On the decompositions of T -quasi-martingales on Riesz spaces∗ Jessica J. Vardy Bruce A. Watson† School of Mathematics University of the Witwatersrand Private Bag 3, P O WITS 2050, South Africa July 11, 2013

Abstract The concept of a quasi-martingale is generalised to the Riesz space setting. Here we show that a quasi-martingale can be decomposed into the sum of a martingale and a quasi-potential. If, in addition, the quasi-martingale and its filtration are right continuous we show that the quasi-martingale can decomposed into the sum of a right continuous martingale and the difference of two positive right continuous potentials. The approach is measure-free and relies entirely on the order structure of Riesz spaces.

1

Introduction

Quasi-martingales were first introduced by H. Rubin in an invited lecture at the Institute of Mathematical Statistics in 1956. In [4] quasi-martingales were formally introduced and defined by Fisk. Fisk gave necessary and sufficient conditions under which a quasi-martingale with continuous sample paths could be decomposed into the sum of a martingale and a process having almost every sample path of bounded variation. Orey, in [14], generalised Fisk’s results to right continuous processes (or F -processes, in Orey’s terminology). Finally, in [15], Rao gave an elegant and greatly simplified proof of Orey’s result. Rao was also able to prove that every right-continuous quasi-martingale can be written as the difference of two positive super-martingales. ∗

MSC2000: 46A40; 47B60; 60G20; 60G48. Keywords: Riesz space; Conditional expectation; Quasimartingale. † Supported in part by NRF grant IFR2011032400120 and by the Centre for Applicable Analysis and Number Theory

1

In this paper, we generalise quasi-martingales to the Riesz space setting. We show that quasi-martingales in a Riesz space can be uniquely decomposed into the sum of a martingale and a quasi-potential (a Riesz decomposition). If, in addition, the quasimartingale is right continuous then the martingale and quasi-potential of this decomposition are also right continuous. Further to this, we show that each right continuous quasi-potential can be decomposed into the difference of two positive potentials. This continues the work of Kuo, Labuschagne and Watson on the generalisation of stochastic process to the Riesz space setting, see [8, 9], other generalisations and studies of martingales and stochastic processes in the setting of Riesz spaces have been given by Boulabiar, Buskes and Triki [2], Dodds, Grobler, Huijsmans and de Pagter [3, 5, 6], Luxemburg and de Pagter [11], Stoica [16, 17], Troitsky [18]. In Section 2 we present, for the reader’s convenience, a summary of results and definitions pertaining to martingales in Riesz spaces. Section 3 focuses on various aspects of right continuity of filtrations and processes. In Section 4 we define quasi-martingales in Riesz spaces and give a decompositions of Reisz space quasi-martingales as the sum of a Riesz space martingale and Riesz space quasi-potential. In Section 5 each Riesz space quasi-potential is shown to be representable as the difference of two positive Riesz space potentials. The paper concludes with an application of these results to spaces of measurable functions. We thank Professor J.J. Grobler and the referee for their valuable suggestions and comments.

2

Preliminaries

The reader is assumed familiar with the notation and terminology of Riesz spaces, for details see [1, 12, 20, 21] or, for the more specific aspects used here, see [8, 9]. All limits are understood to be order limits. We refer the reader to [1] for a characterization of order convergence, in a Dedekind complete Riesz space, in terms of limit superior and limit inferior which is implicitly used at various points.

Definition 2.1 Let E be a Riesz space with a weak order unit. A positive order continuous projection T : E → E, with range R(T ) a Dedekind complete Riesz subspace of E, is called a conditional expectation if T (e) is a weak order unit of E for each weak order unit e of E.

We note that for T a conditional expectation operator (as T is a positive operator), we have that |T f | ≤ T |f |, f ∈ E. In many parts of this paper we require that T is not just positive (T f ∈ E+ for f ∈ E+ ) but that T is strictly positive, by which we mean that T is positive and T f > 0 for all f 6= 0 in E+ . Here E+ denotes the positive cone of E.

