. Representation theorems for multidimensional risk measures are proved. Note that the case of random currency exchange rates is introduced. Two applications of multidimensional coherent risk measures are considered: to the capital allocation problem and to the problem of risk contribution.
The application of multidimensional coherent risk measures to the problem of pricing is considered. I.e., we consider pricing based on No Good Deals (NGD). Dynamic model of currency exchange rates is considered and the sets of fair prices are considered. High and low prices along datum lines, sub- and superhedging along direction are introduced and examples are considered.
The various multidimensional generalizations of one of the most important coherent risk measure as Tail V@R are considered. Three different approaches are considered. Several financial situations in which these functions give different result are considered and the conditions when they coincide are also studied. Another important property as space consistency is introduced. Also law invariance property is given in multidimensional case. The various generalizations of Tail V@R are tested if they are space consistent, law invariant or not. Some necessary and sufficient conditions for multidimensional risk measures to be space consistent and law invariant are given. Three generalizations of Weighted V@R are also considered.