To provide a solid analytic foundation for the module approach to conditional risk measures, this paper establishes a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the (\varepsilon,\lambda)--topology and the locally L^0-- convex topology). It should be also mentioned that D. Filiporvi\'{c --convex modules in D. Filipovi\'{c, M. Kupper, N. Vogelpoth, Separation and duality in locally L^0--convex modules, J. Funct. Anal. 256 (2009) 3996-4029}%DIFDELCMD < ] %%% (briefly,the FKV paper), where they made some important contributions to the subject and presented some good ideas of financial applications. Unfortunately, there are serious shortcomings in the FKV paper. First, most of the principal results in the FKV paper were based on the premise that the locally L ^{0}--convex topology for every locally L ^{0--conditional risk measure (1\leq p\leq+\infty) can be uniquely extended to an L^{0}} --convex module may be induced by a family of L^{0 the FKV paper ever gave a proof of this premise, but there was a hole in this proof, in fact, it remains open up to now whether the premise is valid or not. In this paper we overcome the difficulty by working with random locally convex modules endowed with the locally L^0 --convex topology rather than locally L^0--convex modules. Besides, some basic and key results in the FKV paper are also false so that some more interesting and essential things are covered, so we first correct these mistakes and further give a thorough treatment of random convex analysis .
To provide a solid analytic foundation for the module approach to conditional risk measures, this paper establishes a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the (\varepsilon,\lambda)--topology and the locally L^0-- convex topology). Then, we make use of the advantage of the (\varepsilon,\lambda)--topology and grasp the local property of L^0 --convex , M. Kupper, N. Vogelpoth, Separation and duality in locally L^0--convex modules, J. Funct. Anal. 256 (2009) 3996-4029}%DIFDELCMD < ] %%% conditional risk measures to prove that every L ^{0}--convex L ^{p--conditional risk measure (1\leq p\leq+\infty) can be uniquely extended to an L^{0}} --convex L^{p the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L^{0 --convex conditional risk measures .
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(briefly,the FKV paper), where they made some important contributions to the subject and presented some good ideas of financial applications. Unfortunately, there are serious shortcomings in the FKV paper. First, most of the principal results in the FKV paper were based on the premise that the locally L
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FKV paper ever gave a proof of this premise, but there was a hole in this proof, in fact, it remains open up to now whether the premise is valid or not. In this paper we overcome the difficulty by working with random locally convex modules endowed with the locally L^0
process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L^{0
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topology rather than locally L^0--convex modules. Besides, some basic and key results in the FKV paper are also false so that some more interesting and essential things are covered, so we first correct these mistakes and further give a thorough treatment of random convex analysis
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