2

For example, in the Riesz space E = Lp (Ω, F, P ), 1 ≤ p ≤ ∞, where (Ω, F, P ) is a probability space, T is a Riesz space conditional expectation on E which leaves the weak order unit e = 1 (the constant function with value 1) invariant if and only if T = E[·|Σ] for some Σ a sub-σ-algebra of F. In this case T has range Lp (Ω, Σ, P ), see [8, 9]. If E is a Dedekind complete Riesz space and T is a strictly positive conditional expectation operator on E, then E has a T -universal completion, see [9], which is the natural domain of T , denoted dom(T ) in the universal completion, E u , of E, also see [3, 6, 13, 20]. Here dom(T ) = D − D and T x := T x+ − T x− for x ∈ dom(T ) where u D = {x ∈ E+ |∃(xα ) ⊂ E+ , xα ↑ x, (T xα ) order bounded in E u },

and T x := supα T xα , for x ∈ D with xα ↑ x, (xα ) ⊂ E+ , (T xα ) order bounded in E u .

Definition 2.2 Let E be a Dedekind complete Riesz space with weak order unit and T be a strictly positive conditional expectation on E. The space E is universally complete with respect to T , for brevity T -universally complete, if for each increasing net (fα ) in E+ with (T fα ) order bounded in the universal completion, E u , of E, we have that (fα ) is order convergent.

We recall for the reader’s convenience the following definitions from [8, 9]. Let E be a Riesz space with a weak order unit e and Λ be a partial order index set. A filtration (or stochastic basis) on E index by Λ is a family of conditional expectations, (Tλ )λ∈Λ on E with Tλ Tγ = Tλ = Tγ Tλ for all λ ≤ γ, λ, γ ∈ Λ. An family (fλ ) in E indexed by Λ is adapted to a filtration (Tλ ) on E indexed by Λ if fλ ∈ R(Tλ ) for each λ ∈ Λ. A family of tuples (fλ , Tλ ) indexed by Λ is a (sub, super) martingale if (Tλ ) is a filtration, (fλ ) is adapted to (Tλ ) and fλ (≤, ≥) = Tλ fγ for all λ ≤ γ, λ, γ ∈ Λ. In this paper we will only be concerned with the cases of Λ = [0, ∞) and Λ a countable subset of [0, ∞) with ordering inhertied from R.

3

Right continuity

In order to proceed with the development of a theory of continuous-time quasi-martingales in Riesz spaces continuity conditions need to be imposed on the filtration. In addition to these continuity conditions need to be placed on the process considered. These are outlined below, along with some connections between these different continuity assumptions. The following definition is from Grobler, [5], where an in depth discussion of the properties of continuous time martingales in Riesz spaces can be found.

3

Definition 3.1 Let E be a Dedekind complete Riesz space with weak order unit and (Tt ), t ∈ [0, ∞), be a filtration on E. We say that the filtration, (Tt )t∈[0,∞) , is right \ R(Tt ) for all s ∈ [0, ∞). continuous if R(Ts ) = t>s

Note 3.2 Let E be a Riesz space with weak order unit and conditional expectation T . If in addition E is T -universally complete and (Tt ), t ∈ [0, ∞), is a filtration on E with Tt T = T = T Tt for all t ∈ [0, ∞), then from the Radon-Nikod´ ym Theorem, [19], there is\a conditional expectation operator Ts+ on E which commutes with T and has R(Tt ) for each s ∈ [0, ∞). In this setting the filtration (Tt )t∈[0,∞) is right range t>s

continuous if and only if Ts+ = Ts for all s ∈ [0, ∞). The following two types of right continuity are some of the weakest types and will be used in our final theorem. Definition 3.3 Let E be a Dedekind complete Riesz space with weak order unit and strictly positive conditional expectation operator T . (a) We say that (ft )t∈[0,∞) ⊂ E is sequentially right continuous if for each s ∈ [0, ∞) and each sequence (ti )i∈N with ti ↓i s we have that fti → fs as i → ∞. (b) We say that (ft )t∈[0,∞) ⊂ E is T -sequentially right continuous if for each s ∈ [0, ∞) and each sequence (ti )i∈N with ti ↓i s we have that T |fti − fs | → 0 as i → ∞. It is easily verified for ft ∈ E, t ∈ [0, ∞), that the right continuity of ft (i.e. ft → fs as t ↓ s) implies sequential right continuity of ft which in turn implies T -sequential the T -sequential right continuity of ft . We recall from [7] the following. A sequence (Ti ) of conditional expectation operators on E is called a reverse filtration if Ti Tj = Tj = Tj Ti for all j ≥ i. The sequence of pairs (fi , Ti ) is called a reverse martingale if fj = Tj fi for all j ≥ i. If there is g ∈ E so that T |fi | ≤ g, for all i ∈ N, then the sequence (fi ) is said to be T -bounded. In [7] it was shown that each T -bounded reverse martingale in a T -universally complete Riesz space is convergent. We now use this to develop a lemma needed for the final theorem of the paper. Lemma 3.4 Let E be a T -universally complete Riesz space with weak order unit and (Tt )t∈[0,∞) be a right continuous filtration on E with T Tt = T , where T is a strictly positive conditional expectation operator on E. If (ft , Tt )t∈[0,∞) is a martingale, then ft is sequentially right continuous. Proof: Let s ∈ [0, ∞) and (ti ) be a sequence in [0, ∞) decreasing to s, then (fti , Tti )i∈N is a reverse martingale. Furthermore, for i ∈ N, T |fti | = T |Tti ft1 | ≤ T Tti |ft1 | = T |ft1 |

