The scales of classes of stochastic processes are introduced. New interpolation theorems and boundedness of some transforms of stochastic processes are proved. Interpolation method for generously monotonous processes is entered. Conditions and statements of interpolation theorems concern the fixed stochastic process, which differs from the classical results.

1. Introduction

Interpolation methods of functional spaces are one of the basic tools to get inequalities in parametrical spaces. These methods are widely applied in the theory of stochastic processes (see [1–5] and other).

In this paper classes of stochastic processes are considered, which, in some sense, are analogues of the net spaces which were investigated in [6–8].

Assume that (Ω, 𝔊, P) is a complete probability space. A family G = {𝔊_n} _n⩾1 of σ-algebras 𝔊_n such that 𝔊₁⊆⋯⊆𝔊_n⊆⋯⊆𝔊 is called a filtration.

Let G be a filtration, a sequence {X_n} _n⩾1 of random variables X_n measurable function with respect to the σ-algebra 𝔊_n. Then we say that the set X = (X_n, 𝔊_n) _n⩾1 is a stochastic process.

Let F = {𝔉_n} _n⩾1 be a system of sets satisfying the condition 𝔉₁⊆⋯⊆𝔉_n⊆⋯⊆𝔊. We say that a stochastic process X = (X_n, 𝔊_n) _n⩾1 is defined on a system F = {𝔉_n} _n⩾1 if 𝔉_n ⊂ 𝔊_n, n ∈ ℕ. For a stochastic process X, which is defined on a system F = {𝔉_n} _n⩾1, we define the sequence of numbers

, where

()

We call this sequence a majorant of a process X on a system of sets F.

Let us give some examples of a choice of a system of sets F = {𝔉_n} _n⩾1: 𝔉_n = Ω, in this case the sequence is a sequence of averages of a process X = {X_n}; for 𝔉_n = 𝔊_n−1 it is a majorant of sequence of conditional averages M(X_n∣G_n−1); and for 𝔉_n = 𝔊_n it is a majorant of a process X = {X_n}. The following cases are interesting: (1) 𝔉_n = 𝔊_n∧τ, where τ = τ(ω) is the fixed stopping time, and 𝔊_n∧τ = {A ∈ 𝔊 : A∩{τ = n} ∈ 𝔊_n}; (2) , where τ_n = τ_n(ω), and n ∈ ℕ is the sequence of the stopping times.

We consider the classes of stochastic processes defined on F, which characterize the speed of convergence of sequence to zero.

By N_p,q(F), 0 < p < ∞, 0 < q ⩽ ∞ we denote the set of all stochastic processes X, defined on F for which

()

if 0 < q < ∞ and

()

if q = ∞.

Let us denote

()

for 0 < q < ∞, and

()

for q = ∞, where

()

We consider that

()

where [α] is the integer part of the number α. In particular X_1/2(ω) ≡ 0.

A random variable τ, which takes values in the set (1,2, …, ∞), is called the Markov time of the filtration G = {𝔊_n} _n⩾1, if {ω : τ(ω) = n} ∈ 𝔊_n for any n ∈ ℕ. The Markov time τ, for which τ(ω) < ∞ (a.p.) [9], is called the stopping time.

Let X = (X_n, 𝔊_n) _n⩾1 be a stochastic process and τ be the Markov time. By X^τ we denote the stopped process X^τ = (X_n∧τ, 𝔊_n), where and χ_A(ω) is the characteristic function of the set A.

We assume also that

()

The spaces are spaces of converging stochastic processes, where parameters α, q, and p characterize the speed and the metrics, in which a given process converges.

In this paper we prove a Marcinkiewicz-type interpolation theorem for the introduced space. An interpolation method, essentially related to the properties of the Markov stopping times, is introduced. In the last paragraph the given interpolation method is applied to Besov type space with variable approximation properties. Part of the results were announced in [10].

We write A≲B (or A ≳ B) if A ⩽ cB (or cA⩾B) for some positive constant c independent of appropriate quantities involved in the expressions A and B. Notation A≍B means that A≲B and A ≳ B.

2. Properties of the Spaces N_p,q(F) and

We say [9] that stochastic process (X_n, 𝔊_n) _n⩾1 is a martingale if for every n ∈ ℕ the following conditions hold: (1) E|X_n| < ∞; (2) E(X_n+1∣𝔊_n) = X_n (a.p.). If instead of property (2) it is assumed that E(X_n+1∣𝔊_n)⩾X_n(E(X_n+1∣𝔊_n) ⩽ X_n), then we say that a process is a submartingale (supermartingale).

