On representations of the set of supermartingale measures and applications in discrete time

We investigate some new results concerning the m-stability property. We show in particular under the martingale representation property with respect to a bounded martingale S that an m-stable set of probability measures is the set of supermartingale measures for a family of discrete integral processes with respect to S.


Introduction
The m-stability property plays a primordial role within the theory of dynamic risk measures and in Financial Mathematics in general. In dynamic setting it is crucial that a dynamic risk measure satisfies the recursiveness property when evaluating risk for a financial position, by taking into account the new incoming information in a consistent way. It should also be the case for a pricing mechanism when pricing a financial claim in incomplete markets, avoiding so the creation of arbitrage opportunities in a given time axis. In decision making and in econometrics the m-stability property known by rectangularity is an essential assumption in modeling preferences and utility functions and in constructing set of priors. We refer to [5,6] for more details.
Let us consider a probability space ( , F, (F t ) t=0...T , P) and a set Q of probability measures in P containing at least one equivalent to P, where P (resp., P e ) is the set of P-absolutely continuous (resp., P-equivalent) probability measures. In [2] it was shown that the m-stability assumption on Q is a corner stone in generalizing a number of results from one-period case to multi-period case. We recall in particular that for an m-stable set Q, we get the following: (i) Suppose Q = {Q ∈ P : E Q (X ) = 0} =: M(0, X ) for a vector-valued random variable X = (X 1 , . . . , X k ).
Then Q is the set of martingale measures for the R k -valued adapted process M defined by M i t = E Q t (X i ) := E Q (X i |F t ) for some (or any) Q ∈ Q e and i = 1, . . . , k, where Q e is given by Q e = Q ∩ P e . In this case we can write a Q-supermartingale as the sum of a local martingale and a decreasing process, by applying the well-known theorem of Föllmer and Kabanov [7]. (ii) The set Q , to be redefined later, is the set of martingale measures for a family Y of adapted processes.
Such family Y can be replaced by a finite one if we suppose further that Q is optionally m-stable with respect to a vector-valued bounded random variable V .
Our goal in this paper is to prove the Q-supermartingale decomposition under minimal assumptions on Q. To do that, we will start by proving in Sect. 3 some interesting properties of the set Q st (See Definition 2.2 below). We will state in particular two fundamental assertions: (1) Q-supermartingales are Q st -supermartingales, and (2) Q st is the set of supermartingale measures for a family Y of bounded adapted processes.
In Sect. 4 we suppose the martingale representation property of the filtration with respect to an adapted process S with values in R d . We shall state the existence of a convex cone C of vector-valued adapted processes such that Q st is the set of supermartingale measures for a family of processes of the form α • S := s<. α s . s S with α ∈ C and s S = S s+1 − S s . We apply such result and deduce that any positive (or bounded) Qsupermartingale X can be written as X 0 + α • S − B with B an increasing process and α ∈ C.
In Sect. 5 and further under the assumption Q = Q , we shall prove that Q st is the set of martingale measures for a family Y of bounded adapted processes and the process α • S appearing in the previous decomposition of X is a local Q-martingale, generalizing the theorem of Föllmer and Kabanov.
We prove also under the martingale representation property that (Q st ) = (Q ) st , which is the commutativity property of the two operators Q → Q st and Q → Q .

Notation and review
In this section we recall the definition of the main properties which will be used along this paper, and also some established characterizations of these properties.

Notation
We consider a probability space ( , F, P) and a discrete time filtration (F t ) t∈I with I = {0, . . . , T } and I * = I\{T }. We denote by (resp., c ) the set of all (resp., L 1 -closed convex) subsets in P, with e = {Q ∈ : Q ∩ P e = ∅} and c,e = c ∩ e . For a set Q of probability measures in P, we define the dynamic , the family of acceptance sets A t,u = {X ∈ L ∞ (F u ) : E t (X ) ≤ 0 a.s.} for t < u with A := A 0,T , and the set of Radon-Nikodym densities Z = {Z Q := dQ/dP : Q ∈ Q}, Z e = {Z ∈ Z : Z > 0 a.s.} with Z τ := E τ (Z ) for Z ∈ Z and a stopping time τ .

On the m-stability property
Definition 2. 1 We say that a set of probability measures Q is m-stable with respect to the filtration (F t ) t∈I if for any Z 1 , Z 2 ∈ Z with Z 2 > 0 a.s. and a stopping time τ we have Z : Definition 2.2 For any Q ∈ c,e , we define Q st to be the intersection of all m-stable closed convex subsets in P containing Q, and denote by E st and A st , respectively, the dynamic expectation operator and the acceptance set associated to Q st .
We recall some interesting characterizations of the m-stability property, stated in Delbaen [5].