4

giving that (fti ) is T bounded. The reverse martingale convergence theorem of Korostenski, Labuschagne and Watson, [7], \ can be applied to (fti , Tti )i∈N to give that (fti ) R(Tt ) = R(Ts ). Thus fs = Ts fti → Ts f s = f s is order convergent in E to say f s ∈ t>s

as i → ∞ and fti → fs as i → ∞

4

Quasi-martingales

Definition 4.1 Let E be a Dedekind complete Riesz space with conditional expectation operator T and weak order unit e = T e. Let (Tt )t∈[0,∞) be a filtration on E with T Tt = T = Tt T . We say a process (ft )t∈[0,∞) is a T -quasi-martingale if (ft ) is adapted to (Tt ) and there is M ∈ E+ for which n X

T |fti − Tti fti+1 | ≤ M,

i=1

for all (t1 , t2 , . . . , tn+1 ) ∈ Π. Here Π is the collection of all strictly increasing finite sequences in [0, ∞). If (ft )t∈[0,∞) is a T -quasi-martingale, then we say (ft )t∈[0,∞) is a T -quasi-potential if T |ft | tends to zero, in order, as t tends to infinity.

Theorem 4.2 (Riesz Decomposition Theorem) Let E be a T -universally complete Riesz space where T is a strictly positive conditional expectation operator and e is a weak order unit such that e = T e. Every T -quasi-martingale can be written uniquely as the sum of a martingale and a T -quasi-potential.

Proof: Uniqueness. Suppose the quasi-martingale (ft ) has Riesz decompositions qt + Zt = ft = q t + Z t ,

t ∈ [0, ∞),

(4.1)

where (Zt ) and (Z t ) are quasi-potentials and (qt , Tt ) and (q t , Tt ) are martingales. Rearranging (4.1) gives qt − q t = Z t − Z t .

(4.2)

Applying T to the absolute value of (4.2) and making use of the fact that Zt , Zt are quasi-potentials we get T |qt − q t | = T |Zt − Z t | ≤ T |Zt | + T |Z t | → 0,

(4.3)

as t → ∞. Now (|qt − q t |, Tt ) is a sub-martingale giving that T |qt − q t | is increasing (not necessarily strictly) in t. This along with (4.3) yields T |qt − q t | = 0,

for all

5

t ∈ [0, ∞).

(4.4)

The strict positivity of T and (4.4) give qt = q t and consequently from (4.2), Zt = Z t for all t ∈ [0, ∞). Existence. Let (sn ) be a strictly increasing sequence in [0, ∞) with lim si = ∞. Set i→∞

∆i = fsi − Tsi fsi+1 ,

for i ∈ N.

Let t ∈ [0, ∞),

Yt,i = Tt fsi ,

i ∈ N.

For si ≥ t, Tt ∆i = Yt,i − Yt,i+1 .

(4.5)

Applying T to |Tt ∆i | from (4.5) we get T |∆i | = T Tt |∆i | ≥ T |Tt ∆i | = T |Yt,i − Yt,i+1 |.

