Decomposition of supermartingales indexed by a linearly

Ordered Set

Gianluca Cassese

Università degli Studi di Lecce and University of Lugano April 19, 2006

Abstract We prove a version of the Doob Meyer decomposition for supermartingales with a lin- early ordered index set.

Key words Doob Meyer decomposition, natural increasing processes, potentials, supermartin- gales.

Mathematics Subject Classi…cation (2000): : 28A12, 60G07, 60G20.

1 Introduction.

In this paper we prove a version of the Doob Meyer decomposition for supermartingales indexed by a linearly ordered set . Given a standard probability space ( ;F; P) we consider a family F~ = (F : 2 )of sub algebras ofF with the property that ; "2 and "implyF F". X = (X : 2 ) is a supermartingale if X 2 L¹(F ) for each 2 and ; " 2 and "

implyX P(X"j F ).S denotes the collection of all supermartingalesX which are bounded with respect to the normkXk_S = sup ₂ kX kL¹.

As is well known, the Doob Meyer decomposition was initially established by Doob [2] for su- permartingales indexed byNand later extended by Meyer [6] to the case of right continuous super- martingales indexed byR+ and under the “usual assumptions”. Despite the key importance of this result in the theory of stochastic processes, several situations of interest lead outside of its range, particularly so in applications where it is often more useful to index processes by stopping times.

The case treated in this work may be exempli…ed by taking to be the collection of hitting times arising from a corresponding collection of pairwise nested subsets ofR.

Correspondence to: Gianluca Cassese, Università della Svizzera Italiana, via G. Bu¢ 13, CH-6904, Lugano, Switzerland. e-mail:gianluca.cassese@lu.unisi.ch

The main result of this paper, Proposition 1, provides a necessary and su¢ cient condition for X 2 S to admit a version of the Doob Meyer decomposition in which the intervening increasing process is required to be natural (in the sense of de…nition 2). Although such decomposition may appear somehow “weaker” than the original one, it is established under signi…cantly more general conditions than those usually considered.

Given its repeated use and despite being essentially known, we state here the following criterion for weak convergence inL¹, a slight generalization of [4, lemma 1, p. 441] (throughout the paper we identify sets with their indicators).

Lemma 1For each u in a directed set U let H^u be a sub algebra of F and fu 2L¹(H^u). Set H=S

uHu.hf_uiu2U converges to a Hmeasurable limitf weakly inL¹ if and only if (i).lim_uP(f_uH)exists and is …nite for eachH 2 Hand

(ii).hfuiu2U is uniformlyP integrable.

Moreover,hfuiu2U converges inL¹ whenever (ii) is replaced by (ii’).hfuiu2U is positive and increasing.

Proof Ifhfuiu2Uconverges weakly inL¹then the setffu:u2Ugis compact in that same topology:

(ii) follows from [3, IV.8.11, p. 294]; (i) is obvious. Assume that hfuiu2U satis…es (i). If (ii’) holds then (ii) follows from (i) and the inequality

sup

u2U

P(f_uF) P(f_u0F) + lim

u2UP(f_u f_u0), F 2 F

Assume (i) and (ii), letH 2 H,H⁰ 2 Hand denote byjH H⁰jtheir symmetric di¤erence. Then 2 sup

u

P(jf_uj jH H⁰j) sup

u⁰;u⁰⁰ ufP(f_u0(H H⁰)) P(f_u00(H H⁰))g lim sup

u⁰

P(f_u0(H H⁰)) lim inf

u⁰⁰P(f_u00(H H⁰))

= lim sup

u⁰

P(fu⁰H) lim inf

u⁰⁰P(fu⁰⁰H)

Given thatP(jH H⁰j)can be made arbitrarily small by an accurate choice of H⁰ 2 H and given (ii), Q(h) = lim_uP(f_uh) exists for any h 2 H and, hf_uiu2U being norm bounded, for each h 2 L¹(F). Q 2 ba( H) and, by (ii), Q P. Let f 2 L¹( H) be the corresponding Radon Nikodym derivative. Ifg2L¹(F),

limu P(fug) = lim

u P(fuP(gj H)) =P(f P(gj H)) =P(f g) Under (ii’),P(jf fuj) =P(f fu).

Denote by ₁and 0the indexes associated to the algebrasW

2 F andV

2 F respectively.

By Lemma 1, whenever the supermartingaleX is uniformly integrable we may and will consider the corresponding weak limitsX ₁ andX ₀ i.e. treat as if it admitted a maximal as well as a minimal element.

