Multiparameter Martingales

In this section we will study two multiparameter martingales, both of which are due to R. Cairoli and J. B. Walsh. In particular, we will introduce the maximal inequality for both martingales. All the results stated can be found in Khoshnevisan [15], and we refer to [15]

for a complete description of multiparameter martingale theory.

First let us recall one parameter martingales. One parameter martingale theory can be found in most standard graduate level probability textbooks, for example, [16]. A collection of σ-algebras {Ft}_t≥0in the probability space (Ω, F , P) is called a filtration if Fs⊂ F_tfor all 0 ≤ s ≤ t. A one-parameter process X := {X(t)}t≥0 is said to be adapted to {Ft}_t≥0 if X(t) is measurable with respect to F_t for all t ≥ 0. Then X is a submartingale with respect to the filtration {Ft}_t≥0 if

1. X is adapted to {Ft}_t≥0;

2. X(t) ∈ L¹(P), that is, E[|X(t)|] < ∞, for all t ≥ 0; and 3. E[X(t)|Fs] ≥ X(s) almost surely for all 0 ≤ s ≤ t.

We call X a supermartingale if −X is a submartingale. Moreover, if X is both a supmartingale and supermartingale, then X is a martingale. Classical martingale theory tells us that a nonnegative submartingale X whose sample paths are right-continuous with left limits satisfies Doob’s maximal inequality ([15], Chapter 1, Theorem 1.4.1):

E

 sup

0≤s≤t

X(t)



≤

 p p − 1

p

E[X(t)^p] ∀ t ≥ 0 and p > 1. (3.20) In many cases, we deal only with the natural filtration of X, namely, the filtration {G_t}_t≥0 such that G_t = σ({X_s : 0 ≤ s ≤ t}). In words G_t is the smallest σ-algebra with respect to which {Xs: 0 ≤ s ≤ t} are measurable. According to the towering property of conditional expectations ([16], Theorem 8.5), if X is a submartingale with respect to {F_t}_t≥0 then X is a submartingale with respect to {Gt}_t≥0.

When the index set is R^N+, we lose the total ordering of R that we use to define filtrations. Since there are many partial orderings of R^N+, we can define different types of martingales. A first natural multiparameter extension of one-parameter martingale is an orthomatringale.

Definition 3.13. Let N be a positive integer and F := (F¹, . . . , F^N) be N one-parameter filtrations with F^j = {F_t^j

j}_tj≥0. A multiparameter stochastic process X := {X(t)}_t∈RN + is

29 an orthosubmartingale with respect to F if for each 1 ≤ j ≤ N , and for all fixed non-negative {s_k}_{1≤k≤N,k6=j}, the one-parameter process {X(s₁, . . . , s_j−1, t_j, s_j+1, . . . , s_N)}_tj≥0is a submartingale with respect to F^j. X is called an orthosupermartingale if −X is an orthosubmartingale. If X is both a orthosubmartingale and orthosupermartingale, then X is an orthomartingale.

Example 3.14. Let X_j := {X_j(t_j)}_tj≥0 (1 ≤ j ≤ N ) be N one-parameter martingales with respect to one-parameter filtrations X^j (1 ≤ j ≤ N ). Furthermore, we assume X^j’s are independent of each other. Then the additive process

X(t1, . . . , tN) :=

Here ∨j∈JG^j denotes the σ-algebra generated by the union of σ-algebras {G^j}_j∈J. In partic-ular, the random variables {X_j(s_j) : 0 ≤ s_j ≤ t_j} and {X_k(s_k) : s_k ≥ 0, 1 ≤ k ≤ N, k 6= j}

are measurable with respect to F_t^j

j.

Similar to the one-parameter case, we deal with the natural filtrations of multiparameter processes. In the case of othorsubmartingales, these natural filtrations are called ortho-histories.

Definition 3.15. Let {X(t)}_t∈RN

+ be a multiparameter process. The orthohistories generate by X is the collection of filtrations {H^j}_1≤j≤N such that

H^j = {H^j_tj}_tj≥0 and H^j_t

j = σ({X(s) : 0 ≤ sj ≤ t_j, sk≥ 0 for k 6= j}). (3.23) Lemma 3.16. Let X := {X(t)}_t∈RN

+ be an orthosubmartingale with respect to one-parameter filtrations F¹, . . . , F^N. Then X is an orthosubmartingale with respect to its orthohistories H¹, . . . , H^N.

Proof. We refer to Chapter 1 of [15], Lemma 2.6.1 for a proof.

The above lemma shows that the orthohistories of an orthosubmartingale X is the smallest collection of σ-algebras with respect to which X is still an orthosubmartingale.

This frees us from concerning the underlying filtrations of an orthosubmartingale in many cases. Therefore we will refer to the orthohistories if the collection of filtrations is not given explicitly for an orthosubmartingale.