(4.6)

Let M ∈ E+ be as in Definition 4.1, then T

i≤n X

|Yt,i − Yt,i+1 | ≤

si ≥t

i≤n X

T |∆i | ≤ M,

si ≥t

P for all n ∈ N. Thus, from the T -universal completeness of E, (Yt,i − Yt,i+1 ) is absolutely convergent in E and, as this a telescoping series, (Yt,i ) is convergent in E as i → ∞. Denote qt = lim Yt,i . (4.7) i→∞

Here qt is independent of the choice of sequence (si ). To see this, let (ui )i∈N and (vi )i∈N be two increasing, unbounded sequences in [0, ∞) and lim Tt fui = q˜t

i→∞

and

lim Tt fvi = q t .

i→∞

Let (si )i∈N be an increasing sequence with (ui )i∈N and (vi )i∈N as subsequences, then we can define qt by lim Tt fsi = qt . (4.8) i→∞

By the uniqueness of limits and the construction of (si )i∈N , we have that q˜t = qt = q t . We now show (qt , Tt ) is a martingale in E. Let s ≤ t. From the definition of qt and as Ts is order continuous, it follows that Ts qt = Ts lim Yt,i = lim Ts Yt,i . i→∞

i→∞

(4.9)

Since (Tt ) is a filtration Ts Tt = Ts and the definition of Yt,i gives Yt,i = Tt fsi for si ≥ t ≥ s, from which it follows that Ts Yt,i = Ts Tt fsi = Ts fsi = Ys,i .

6

(4.10)

Taking i tending to infinity in (4.10) along with (4.9) and the definition of qt gives Ts qt = qs for s ≤ t. Thus (qt , Tt ) is a martingale. It remains to show that (ft − qt ) is a T -quasi-potential, that is T |ft − qt | → 0 as t → ∞. Towards this we begin by verifying that (fsi − qsi ) is a T -quasi-potential, i.e. T |fsi − qsi | → 0 as i → ∞. As E is Dedekind complete and

n X

T |∆i | ≤ M for all n ∈ N it follows that

i=1

xk =

T |∆i |

i=1

is convergent. Thus xk ↓ 0 as k → ∞ where ∞ X

∞ X

T |∆i |.

i=k

For si ≥ sk , from (4.5), Tsk ∆i = Ysk ,i − Ysk ,i+1 . From this, (4.6) and the fact that Tsk is a positive operator with T Tsk = T , we have T |∆i | = T Tsk |∆i | ≥ T |Tsk ∆i | = T |Ysk ,i − Ysk ,i+1 |. Hence lim

n→∞

n X

T |Ysk ,i − Ysk ,i+1 | =

∞ X

T |Ysk ,i − Ysk ,i+1 | ≤ xk ,

k ∈ N.

(4.11)

i=k

i=k

Now as Ysk ,k = fsk , n X

(Ysk ,i − Ysk ,i+1 ) = Ysk ,k − Ysk ,n+1 = fsk − Ysk ,n+1 .

(4.12)

i=k

Combining (4.11) and (4.12) gives lim sup T |fsk − Ysk ,n+1 | ≤ n→∞

∞ X

T |Ysk ,i − Ysk ,i+1 | ≤ xk .

i=k

But Ysk ,n+1 → qsk as n → ∞ and so T |fsk − qsk | ≤ xk ↓ 0, as k → ∞, showing that (fsk − qsk ) is a T -quasi-potential. It remains only to show that (ft − qt ) is a T -quasi-potential. As T |fs − Ts ft | ≤ M , for each t > s, and Ts ft → qs as t → ∞ we have 0 ≤ lim sup T |fs − qs | = h ≤ M, s→∞

exists in E. Let (t1 , . . . t2n+1 ) ∈ Π then, from the definition of a T -quasi-martingale, n X i=1

T |ft2i − Tt2i ft2i+1 | ≤

2n X i=1

7

T |fti − Tti fti+1 | ≤ M.

(4.13)

Now consider (4.13) with t2n+1 successively replaced by the elements of an increasing sequence (sj ) with t2n < s1 with limit infinity, then T |ft2i − Tt2i fsj | +

n−1 X

T |ft2i − Tt2i ft2i+1 | ≤ M.

(4.14)

i=1

Taking the limit as j tend to inifinity in (4.14) gives T |ft2i − qt2i | +

n−1 X

T |ft2i − Tt2i ft2i+1 | ≤ M,

(4.15)

i=1

for all (t1 , . . . t2n ) ∈ Π. Now taking the limit supremum of (4.15) as t2n → ∞ in [0, ∞), we have n−1 X h+ T |ft2i − Tt2i ft2i+1 | ≤ M, (4.16) i=1

for all (t1 , . . . t2(n−1)+1 ) ∈ Π. This process can now be successively repeated to give nh ≤ M . Since n was arbitrary in the initial choice of (t1 , . . . t2n+1 ) ∈ Π, we have 0 ≤ nh ≤ M for all n ∈ N, but E is an Archimedean Riesz space so h = 0. Thus T |ft − qt | → 0 as t → ∞ with t ∈ [0, ∞).