Proposition 2.3
Let Q ∈ c,e . Then the following assertions are equivalent: For any t ∈ I * , F ∈ F t and Z , Z 1 , Z 2 ∈ Z with Z 1 > 0 and Z 2 > 0 a.s., we have (3) E satisfies the recursiveness property, i.e., for any X ∈ L ∞ and for any stopping times τ ≤ σ we have E τ (E σ (X )) = E τ (X ), (4) E is time consistent, i.e., for any X, Y ∈ L ∞ and for stopping times for any X ∈ L ∞ , the process E . (X ) is a Q-supermartingale, (6) for any stopping time τ > 0 and A := A 0,T we have A = A 0,τ + A τ,T .

On the optional m-stability property
This subsection is devoted to recalling the definition of the optional m-stability. This concept was first introduced by Jacka et al. in [8]. We shall say that an Definition 2.4 (See Jacka and Berkaoui [8]) We say that Q is optionally m-stable with respect to a viable random vector V if for all Characterizations of this property can be found in [1,8].

On the smallest set of martingale measures
Here we recall the definition of the set Q associated with Q, and also Theorem 2.1 in [3], stating some properties of Q . For a set Q of probability measures and its acceptance set A, we define We prove easily that the acceptance set A of Q is given by the closure in weak star sense of lin(A) + L ∞ − . By supposing that Q is m-stable, the next theorem will state mainly that Q is m-stable and that Q is the set of martingale measures for a family of bounded adapted processes.

Intermediate results
Now we investigate some properties of the mapping Q → Q st .

Properties
First we express the triplet Thanks to Assertion (2) in Proposition 2.3 we deduce by induction on t = 0 . . . T − 1 that all R t ∈ Z st and since Z = R T −1 we deduce that B ⊆ Q st and, therefore, co(B) ⊆ Q st . For the direct inclusion we remark that B is m-stable and Q ⊆ B, so Q st ⊆ co(B).
(2) Fix s < t and take X ∈ A st s,t with E st s (X ) = 0. We have A st s,t ⊆ A st , then there exists some X u ∈ A u,u+1 for u ∈ I * such that X = X 0 + · · · + X T −1 . We apply E st s on both parts of the equality and obtain that 0 Then we have the following: Proof Assertions (1) and (2) are trivial. (3) We shall show that ( for all t ∈ I * , with the closure taken in weak star sense in L ∞ . The inverse inclusion is trivial. Let us prove the direct one. We know that the dynamic expectation operator We remark that X i ∈ A i t,t+1 for i = 1, 2 and deduce the result. (4,i) Let Z 1 , Z 2 ∈ Z with Z 2 > 0 a.s. and a stopping time τ . Then there exists an integer n such that Z 1 , Z 2 ∈ Z n and since Q n is m-stable we deduce that Z : Q and sincẽ Q is an m-stable closed convex set we deduce that Q st ⊆Q. For the other inclusion we have Q n ⊆ Q which impliesQ ⊆ Q st .

Link with the concept of supermartingale
We investigate the relationship of the m-stability property with the concept of supermartingale. We start by giving the definition of a supermartingale measure. Definition 3. 3 We say that a probability measure Q ∈ P is a supermartingale measure for a family Y of adapted processes if each element Y ∈ Y is a Q-supermartingale. We denote by M sp (Y) the set of all supermartingale measures for the family Y, and we will say that Q is a set of supermartingale measures if it is the set of supermartingale measures for a family Y of bounded adapted processes. Now we state results related to that. Proof (1) we know from Assertion (4) in Proposition 2.3 that the process E st for a family Y of bounded adapted processes with Q ⊆Q. SinceQ is an m-stable closed convex set, we deduce that Q st ⊆Q.

