FKG INEQUALITY FOR BROWNIAN MOTION AND STOCHASTIC DIFFERENTIAL EQUATIONS

The purpose of this work is to study some possible application of FKG inequality to the Brownian motion and to Stochastic Differential Equations. We introduce a special ordering on the Wiener space and prove the FKG inequality with respect to this ordering. Then we apply this result on the solutions $X_t$ of a stochastic differential equation with a positive coefficient $\sigma$ , we prove that these solutions $X_t$ are increasing with respect to the ordering, and finally we deduce a correlation inequality between the solution of different stochastic equations.


Introduction
The FKG inequality is a correlation inequality for monotone functions. It is named after Fortuine, Ginibre and Kasteleyn, who gave a rigorous formulation and established sufficient condition for its validity [3]. In the following years a lot of works were inspired by this inequality, new inequalities generalizing FKG were discovered and a great deal of applications were found, like in the field of statistical mechanics, which it was born for, or in different fields: for example the FKG inequality for the optimal transportation problems [2], or the applications of FKG inequality to cellular automata [7]. In [3] Fortuine, Kasteleyn and Ginibre found some sufficient condition to build spaces where the inequality (1) holds . This immediately permitted the application of this inequality to the rigorous analysis of percolation and ferromagnetic models. Many generalizations followed: Holley [5] introduced an inequality on convex dominations of measures, Preston [8] passed from discrete to continuous spin models, Kamae, Krengel and O'Brien [6] made a work on partially ordered Polish spaces and Ahlswede and Daykin [1] found a brilliant generalization of Holley's work [5] introducing a combinative inequality. This work aims at proving the FKG inequality for the Wiener space with a special ordering on the increments. Then we apply this FKG inequality to prove (theorem 7) a correlation inequality between the solutions of two stochastic equations.

Preliminaries
Let us start recalling the definition of FKG space. Let from now on (Ω, F, P, ≥) be a partially ordered probability space, where ≥ is an order relation on the set Ω. A function f : Ω → R is said to be increasing or ≥ increasing if for all That is the same of saying an event is increasing if its indicator function I A is so. holds.
Proposition 1 Let (Ω, F, P, ≥) be a partially ordered probability space, then it satisfies the FKG inequality if and only if for all A, B ∈ F increasing sets, the inequality is verified.
be a partially ordered measurable space, let X : Ω → E be a r.v. and let µ = X(P) be its law. The r.v. X is said to satisfy FKG if the space (E, E, µ, ≥) does.

FKG and the increasing functions
A first way to prove that the space satisfy FKG is to show that it is the increasing image of an FKG space. Let (Ω 1 , F 1 , P 1 , ≥ 1 ) and (Ω 2 , F 2 , P 2 , ≥ 2 ) be two partially ordered spaces.

Proof.
Let A 2 , B 2 be two increasing events in F 2 and let us define and therefore ω 2 ∈ A 1 . A 1 and B 1 increasing implies that they satisfy the inequality P 1 (A 1 ∩B 1 ) ≥ P 1 (A 1 )P 1 (B 1 ) and since P 2 = f (P 1 ) then it also holds: P 2 (A 2 ∩B 2 ) ≥ P 2 (A 2 )P 2 (B 2 ).

FKG inequality on product spaces
Another way of building FKG spaces from FKG spaces is that of making their product. Let (Ω i , F i , P i , ≥ i ) i∈Γ be a family of partially ordered probability spaces, and let us define the space (Ω, F, P, ≥) in this way: With the last definition we mean that for all u, v ∈ Ω From now on we will denote this product in this way: Proof. See appendix.
This theorem allows to tie the FKG spaces of classic literature together. In this way we easily originate examples of FKG spaces as it is showed in the next section.

Examples
Remark 1.1 Every totally ordered probability space satisfies FKG.
Proof. Let (Ω, A, P, ≥) be totally ordered and let A, B ∈ A be two increasing events, then either A ⊆ B or B ⊆ A; therefore P(A ∩ B) ≥ P(A)P(B).
Example 1 Every probability measures µ on R satisfies the FKG inequality. Therefore if f and g are increasing and integrable the following inequality holds: Example 2 Every real random variable satisfies FKG and, by theorem 3, so does their product.