5

Quasi-potentials

Recall that a process Xt is a potential if (Xt , Tt ) is a super-martingale and T |Xt | → 0 as t → ∞.

Theorem 5.1 Let E be a T -universally complete Riesz space with weak order unit where T is a strictly positive conditional expectation operator. Let (Tt )t∈[0,∞) be a filtration on E, with Tt T = T = T Tt , t ∈ [0, ∞). If (Xt )t∈[0,∞) ⊂ E is a T -sequentially right continuous T -quasi-potential adapted to (Tt )t∈[0,∞) then there exist positive potentials Xtp , Xtm such that Xt = Xtp − Xtm

for all t ∈ [0, ∞).

(5.17)

Proof: For k, n ∈ N ∪ {0} define ∆(k, n) = Tk2−n (Xk2−n − X(k+1)2−n ) = Xk2−n − Tk2−n X(k+1)2−n . By the definition of a T -quasi-potential there exists M ∈ E+ so that κ X

T |∆(k, n)| ≤ M

k=0

8

(5.18)

for all n, κ ∈ N. From the T -universal completeness of E,

∞ X

|∆(k, n)| converges,

k=i

giving the convergence of ∞ X

∆(k, n)

and

k=i

∞ X

∆± (k, n).

(5.19)

k=i

Here ∆± (k, n) = 0 ∨ (±∆(k, n)) and we can define P ± := Tt k≥b2n tc+1 ∆± (k, n) ∈ R(Tt ), Xt,n

(5.20)

for t ∈ R, n ∈ N, where bxc is the greatest integer less than or equal to x. For s ≤ t, Ts Tt = Ts so X X + + ∆+ (k, n) = Xs,n , ∆+ (k, n) ≤ Ts Ts Xt,n = Ts k≥b2n sc+1

k≥b2n tc+1

± which with (5.20) gives that (Xt,n , Tt ) is a super-martingale for each n ∈ N. Further± more, as Xt,n ≥ 0 and T Tt = T , by (5.19) X ± T Xt,n =T ∆± (k, n) → 0, k≥b2n tc+1 + − as t → ∞. Thus Xt,n and Xt,n are potentials. ± We now show Xt,n to be increasing in n. Consider t ∈ [i2−n , (i + 1)2−n ) for some i ∈ {0, 1, 2, . . . } then X ± Xt,n = Tt ∆± (k, n). (5.21) k≥i+1

Now either t ∈ [2i/2n+1 , (2i + 1)/2n+1 ) or t ∈ [(2i + 1)/2n+1 , (2i + 2)/2n+1 ) so  2i 2i+1   X , 2n+1  Tt ∆± (2i + 1, n + 1), t ∈  2n+1 ± Xt,n+1 = Tt ∆± (k, n + 1) + 2i 0, t ∈ 22i+1 n+1 , 2n+1 k≥2i+2 X ≥ Tt ∆± (k, n + 1) k≥2i+2

= Tt

X

 ∆± (2k, n + 1) + ∆± (2k + 1, n + 1) .

k≥i+1

But f ± + g ± ≥ (f + g)± for all f, g ∈ E so ∆± (2k, n + 1) + ∆± (2k + 1, n + 1) ≥ (∆(2k, n + 1) + ∆(2k + 1, n + 1))± and t < (i + 1)2−n ≤ 2k2−(n+1) for k ≥ i + 1, so Tt = Tt T2k2−(n+1) giving X ± Xt,n+1 ≥ Tt (∆(2k, n + 1) + ∆(2k + 1, n + 1))± k≥i+1

=

X

Tt T2k2−(n+1) (∆(2k, n + 1) + ∆(2k + 1, n + 1))± .

k≥i+1

9

As T2k2−(n+1) (f ± ) ≥ (T2k2−(n+1) f )± from the previous display it follows that X ± Xt,n+1 ≥ Tt (T2k2−(n+1) {∆(2k, n + 1) + ∆(2k + 1, n + 1)})± k≥i+1

= =

X k≥i+1 ± Xt,n

Tt ∆± (k, n) from (5.21).

