Research On Loss-based Risk Measures And Cash Sub-additive Risk Measures And The Related Problems

Today,the risk measures and its related researches have become one of the hot issues in many fields such as mathematics,finance and economy.Some scholars even have named the research of risk measures as "the third financial revolution on Wal-1 Street." In recent years,the multidimensional risk measures and the set-valued risk measures have become the hot issues in the research of risk measures.This is because that in the research of multidimensional risk measures,it is allowed to have the in-terdependence among the multiple financial positions in the investment portfolio and this relationships are again reflected in the set-valued risk capital requirement.Burgert and Ruschendorf[21]first studied the scalar multidimensional coherent and convex risk measures.Jouini et al.[62]first studied the set-valued multidimensional risk measures.On the other hand,since Artzer et al.[11]first introduced the scalar coherent risk mea-sures from an axiomatic perspective,the study of the scalar risk measures never ceased.After that,Delbaen[31]studied the definition and properties of the coherent risk mea-sures which are defined on the general probability space,Follmer and Schied[46],and independently,Frittelli and Rosazza Gianin[49]introduced the broader class:convex risk measure,and thus derived a lot of new research.This doctoral dissertation focuses on the loss-based risk measures and cash sub-additive risk measures and its related problems.The details are as follows:For the first time,Cont et al.[28]defined the coherent and convex loss-based risk measures based on axiomatic approach from the perspective of financial market's reg-ulators,and study the dual representations.In the present paper,we will extend the loss-based risk measures to multidimensional and set-valued cases.Specifically,in the multidimensional cases,we will based on the axiomatic approach and make use of the results of Cheridto and Li[25]to get the dual representations of multidimensional co-herent and convex loss-based risk measures.In the of set-valued cases,we will make use of the set-valued convex duality theory framework established by Hamel[53]to obtain the dual representation of set-valued convex loss-based risk measures.We note that each convex risk measure can deduce a convex loss-based risk measure.However,in general,a convex loss-based risk measure may not be deduced from a convex risk measure.Thus we studied the sufficient and necessary condition under which a convex loss-based risk measure can be deduced from a convex risk measure.Taking the time-value of money into account,El Karoui and Ravanelli[37]in-troduce a new kind of risk measures from the axiomatic approach,which is the well known cash sub-additive risk measures nowadays.The proposed of this kind of risk measures,makes people could consider the change of exchange rate in the process of quantifying the risk.This is a special characteristic that the previous risk measure did not have.It is also this feature that makes the cash sub-additive risk measures more in line with the actual financial market.For those with long time spans and large ex-change rate fluctuations,the cash sub-additive risk measures is more appropriate than the usual risk measures.As pointed out by Cerreia-Vioglio et al.[22],when there is un-certainty about interest rates,the cash additivity assumption,also known as translation invariance,becomes problematic and will be replaced by cash sub-additivity.Parkas et al.[38]pointed out that when the eligible assets axre default bonds,cash sub-additivity will appear.Mastrogiacomo and Rosazza Gianin[73]study the time consistency of the dynamic cash sub-additive risk measures.In this paper,we will study the dual represen-tations of set-valued additive risk measures from both static and dynamic perspectives separately.In the research of risk measures,we usually use the prmoment limited space of random variables LP or its subspace L? to describe the space of financial risk positions.Frittelli and Rosazza Gianin[49]for the first time give a rough representation of the convex risk measure based on LP.Later,Cherny[23]and Rockafellar et al.[83]also make the contributions in this direction.For a more detailed dual representations of convex risk measures on LP,see Kaina and Riischendorf[63],Cheridito and Li[25]and Filipovic and Svindland[45].On the other hand,the space for financial risk positions space,Cheridito and Li[25]studied the risk measures based on Orlicz Heart,Arai[8]studied can also be expanded in a more general direction.Ruszcynski and Shapiro[85]studied the risk measures based on general vector the convex risk measure on Orlicz s-pace,Kountzakis[66]studied the conherent and convex risk measures on Banach space,Konstantinides and Kountzakis[67]further study the risk measures in ordered linear normative space.In this paper,we will study the risk measures in a completely new space,i.e.the variable exponent Bochner-Lebesgue space,which is denoted by LP(·),where the order p(.)is no longer a fixed positive number,but a random variable.In other words,this is the uncertainty of the order.We expect to use this uncertainty of the order to describe the uncertainty of the financial market.on this new space of financial risk positions,we will study several common risk measures:coherent risk mea-sures,convex risk measures,dynamic risk measures and cash sub-additive risk measures.