Theorem 3.5
Let Q ∈ c,e . Then we have the following: Proof (1) It is straightforward from the definitions of spm(Q) and A t,t+1 for t ∈ I * .
(2) The inclusion B t ⊆ A t,t+1 is straightforward. For the direct inclusion let X ∈ A t,t+1 and define the process Y by Y s = X for s > t and Y s = 0 for s ≤ t. Then Y ∈ spm(Q) by (1) and X = Y t+1 − Y t . (3) We apply Assertion (1) and the fact that A t,t+1 = A st t,t+1 for all t ∈ I * . (4) LetQ ⊆ P satisfying spm(Q) = spm(Q). Then from Assertion (2) we get A t,t+1 =Ã t,t+1 for all t ∈ I * and, therefore, A st =Ã st . We conclude thatQ ⊆Q st = Q st . (5) We remark that X ∈ m(Q) if and only if ±X ∈ spm(Q). We use Assertion (3) and deduce the result.
An immediate consequence of Theorem 3.5 is as follows: Corollary 3.6 Let Q 1 , Q 2 ∈ c,e . Then the following three assertions are equivalent: For a set B ⊆ L 0 we denote lin(B) = B ∩ −B and for Q ⊆ P we denote Q = {Q ∈ P : E Q (X ) = 0 for all X ∈ lin(A)}. The set Q was first introduced by Berkaoui in [2] and defined properly in [3]. We refer to Theorem 2.1 in [3] for more details on this set. Theorem 3.7 Let Q ∈ c,e . Then we have the following: t+1 ) for all t ∈ I * }. Proof (1) It is straightforward from the definitions of m(Q) and lin(A t,t+1 ) for t ∈ I * .
(2) The inclusion D t ⊆ A t,t+1 is straightforward. For the direct inclusion let X ∈ lin(A t,t+1 ) and define the process Y by Y s = X for s > t and Y s = 0 for s ≤ t. Then Y ∈ m(Q) and X = Y t+1 − Y t . (3) Since m(Q) = m(Q st ) it suffices to show that m(Q) = m(Q ). We apply Theorem 2.5 to deduce that lin(A t,t+1 ) = lin(A t,t+1 ) for all t ∈ I * . The result is concluded thanks to Assertion (1). (4) LetQ ∈ such that m(Q) = m(Q), then from Assertion (2) we get lin(A t,t+1 ) = lin(Ã t,t+1 ) for all t ∈ I * and, therefore, (A st ) = (Ã st ) thanks to Theorem 2.5. We conclude thatQ ⊆ (Q st ) .
An immediate consequence of Theorem 3.7 is as follows: Then the following three assertions are equivalent:

Main results
In this section we precise further the results of Theorem 3. 4. In what follows, we suppose the assumption MRP(S): the filtration (F t ) t∈I satisfies the martingale representation property with respect to a bounded martingale S with values in R d , which means that any square integrable martingale X can be written as the sum E(X ) + α • S for some vector-valued adapted process α.

has a finite number of extreme points, then Q st = M sp (g • S) for a matrix-valued adapted process g.
We state first in the next theorem a similar version of Theorem 4.1 in the one-period model. We say that Proof (1) We will show that Q is optionally m-stable with respect to S : Then for all X ∈ A and thanks to the assumption MRP(S), there exists some vector-valued F 0 -measurable random variable α 0 such that X = E 0 (X ) + α 0 . S and, therefore, By applying Theorem 2.17 in [8] (2) We define the set K to be the closure of C 0 in L 0 (R d ) with respect to the topology of convergence in measure and verify that Thanks to Theorem 4.6 in [9] there exists a random closed convex cone W in R d such that α 0 ∈ K if and only if α 0 ∈ L 0 (F 0 ; R d ) and α 0 ∈ W a.s. which means that α 0 .Y i ≤ 0 a.s. for i = 1 . . . k. We denote by g 0 = (g 1 0 , . . . , g d 0 ) the generating family of W , i.e., for all α 0 ∈ W , there exists positive F 0 -measurable scalar random variables λ 1 , . . . , λ d such that α 0 = λ.g 0 = d i=1 λ i g i 0 . We shall prove that Q = M sp (0; g 0 . S). For the direct inclusion we have E Q 0 (g 0 . S) ≤ 0 for all Q ∈ Q since g 0 ∈ K . For the inverse inclusion let Q ∈ M sp (0; g 0 . S) and X ∈ A, then there exists a sequence α n 0 ∈ C 0 such that X = lim n→∞ α n 0 . S with the limit taken in weak star topology. There exists then a sequence λ n ≥ 0 such that α n 0 = λ n .g 0 ; therefore, α n 0 . S = λ n .g 0 . S and then E Q 0 (X ) = lim n→∞ E Q 0 (λ n .(g 0 . S)) = lim n→∞ λ n .E Q 0 (g 0 . S) ≤ 0. So Q ∈ Q. Now we prove Theorem 4.1.
Proof We shall show first that for each t ∈ I * , the assumption MRP(S t , S t+1 ) is satisfied on the one-period model {t, t + 1} with t S := S t+1 − S t . Indeed for a process X = (X t , X t+1 ) with E t (X t+1 ) = X t we define the process Y by Y s = E s (X t+1 ) for s ∈ I and remark that Y is a martingale. So there exists a process α such that Y = Y 0 + α • S. In particular, we get X t+1 We denote by Q t for t ∈ I * , the set of probability measures Q ∈ P, defined on ( , F t+1 ) such that . For Assertion (1) we apply Assertion (1) in Theorem 4.2 and obtain that (2) we shall show that Q t has a finite number of F t -extreme points. Let (Q 1 , . . . , Q k ) be the extreme points of Q with their respective densities (Z 1 , . . . , Z k ). Then the probability measures (Q t,1 , . . . , Q t,k ), defined on F t+1 by their respective densities (Z t,1 , . . . , Z t,k ) :   4 We consider the one-period model with = {ω 1 , ω 2 }. We verify easily that Q := [(1/2, 1/2), (1/3, 2/3)] = M sp (0; X ) since Q has only two extreme points with X = (X 1 , X 2 ), X 1 = (1, −1) and X 2 = (−2, 1).