± Since (Xt.n ) is an increasing sequence in E bounded above by M , making use of the Dedekind completeness of E, we can define + + Xtp := sup Xt,n = lim Xt,n , n→∞

n

Xtm

:=

− sup Xt,n n

− = lim Xt,n . n→∞

Here (Xtp , Tt ) and (Xtm , Tt ) are super-martingales as they are the suprema of supermartingales. Now for t ∈ [0, ∞), ∞ X

− + Xt,n − Xt,n = Tt

∆(j, n)

j=b2n tc+1 ∞ X

= Tt

Tj2−n Xj2−n − X(j+1)2−n



j=b2n tc+1 ∞ X

 Tt Tj2−n Xj2−n − X(j+1)2−n .

=

j=b2n tc+1

As (b2n tc + 1)/2n > t, we have ∞ X

+ − Xt,n − Xt,n =

Tt Xj2−n − X(j+1)2−n



j=b2n tc+1

and thus from the order continuity of Tt ,  + − Xt − (Xt,n − Xt,n ) = lim Tt Xt − k→∞

k X

  Xj2−n − X(j+1)2−n 

j=b2n tc+1

 = Tt Xt − X(b2n tc+1)2−n + lim Tt X(k+1)2−n . k→∞

Using that (Xt ) is a T -quasi-potential, T Tt = T , the positivity of Tt and the order continuity of T gives  + − T |Xt − (Xt,n − Xt,n )| ≤ T |Tt Xt − X(b2n tc+1)2−n | + lim T |Tt X(k+1)2−n | k→∞

≤ T Tt |Xt − X(b2n tc+1)2−n | + lim T Tt |X(k+1)2−n | k→∞

= T |Xt − X(b2n tc+1)2−n | + lim T |X(k+1)2−n | k→∞

= T |Xt − X(b2n tc+1)2−n |

10

Now taking n tending to infinity and using the T -sequential right continuity of Xt gives T |Xt − (Xtp − Xtm )| = 0 From the strict positivity of T we now have that Xt = Xtp − Xtm . Finally, we show that Xtm and Xtp are potentials. Recall X ± Tt ∆± (k, n) Xt,n =

(5.22)

k≥b2n tc+1

is increasing in n. From the definition of ∆(k, n), T ∆(2k, n + 1)± + T ∆(2k + 1, n + 1)± ≥ T (∆(2k, n + 1) + ∆(2k + 1, n + 1))± = T T2k2−(n+1) (∆(2k, n + 1) + ∆(2k + 1, n + 1))± ≥ T (T2k2−(n+1) (∆(2k, n + 1) + ∆(2k + 1, n + 1)))± = T ∆(k, n)± giving that b2n tc ± Ht,n

:=

X

T ∆± (k, n)

(5.23)

k=0 ± is increasing in n. From its definition, it is also clear that Ht,n is increasing in t. From ± (5.18), Ht,n ≤ M for all t and n, thus ± ± ± lim lim Ht,n = sup Ht,n = lim lim Ht,n .

t→∞ n→∞

(5.24)

n→∞ t→∞

t,n

Here the Dedekind completeness of E has been used to ensure the existence of all terms in the above equalities. Now, by (5.22), ± ± ± T Xt,n + Ht,n = lim Ht,n . t→∞

Letting n → ∞ in the above equation gives p/m

T Xt

± ± + lim Ht,n = lim lim Ht,n . n→∞

n→∞ t→∞

Taking t → ∞ in the above and noting (5.24) gives lim T |Xtp | = 0 = lim T |Xtm |. t→∞

Thus Xtp and Xtm are potentials.

t→∞

From Lemma 3.4 and Theorems 4.2, 5.1 we have the following decomposition theorem.

Theorem 5.2 Let E be a T -universally complete Riesz space with weak order unit where T is a strictly positive conditional expectation operator. Let (Tt )t∈[0,∞) be a right continuous filtration on E, where Tt T = T = T Tt , t ∈ [0, ∞). If (Xt )t∈[0,∞) ⊂ E is a T -sequentially right continuous T -quasi-martingale then (Xt )t∈[0,∞) can be decomposed into the sum of a martingale with the difference of two positive potentials.