Similar to one-parameter submartingales, there is a maximal inequality for orthosub-martingales. In order to state it, we need to introduce some ordering on the index set. For s = (s₁, . . . , s_N) and t = (t₁, . . . , t_N), define

s 4 t if and only if sj ≤ t_j ∀ 1 ≤ j ≤ N. (3.24) For a general multiparameter process {X(t)}_t∈R^N

+, sup_04s4tX(t) may not be a random variable. In order to solve this technical issue, we will assume the process is separable. A stochastic process {X(t)}_t∈R^N

+ is said to be separable if there exists an at most countable collection T ⊂ R^N+ and a null set Λ ⊂ Ω such that for all closed sets A ⊂ R^d and all open sets I ⊂ R^N+ of the form I = Π^N_j=1(a_j, b_j), where a_j ≤ b_j are rational or infinite for all 1 ≤ j ≤ N ,

{ω : X_s(ω) ∈ A for all s ∈ I ∩ T } {ω : Xs(ω) ∈ A for all s ∈ I} ⊂ Λ. (3.25) Fortunately, the assumption of separability does not cost us too much.

Theorem 3.17 (Doob’s Separability Theorem). Every multiparameter process {X(t)}_t∈RN +

has a separable modification, that is, there exists a separable process {Y (t)}_t∈RN

+ such that P{X(t) = Y (t)} = 1 for all t ∈ R^N₊.

Proof. We refer to Chapter 5 of [15], Theorem 2.2.1 for a proof.

Lemma 3.18 (Cairoli’s Strong (p, p) Inequality). Let {X(t)}_t∈R^N

+ be a separable, nonneg-ative orthosubmartingale with respect to one parameter filtrations F¹, . . . , F^N. Then

E

 sup

04s4t

X(t)



≤

 p p − 1

N p

E[X(t)^p] ∀ t ∈ R^N+ and p > 1. (3.26) Proof. We refer to Chapter 1 of [15], Theorem 2.3.1 for a proof.

The martingale property of an orthomartingale is essentially that of a one-parameter martingale. Next we introduce a type of genuine multiparamter martingales. Recall the ordering s 4 t defined in (3.24). A collection of σ-algebras F = {Ft}_t∈RN

+ is a multiparameter filtration if Fs ⊂ F_t for all s 4 t. For simplicity, we still call F a filtration. An N -parameter process {X(t)}_t∈RN

+ is adapted to the filtration F if X(t) is measurable with respect to Ftfor all t ∈ R^N+.

31 Definition 3.19. A multiparameter stochastic process X := {X(t)}_t∈R^N

+ is an multipa-rameter submartingale with respect to the filtration F if

1. X is adapted to F ;

2. X(t) ∈ L¹(P), that is, E[|X(t)|] < ∞, for all t ∈ R^N+; and 3. E[X(t)|Fs] ≥ X(s) almost surely for all 0 4 s 4 t.

X is called an multiparameter supermartingale if −X is an multiparameter submartin-gale. If X is both a multiparameter submartingale and supermartingale, then X is an multiparameter martingale. For simplicity, we will drop the term multiparameter whenever there is no ambiguity of doing so.

There is a close connection between martingales and orthomartingales. Let F :=

{F_t}_t∈RN

+ be a filtration. Then we can extract one-parameter filtrations F^j := {F_t^j

j}_tj≥0 (1 ≤ j ≤ N ) from F . We define

F_t^j

j := _

s∈R^N₊,sj=tj

F_s, (3.27)

and call F¹, . . . , F^N the marginal filtrations of F . Proposition 3.20. Let X := {X(t)}_t∈RN

+ be adapted to the N -parameter filtration F . If X is an orthosubmartingale with respect to the marginal filtrations of F , then X is a submartingale with respect to F .

Proof. We refer to Chapter 1 of [15], Proposition 3.2.1 for a proof.

Unfortunately, the converse of the above proposition is not true in general. We refer to section 3.3 of Khoshnevisan [15] for a counterexample where a multiparameter process is a martingale but not an orthomartingale. However, with additional assumptions on the filtration, a multiparameter process is guaranteed to be an orthomartingale. One such sufficient condition is the conditional independence of filtrations.

Definition 3.21. Let G₀, G₁ and G₂be σ-algebras. G₁ and G₂ are said to be conditionally independent given G0 if

E[Y1Y2|G₀] = E[Y1|G₀]E[Y2|G₀] a.s., (3.28) for all bounded random variables Y₁ and Y₂ measurable with respect to G₁ and G₂, respec-tively.

Proposition 3.22 (Cairoli and Walsh). Let F := {Ft}_t∈RN

+ be an N -parameter filtration and X := {X(t)}_t∈RN

+ be a process that is adapted to F . Assume F_tand F_sare conditionally independent of F_tfs for all s, t ∈ R^N+, where t f s = (min(ti, si))1≤i≤N. Then X is a submartingale with respect to F if and only if X is an orthosubmartingale with respect to the marginal filtrations of F .

Proof. We refer to Chapter 1 of [15], Theorem 3.5.1 for a proof.

An immediate and important consequence of the above proposition is a maximal inequal-ity for submartingales with respect to a filtration that has the conditional independence property. It follows from the above proposition and Lemma 3.18.

Corollary 3.23 (Cairoli’s Strong (p, p) Inequality). Let X := {X(t)}_t∈R^N

+ be a separa-ble, nonnegative submartingale with respect to a N -parameter filtration F := {F_t}_t∈RN

+. Assume Ft and Fs are conditionally independent of F_tfs for all s, t ∈ R^N+, where t f s = (min(ti, si))_1≤i≤N. Then

E

 sup

04s4t

X(t)



≤

 p p − 1

N p

E[X(t)^p] ∀ t ∈ R^N+ and p > 1. (3.29) Proof. We refer to Chapter 7 of [15], Theorem 2.3.2 for a proof.