The relations among the three kinds of conditional risk measures

Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations

“…In [35], it is already proved that L p F0 (F T , R d ) = H cc (L p (F T , R d )). In fact, L p F0 (F T , R d ) is a complete RN module with base (Ω, F 0 , P ), for any element g ∈ L p F0 (F T , R d ) there exist some ξ ∈ L 0 (F 0 ) and x ∈ L p (F T , R d ) such that g = ξ · x.…”

“…To link Example 7.6 with the work of [35], we give the following final remark: Remark 7.7. In Example 7.6, let d = 1 and g be independent of y such that g(ω, t, ·) : R n → R is a convex function and g(ω, t, 0) = 0 for each (ω, t) ∈ Ω × [t 0 , T ].…”

On the kernel space for an L0‐linear function

“…On the basis of the above stratification structures, the theory of R N modules together with the theory of random conjugate spaces has obtained some substantive and deep advance. ()…”

“…On the basis of the above stratification structures, the theory of RN modules together with the theory of random conjugate spaces has obtained some substantive and deep advance. [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] Now, let us study the notion of a stratification of an L 0 -linear function. Let S be an RN module over K with base (Ω,  , P); the mapping ∶ S → L 0 ( , K) is called an L 0 -linear function if f( x + y) = f(x) + f(y) for any x, y in S, and , in L 0 ( , K).…”

On Continuous Module Homomorphisms Between Random Locally Convex Modules

“…Then, based on these, lots of new and basic researches have recently been done in [11,12,13,17,18,19,20,21,22,23]. In this process, Guo first deeply considers the problem of applying the theory of RN modules and random locally convex modules to L 0 -conditional risk measures, for example, the Fenchel-Moreau dual representation theorem and the continuity and subdifferentiability theorems for L 0 -convex functions were pointed out in a proper form in [9], subsequently, Guo found that the study of L p -conditional risk measures can be incorporated into that of L p F (E)-conditional risk measures and further established a complete random convex analysis over random locally convex modules under the two kinds of topologies in [14].…”