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Proof: From Theorem 4.2 Xt = qt + Zt where (qt , Tt ) is a martingale and (Zt ) is a T -quasi-potential. From Lemma 3.4, qt is sequentially right continuous and hence T sequentially right continuous. By assumption Xt is T -sequentially right continuous, making Zt T -sequentially right continuous. Hence Theorem 5.1 is applicable to Zt , giving that Zt can be expressed as the difference of two positive potentials. In a measure space setting Theorem 5.2 extends the results of Rao as follows. Let (Ω, F, µ) be a measure space and F0 be a sub-σ-algebra of F. Let E denote the Riesz space of all µ-a.e. defined equivalence classes of F-measurable functions on Ω for which the conditional expectation E[|f ||F0 ] exists as an a.e. finite valued function. For Ft a right continuous filtration with F0 ⊂ Ft ⊂ F for t ∈ [0, ∞), let Xt ∈ E, t ∈ [0, ∞), be adapted to Ft and there exists Y ∈ E so that for each finite increasing sequence 0 ≤ t1 < · · · < tn+1 , n ∈ N, n X

E[|Xti − E[Xti+1 |Fti ]|F0 ] ≤ Y.

i=1

If for each t ∈ [0, ∞) and each sequence (tn ) ⊂ [0, ∞) with tn ↓ t we have lim E[|Xtn − Xt ||F0 ] = 0,

n→∞

then there exists a a martingale Yt and positive super-martingales, Xt± so that Xt = Yt + Xt+ − Xt− and lim E[Xt± |F0 ] = 0. t→∞

References [1] Y.A. Abramovich, C.D. Aliprantis, An invitation to operator theory, American Mathematical Society, 2002. [2] K. Boulabiar, G. Buskes, A. Triki, Results in f -algebras, Positivity, Trends in Mathematics (2007), 73-96. [3] P.G. Dodds, C.B. Huijsmans, B. de Pagter, Characterizations of conditional expectation-type operators, Pacific J. Math., 141 (1990), 55-77. [4] D.L. Fisk, Quasi-martingales, Trans. Amer. Math. Soc, 120 (1965), 369 - 389. [5] J.J. Grobler, Continuous stochastic processes on Riesz spaces: the Doob-Meyer decomposition, Positivity, 14 (2010),731-751. [6] J.J. Grobler, B. de Pagter, Operators representable as multiplicationconditional expectation operators, J. Operator Theory, 48 (2002), 15-40. [7] M. Korostenski, C.C.A. Labuschagne, B.A. Watson, Reverse martingales in Riesz spaces, Operator Theory, Advances and Applications, 195 (2009), 213230. [8] W.-C. Kuo, C.C.A. Labuschagne, B.A. Watson, Discrete time stochastic processes on Riesz spaces, Indag. Math. N.S., 15 (2004), 435-451.

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[9] W.-C. Kuo, C.C.A. Labuschagne, B.A. Watson, Conditional expectations on Riesz spaces, J. Math. Anal. Appl., 303 (2005), 509-521. [10] C.C.A. Labuschagne, B.A. Watson, Discrete stochastic integrals in Riesz spaces, Positivity, 14 (2010), 859-875. [11] W.A.J. Luxemburg, B. de Pagter, Representations of positive projections II, Positivity, 9 (2004), 569-605. [12] W.A.J. Luxemburg, A.C. Zaanen, Riesz Spaces I, North Holland, 1971. [13] J. Neveu, Discrete-parameter martingales, North Holland, 1975. [14] S. Orey, F -Processes Proc. Fifth Berkeley Symp. on Stat. and Prob., Volume 2, Part 1, 301 - 313. [15] K. M. Rao, Quasi-Martingales, Math. Scand., 24 (1969), 79 - 92. [16] G. Stoica, Martingales in vector lattices, Bull. Math. Soc. Sci. Math. Roumanie. (N.S.), 34(82) (1990), 357-362. [17] G. Stoica, Martingales in vector lattices II, Bull. Math. Soc. Sci. Math. Roumanie. (N.S.), 35(83) (1991), 155-157. [18] V. Troitsky, Martingales in Banach lattices, Positivity, 9 (2005), 437-456. [19] B. A. Watson, An Andˆ o-Douglas type theorem in Riesz spaces with a conditional expectation, Positivity, 13 (2009), 543 - 558. [20] A.C. Zaanen, Riesz Spaces II, North Holland, 1983. [21] A.C. Zaanen, Introduction to Operator Theory in Riesz Space, Springer Verlag, 1997.

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On the decompositions of T-quasi-martingales on Riesz spaces

On the decompositions of T-quasi-martingales on Riesz spaces. ∗. Jessica J. Vardy. Bruce A. Watson. †. School of Mathematics. University of the Witwatersrand. Private Bag 3, P O WITS 2050, South Africa. July 11, 2013. Abstract. The concept of a quasi-martingale is generalised to the Riesz space setting. Here we show ...

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