Applications
We suppose again that the assumption MRP(S) is satisfied along this section and investigate the decomposition of Q-supermartingales. Theorem 5.1 Let Q ∈ c,e . Then any bounded Q-supermartingale X can be decomposed as follows: X = X 0 + α • S − C with C an increasing process and the process α • S is a Q-supermartingale.
Proof For t ∈ I * we denote by Q t for t ∈ I * , the set of probability measures Q ∈ P, defined on ( , t+1 . It has been proved in the proofs of Theorems 4.1 and 4.2 that the assumption MRP(S t , S t+1 ) is satisfied on the one-period model {t, t + 1} with t S := S t+1 − S t and that Q t is optionally m-stable with respect to t S. We apply then Proposition 3.3 in [2] and obtain that So for a bounded Q-supermartingale X we have X t+1 − X t ∈ A t,t+1 for each t ∈ I * , and then there exists some α t ∈ K t and B t ∈ L 0

Corollary 5.2
Let Q ∈ c,e . Then any bounded random variable Y can be written as follows: Next we investigate the case where the process α • S appearing in Theorem 5.1 is a local Q-martingale. We generalize then the result of Föllmer and Kabanov [7].

Theorem 5.4
Let Q ∈ c,e such that Q = Q . Then any bounded Q-supermartingale X can be decomposed as follows: X = X 0 + α • S − C with C an increasing process and the process α • S is a local Q-martingale.
To prove Theorem 5.4, we state first some preliminary Lemmas.

Lemma 5.5
Let Q ∈ c,e . Then A ⊆ K + L 0 − where K is the closure in L 0 of lin(A).
Proof We shall prove that the set K + L 0 − is closed in L 0 . We prove it first for the one-period model. We suppose then that T = 1 and F 0 is not necessarily trivial. Let us define D := {α ∈ L 0 (F 0 ; R d ) : α. S ∈ K } and verify easily that D is an F 0 -stable closed vector space in L 0 (F 0 ; R d ). So thanks to Lemma A.4 in [11] and Lemma 2.5 in [4] there exists a generating family g = (g 1 , . . . , g d ) of the set D. We deduce that K = {α.Y : α ∈ L 0 (F 0 ; R d )} with Y = g. S. We apply Lemma 2.1 in [10] to conclude that K + L 0 − is closed in L 0 . Now for the multi-period case we proceed as follows: for all t ∈ I * we define D t = {α t ∈ L 0 (F t ; R d ) : α t . t S ∈ K }, g t = (g 1 t , . . . , g d t ) the generating family of D t , Y t = g t . t S and B t = {α t .Y t : α t ∈ L 0 (F t ; R d )}. We shall prove by induction on t = T − 1, . . . , 0 that the set B t := B t + · · · + B T −1 + L 0 − is closed in L 0 . For t = T − 1 we apply the one-period case. Now we suppose that the sets B t+1 , . . . , B T −1 are closed, we shall prove that B t is closed. We remark that B t = B t + B t+1 and follow the proof of Proposition 3.3 in [2]. Define the set N := {α t ∈ L 0 (F t ; R d ) : −α t .Y t ∈ B t+1 a.s.}.
We prove easily that N is a closed F t -stable convex cone in L 0 (F t ; R d ). We prove now that it is a vector space. Let α t ∈ N so there exists α s ∈ L 0 (F s ; R d ) for s = t + 1 . . . T − 1 and z ∈ L 0 + such that α t .Y t + . . . + α T −1 .Y T −1 = z ≥ 0; therefore, z = 0 and α t .Y t = −α t+1 .Y t+1 + . . . − α T −1 .Y T −1 ∈ B t+1 which means that −α t ∈ N . We denote by N ⊥ the orthogonal vector space of N . The decomposition of B t becomes