2 Uniformly Integrable Potentials.

A stochastic process A = (A : 2 )is increasing if P(A" A A ₀ = 0) = 1 for " ; it is integrable ifsup₂ P(A )<1. Let

D=f 1 : : : _N : ₁= ₀; _N = ₁; N2Ng (1) and ford=n

d

1 : : : ^d_Ndo

2 Dand] ; "] = ⁰2 : < ⁰ " let

P^d= (

F₀f 0g [

N[d 1

n=1

F_ni

d n; ^d_n+1i

:F₀2 F 0; F_n2 F ^d_n; )

(2) Let moreover

P = (

F0f ⁰g [ [N

n=1

Fn] n; "n] :F02 F0; Fn2 F ^d_n; ^d_n; "n2 ;1 n N; N 2N )

(3) De…nition 1Let d2 D andA= (A : 2 ).A is (i)d-predictable (ii) predictable or (iii) weakly predictable if (i)AisP^dmeasurable (ii)Ais Pmeasurable or (iii) for each 2 ,A is measurable with respect to the algebra F =W

< F . To each andU 2

2

L¹ we can associate the predictable process

P^d(U) =P(U ₀j F 0) +

NXd 1

n=1

P U ^d

n+1 F ^d_n i

d n; ^d_n+1i

(4) Identifyf : !Rwithf : !Rand letDbe directed by inclusion.

De…nition 2An increasing, integrable processA is natural if P b

Z

f dA = lim

d2DP Z

P^d(b)f dA, b2L¹; f 2 P (5) De…nition 3A supermartingale X= (X : 2 )is Doob Meyer supermartingale, X2 S^m, if for each 2 ,X =P(Mj F ) A whereM 2L¹ andA is increasing, integrable and natural.X is weakly Doob Meyer, X2 S^w, if Ais weakly predictable rather than natural.

Remark 1De…nition 2 di¤ers from the traditional one (e.g. [7, VII, D18, p. 111]) inasmuch P^d(f) need not converge nor need its would be limit share the properties of the predictable projection.

However, in the special case in which =R+,Ais right continuous and the usual conditions hold, (5) implies thatAis predictable. (This follows from [1, theorem VI.61, p. 126] upon selecting a sequence hdnin2Nsuch that each dyadic rational belongs to somedn andP(bA ₁) = limnPR

P^dn(b)dA).

As sual, a potential is a positive supermartingale such thatlim P(X ) = 0.

Lemma 2Letk >0 andX be a potential such thatP(X k) = 1for all 2 . Then there exists a sequence hX^rir2N inS^w such thatX X^r 0and limrkX X^rk_S = 0.

Proof Consider the casek= 1, …x d=n

d

1 : : : ^d_Ndo

2 D and >1and adopt throughout the proof the conventions 0/ 0 = 0and Q

f?g= 1. For1 n < Nd let h^d_n=

Yn

i=1

X d i

P X ^d

i+1 F^d_i

(6)

For each1 m n < N_d and up to aP null set 1 h^d_m h^d_n X ₀ ⁿY¹

i=1

X ^d

i+1

P X ^d

i+1 F^d_i

0 (7)

Given thatX ^d

Nd = 0,

P h^d_{Nd 1} F^d_n = ¹P P X ^d

Nd 1 F ^d_{Nd 2} h^d_{Nd 2} F ^d_n

= ¹P X ^d

n+1 F ^d_n h^d_n

= 1 X ^d

n

1 h^d_{n 1} i.e.

h^d_{n 1}X d

n= P h^d_{n 1} h^d_{Nd 1} F^d_n (8)

Let

A^dd

n+1 = 1 h^d_n ; 1 n < Nd (9)

De…ne alsoA^d = A^d : 2 andX^d= X^d : 2 by A^d =

NXd 1

n=1

A^dd n+1

n d

n < ^d_n+1o

and X^d=P A^dd

Nd F A^d (10)

By (7)A^d is increasing and bounded by ; by (6)A^d isd-predictable; by (8)X^d₀=X ₀ and X^d= P h^d_n h^d_{Nd 1} F =h^d_nP X ^d

n+1 F , 1 n < Nd; ^d_n < ^d_n+1 (11) We thus conclude from (11) thatX X^d and

X X^d = inf

n: ^d_n+1 P X h^d_nX ^d

n+1

1+ inf

n: ^d_n+1 P X X ^d

n+1 (12)

EndowingL¹with the weak topology and

2 L¹with the corresponding product topology, the set Z 2

2 L¹: Z 0 is compact and containsA^d. Moving to a subnet if necessary (still indexed byD), A^d _d

2D converges to a limitA . LetX =P A

1 F A andX = (X : 2 ). For each 2 ,A isF measurable, by Lemma 1, and0 X X ; moreover, by (12)

kX Xk_S = sup

2

P(X X ) = sup

2

dlim2DP X X^{d 1} If k >0 is arbitrary and X 2 S^w is such that 0 X k ¹X and X k ¹X

S

1 then kX 2 S^w,0 kX X andkkX Xk_S k ¹: the claim follows from being arbitrary.