FKG inequality and Brownian motion
In this section the FKG inequality is proved for the standard Brownian motion (W t ) t∈[0,T ] on the canonical Wiener space (Ω, A, P) endowed with a special partial order.
Here Ω = C 0 ([0, T ]; R) is the space of continuous function vanishing at zero, A = B(Ω) is the Borel σ − algebra induced by the uniform convergence topology and P is the Wiener measure. Moreover, W t denotes the canonical process, W t (ω) = ω(t). The first thing we must do now is to introduce an ordering on (Ω, A, P). The right choice of such an ordering is very important and it can be useful to recall the following remark.
Remark 2.1 Let ≥ 1 , ≥ 2 be two ordering relation on Ω and let ≥ 2 be finer than ≥ 1 , that , then it is increasing also for ≥ 1 . Summarizing, on the space (Ω, A, P, ≥ 1 ) there are more increasing functions and it is more difficult to prove the FKG inequality.
This means that a too fine ordering could lead to a too weak statement, and a too weak ordering could lead to a space that doesn't satisfy FKG anymore. In this work we choose the natural order for additive processes based on path increments.
Definition 2.1 Given ω 1 , ω 2 ∈ Ω, we say that ω 1 ≤ ω 2 if and only if for all This ordering is less fine than the ordering induced by the direct comparison of the trajectories (namely , and for remark 2.1 we obtain a stronger statement. Moreover the choice of this ordering is necessary for the purposes of section 3. Theorem 4 For the Wiener space (Ω, A, P, ≥) with the ordering ≥ of definition 2.1, the FKG inequality holds.
Proof. The idea of the proof is to gradually proceed towards the σ-algebra A. We shall verify (2) at first on a suitable sub-algebra B and then we will proceed with density arguments on all A. Let n be an integer n ≥ 2.
..,n} . Let finally A H ⊂ A the sub-σ-algebra generated by X H and let ≥ H be the ordering induced by . . , n} ). Then, as it was showed in example 2, X H satisfies FKG and also for (Ω, A H , P, ≥ H ) the FKG inequality holds. We also have ω 1 ≤ ω 2 implies ω 1 ≤ H ω 2 and then, ∀A ∈ A, if A is ≥H increasing then A is ≥ increasing. Generally the converse is not true. But as it is showed in proposition 11 the following lemma holds: The following conditions are equivalent: In order to complete the proof we have to show that it is possible to approximate increasing events of A with increasing events of B. This is a less trivial fact and it is done by the following lemma: Lemma 6 ∀ε > 0 and ∀A ∈ A increasing ∃B ∈ B increasing such that P(A B) ≤ ε .
P roof of Lemma 6 : B is an algebra and a basis of A, then B is dense in A that means: ∀ε > 0 ∀A ∈ A ∃B ∈ B such that P(A B) ≤ ε . Now we want to show that if A is increasing then it is possible to approximate A with increasing events of B. Let us fix now ε > 0 and A ∈ A increasing: then, from what we said before, we can choose a partition 0 = t 0 < t 1 < · · · < t n = T with The set E and F are two linear subspaces of Ω , Ω = E ⊕ F and every element of Ω can be written in an unique way as the sum of an element of E and one of F . We can define two maps L : Ω → E and Y : Ω → F such that ∀ω ∈ Ω we have W (ω) = L(ω) + Y (ω).
Let now E and F be the traces A on E and F : we can easily verify that the maps L and Y are random variables on (Ω, A, P) taking values in (E, E) and (F, F), with L −1 (E) = A H and L ,Y independent random variables [9]. Let P 1 = L(P) and P 2 = Y (P) . Then the application is bijective, bi-measurable and it preserves the measure. Let now ϕ : Ω → [0, 1] be the application ϕ(ω) = P 2 Y A ∩ L −1 (L(ω)) .
If ω = e + f with e ∈ E and f ∈ F then ϕ(ω) = F I A (e, g)dP 2 (g) . Now we only have to show that if A is increasing then ϕ is increasing. Let ω 1 ≤ ω 2 be two trajectory with

This expression shows that ϕ is a version of the conditional expectation of I
ω 1 ≤ ω 2 implies e 1 ≤ e 2 and e 1 + g ≤ e 2 + g for all g ∈ F . This means I A (e 1 , g) ≤ I A (e 2 , g) for all g ∈ F and finally ϕ(ω 1 ) ≤ ϕ(ω 2 ) ϕ increasing implies B increasing and this finishes the proof of the lemma and of the theorem.