We obtain next the following characterization where upper bar denotes the closure in the norm topology ofS andS^u denotes uniformly integrable supermartingales.

Theorem 1S^w=S^u.

Proof S^m S^u asS^w S^u andS^u is closed. LetY 2 S^u and X+M its Riesz decomposition as the sum of a martingale M = (P(Y₁j F ) : 2 )and a potential X, both uniformly integrable.

For each k > 0 the process X^k = X^k: 2 , with X^k = X ^k, is a potential bounded by k. Then Y^k = M +X^k 2 S^w by Lemma 2 and limk Y Y^k

S = limksup P X X^k limksup P(Y fY > kg) = 0.

The classical Doob Meyer decomposition (see [1], [6] and [7]) does not follow from uniform integrability but requires the classDproperty [7, VII.T29]. Meyer himself considered this condition as “not very easy to handle”and its relationship with uniform integrability as yet unclear [6, p. 195]¹. Theorem 1 contributes (if at all) to this discussion by clarifying the role of uniform integrability for the Doob Meyer decomposition.

3 The ClassD Property.

Proposition 1 justi…es the following terminology.

De…nition 4A stochastic processX is of class D if the collection

D (X) = (X

n2N

X ^d

nFn : ^d_n2 ; ^d_n ^d_n+1< ₁; Fn2 F ^d_n; FnFn⁰ =?; n; n⁰ 2N )

(13) is uniformly integrable.

If X is a uniformly integrable potential, …x an increasing sequence d^X = D

X n

E

n2N such that

X

1 = ₀, ^X_n < ₁ and lim_nP X X

n = 0 and denote by DX the collection of all increasing sequences in formed by adding tod^X a …nite subset of . LetDX be directed by inclusion.

Proposition 1Let X2 S.X is Doob Meyer if and only ifX is of class D .

Proof LetX =P(Mj F ) A be the Doob Meyer decomposition ofX 2 S^mandA ₁ be theL¹ limit ofA. Ifh=P

n2NX _nFn 2D (X)thenP(jhj) P(jMj+A ₁)and P(jhj fjhj> cg) =PX

n2N

jX _njn F_n X d

n > co PX

n2N

(jMj+A ₁)fF_njX _nj> cg

=P((jMj+A ₁)fjhj> cg)

1 An attempt to investigate the relationship between uniform integrability and the class Dproperty was done in [6, proposition 1, p. 195]. A uniformly integrable supermartingale not of classDwas later constructed in [5, pp. 61-2] where also a characterization for right continuous supermartingales was obtained (see. p. 59).

X is then of classD sinceP(jhj> c) c ¹P(jMj+A ₁).

Assume now thatX is of classD and,a fortiori, uniformly integrable. By Riesz decomposition we can actually focus on the case in whichX is a potential of classD . For eachd=D

d n

E

n2N2 DX

let F^^d = F^d_n:n2N and denote by S(d) the class of L¹ bounded supermartingales on F^^d and by ^d₁ the index assigned to the algebra W

n2NF ^d_n. The potential X^^d = X ^d

n:n2N is uniformly integrable: for each r 2 N there exists X^^d;r 2 S^m(d) such that 0 X^^d;r X^^d and

X^^d;r X^^d

S(d) 2 ^r, by Lemma 2. LetA^^d;r be the increasing, integrable and weakly predictable process associated withX^^d;r andA^^d;rd

1 itsL¹limit. The processesX^^d,X^^d;r andA^^d;r de…ned onF^^d extend to processesX^d,X^d;r andA^d;r onF~ by letting

X^d=X ₀f ⁰g+X

n2N

P X ^d

n+1 F n

d

n< ^d_n+1o

(14)

X^d;r= ^X^d;r₀ f ⁰g+X

n2N

P X^^d;rd

n+1 F n

d

n < ^d_n+1o

(15) and

A^d;r=X

n2N

A^^d;rd n+1

n d

n< ^d_n+1o + ^A^d;rd

1

\

n

n

> ^d_no

(16) ThenX^d;r,X^d2 S are such that

X X^d X^d;r =P A^d;rd

1 F A^d;r, 2 (17)

X^d X^d;r

S = X^^d;r X^^d

S(d) 2 ^r and X^d X = 0, 2d (18) A^d;ris increasing, integrable andd-predictable: thenh^q =P

n2NX ^d

n

n A^d;rd

n+1> q A^d;rd n

o

2D (X).