FKG inequality for stochastic differential equations.
In this section we prove that the solution X t of a stochastic equation with quite general coefficients is a random increasing variable on the space (Ω, A, P, ≥). In this way we may use the FKG inequality for Brownian motion to show a correlation inequality for the random variables X t and Y s , solutions of two different equation, evaluated at different times. Let (Ω, A, P, ≥) be the W iener space, with the ordering of definition 2.1. Let X t , Y t be the solution of the stochastic equations This result is a direct conseguence of the FKG inequality for the space (Ω, A, P, ≥) and the following lemma, which we believe is of conceptual interest in itself and may find other applications.
Lemma 8 Let b be Lipschitz continuous, let σ be differentiable with Lipschitz derivative. Assume there exist two positive constants , M such that ≤ σ ≤ M . Then there exists a stochastic process (X t ) t∈[0,T ] on (Ω, A, P), with X t (ω) defined for all ω ∈ Ω, such that; (i) X t is a solution of (3) (ii) for every t ∈ [0, T ], X t is an increasing function on (Ω, A, P, ≥).

Proof of lemma 8
The idea of the proof is to obtain a simpler differential equation with an increasing change of variable from X t to Z t . Let and G = F −1 . By the above assumption on σ, the function F and G are C 2 (R, R),increasing, bijective and Lipschitz continuous, G (z) = σ(G(z)), G (z) = σ(G(z)) · σ (G(z)) . Let us define: By the assumptions on b and σ we obtain b locally Lipschitz continuous. Let z 0 = F (x 0 ) and let Z be the solution of the following equation: If L is a common Lipschitz constant for b, σ and G. Then By proposition 10 the integral equation (5) have one and only one solution through R. Moreover by the comparison theorem 9 we can deduce that the random variable Z(t) (fixed t) is increasing in the means of definition 2.1. If we define X t : Ω → R, X t = G(Z t ) then X t is increasing on (Ω, A, P, ≥). It remain to be proof that the random variable X t satisfies the stochastic equation (3). By the Itô's formula this completes the proof of the lemma.
Theorem 9 Let us consider the integral equations: Where W 1 , W 2 are continuous functions and b 1 , b 2 are locally Lipschitz functions. Let us suppose the existence of the solutions in an interval [0, T ]. If we assume the "comparison" hypotheses: Then for all t ∈ [0, T ] we have the inequality Proof. See appendix.
Proposition 10 Let W t be continuous.Let b be locally Lipschitz. If there exist two positive constant k 1 and k 2 such that | b(z)| ≤ k 1 + k 2 z ∀z ∈ R, then the integral equation: (Z)(t) := z 0 + t 0 b(Z(s))ds + W t has one and only one solution through R.

Product spaces
Proof. of theorem 3 We will proceed by induction on the cardinality of Γ, proving firstly the finite case, then the enumerable one and finally the generic one. . It is easy to check that f 1 and g 1 are increasing with respect to ≥ 1 .Then Let p n : Ω →Ω n be the projection on the first n coordinates, and p n : Ω → Ω n the projection on the other ones. If we define f n , g n :Ω n → R in this way: f n (x) := P n (p n (p −1 n (x) ∩ A)) , g n (x) := P n (p n (p −1 n (x) ∩ B)) then it easily follows that f n and g n are increasing, bounded and they satisfy the correlation inequality By the Lévy's Upward Theorem we have f n → I A and g n → I B in L 1 and a.s.

Lemma 5
With the notation of Lemma 5 the following proposition holds.
Proposition 11 Let A ∈ A H . Then A is ≥ increasing if and only if A is ≥H increasing.

Proof.
To prove that A ≥H increasing implies A ≥ increasing it is sufficient to remark that ≥ H is finer than ≥.
Remark 4.1 Let A ∈ A and A be ≥H increasing. Then, A ∈ A H .

Proof.
It is sufficient to notice that A = P −1 n (P n (A)). With the projection. P n := (W t1 , . . . , W tn ).