P A^d;r

1

n A^d;r

1 > qo

=X

n2N

P A^d;r

1

n A^d;rd

n+1> q A^d;rd n

o

X

n2N

P X ^d

n+A^d;rd n

nA^d;rd n+1

> q A^d;rd n

o

X

n2N

P X d n+q n

A^d;rd n+1

> q A^d;rd n

o (19)

=PX

n2N

X ^d

n

n A^d;rd

n+1

> q A^d;rd n

o

+qP A^d;r

1 > q

=P h^qn A^d;r

1 > qo

+qP A^d;r

1 > q Following [9, lemma 2] rather strictly, we deduce from (19)

P h^qn A^d;r

1 > qo

P A^d;r

1 q n A^d;r

1 > qo

P A^d;r

1 q n A^d;r

1 >2qo

qP A^d;r

1>2q (20)

which, combined again with (19), delivers P A^d;r

1

n A^d;r

1 >2qo

P h^2qn A^d;r

1>2qo

+ 2qP A^d;r

1 >2q P h^2qn

A^d;r

1> qo

+ 2P h^qn A^d;r

1 > qo 3 sup

h2D (X)

P hn A^d;r

1 > qo

P A^d;r

1 > q q ¹P A^d;r

1 q ¹P(X ₀) and the classD property imply that for each > 0 andqsu¢ ciently highsup_d2DX;r2NP A^d;r

1

nA^d;r

1 >2qo . The collectionn

A^d;r

1 : d2 DX; r2No

is thus uniformly integrable. For …xedd2 DX a subse- quence ofD

A^d;r

1

E

r2N(still indexed byr) admits a limitA^d

1 in the weak topology ofL¹[7, II.T23], [3, I.7.9]. De…neA^d by letting

A^d=P A^d₁ F X^d, 2 (21)

For eachF 2 F we get from (17) and (18) P A^dF = lim

r P A^d;r

1 X^d;r F = lim

r P A^d;rF

By Lemma 1 A^d is increasing, integrable and d-predictable; moreover, A^d converges in L¹ to A^d

1

as X^d is a potential. This same argument applied to the net, A^d

1 d2D^X delivers an increasing, integrable and weakly predictable processAsuch that

A =P(A ₁j F ) X , 2 (22)

Ifh2L¹ F ^d_n , then (21), (18) and (22) imply P h A^dd

n+1 A^dd

n = P h X^dd

n+1 X^dd

n = P h X ^d

n+1 X ^d

n =P h A ^d

n+1 A ^d

n

so that forb2L¹₊ andf 2 P P b

Z

f dA = lim

d P b Z

f dA^d = lim

d P Z

P^d(b)f dA^d= lim

d P Z

P^d(b)f dA

After the work of Mertens [8, T2] the Doob Meyer decomposition of a supermartingale indexed byR+ is known to exist even in the absence of the usual assumptions on the …ltration and the right continuity of trajectories, providedX is an optional, strong supermartingale of classD [1, theorem 20, p. 414]. Although our supermartingale decomposition does not compare exactly to that of Doob and Meyer, its existence does not require but the class D property, a condition considerably less restrictive than what usually assumed.

References

1. C. Dellacherie, P. A. Meyer,Probabilities and Potential B, North-Holland, Amsterdam, 1982.

2. J. Doob,Stochastic Processes, Wiley, New York, 1953.

3. N. Dunford, J. Schwartz,Linear Operators. General Theory, Wiley, New York, 1988.

4. L. L. Helms,Mean Convergence of Martingales, Trans. Amer. Math. Soc.87(1958), 439-446.

5. G. Johnson, L. L. Helms,Class D Supermartingales, Bull. Amer. Math. Soc.69(1963), 59-62.

6. P. A. Meyer,A Decomposition Theorem for Supermartingales,Ill. J. Math.6(1962), 193-205.

7. P. A. Meyer,Probability and Potentials, Blaisdell, Waltham Mass., 1966.

8. J. F. Mertens,Processus Stochastiques Généraux et Surmartingales, Z. Wahrsch. Verw. Geb. 22(1972), 45-68.

9. K. M. Rao,On Decomposition Theorems of Meyer, Math. Scand.,24(1969), 66-78.