Definition 1. Let F = {𝔉_n} _n be a fixed system of sets, X = (X_n, 𝔊_n) _n be a stochastic process defined on F. We say that a process X belongs to the class W(F) if there exists a constant c such that for every k ⩽ m and for every A ∈ 𝔉_k

()

This inequality implies that for every k ⩽ m. The class W(F) contains martingales, nonnegative submartingales, and nonpositive supermartingales. The stochastic process from W(F) we call generalized monotone.

Lemma 2. Let α > 0, 0 < q ⩽ ∞, and 1 < p < ∞. If , then there exists a random variable X_∞ such that (a.p.).

Proof. Let L_p,∞(Ω) be the Marcinkiewicz-Lorentz space and 2^ν−1 ⩽ n < 2^ν. Using the equivalent norm of L_p,∞(Ω) spaces (see [6]) and measurability of function X_n with respect to σ-algebra 𝔊_n, we get the following:

()

Taking into account that , for α > 0, we have .

But , therefore by the Doob theorem ([11]), the process X_n converges almost surely.

Lemma 3. Let α > 0, 0 < q ⩽ ∞, 1 < p < ∞, and X = (X_n, 𝔊_n) _n⩾1 ∈ W(G). Then

()

where

()

Proof. The existence of X_∞ follows from Lemma 2. Further, we have

()

Therefore, using Lemma 8, we obtain

()

The reverse inequality follows from the expression:

()

The lemma is proved.

Lemma 4. Let X ∈ W(F). Then

(1)
for 0 < q ⩽ q₁ ⩽ ∞,
()

()
(2)
for ε > 0, 0 < q, q₁ ⩽ ∞,
()

()

where

depend only on the indicated parameters.

Proof. Let us prove inequalities (16), (18). The proof of inequalities (17), (19) is similar. Let ε > 0. By Minkowski’s inequality and by the generalized monotonicity of a process X = (X_n, 𝔊_n) _n⩾1 we get the following:

()

To prove the second statement it is enough to show that

and apply the first statement. Since p₁ < p, we have the following:

()

Remark 5. Properties of the N_p,q(F) spaces given in Lemma 4 show that the second parameter q is weak with respect to the first p. These properties of the spaces are important in the interpolation.

Lemma 6. Let 0 < p < ∞, a > 1. If X ∈ W(F), then for 0 < q < ∞

()

and for q = ∞

()

Proof. Using the generalized monotonicity of a process X, we have the following:

()

One can prove the reverse estimate in a similar way.

Lemma 7 (Hölder inequality). Let 0 < p₁, p₂, q < ∞, 0 < t, s₁, s₂ ⩽ ∞ and (1/q) = (1/p₁)+(1/p₂), (1/t) = (1/s₁)+(1/s₂). If stochastic processes and , then XY = (X_nY_n, 𝔊_n) ∈ N_q,t(F) and

()

Proof. Since Y_n is measurable with respect to an algebra 𝔊_n, we have and hence

()

We will need the following Hardy-type inequalities.

Lemma 8. Let s⩾1, ν > 0, α > 0, β > 0, and γ > 0; then for a nonnegative sequence a = {a_k} _k the following inequalities hold:

()

3. Interpolation Method for Stochastic Processes

Let

be a transform that transforms a stochastic process X, which is defined on the system

, to the stochastic process

, which is defined on the system

. We say that the transform T is quasilinear if there exists a constant C > 0 such that for any n ∈ ℕ the following inequality holds almost surely:

()

It is known ([9]) that if a process X = (X_n, 𝔊_n) _n⩾1 is a martingale (submartingale), then the process X^τ = (X_n∧τ, 𝔊_n) _n⩾1 is also a martingale (submartingale).

Denote and .

The transforms X^τ and X^* of the stochastic process X are examples of quasilinear transforms.

Let A = (A₀(F), A₁(F)) be a pair of quasinormed own subspaces of linear Hausdorff stochastic processes spaces 𝔑(F), which is defined on a probability space (Ω, 𝔉, P) with a filtration F = {𝔉_n} _n⩾1. Obviously, this pair is compatible pair and hence the scale of interpolation spaces is defined with respect to the real method ([12]).

Moreover, let for 0 < θ < 1, 0 < q < ∞

()

and for q = ∞

()

where

()

is the Peetre functional.

Let

be a sequence of stopping times with respect to a filtration F and A(F) = (A₀(F), A₁(F)) be a pair of quasinormed own subspaces 𝔑(F). Let X ∈ 𝔑(F) and t ∈ (0, ∞). We define the following:

()

Here the infimum is taken over all stopping times from R. Moreover for 0 < q < ∞

()

and for q = ∞

()

Theorem 9. Let (A₀(F), A₁(F)), (B₀(Φ), B₁(Φ)) be two compatible pairs of stochastic processes and let R = {τ(ω)} be some fixed family of Markov times with respect to a filtration F. If T is a quasilinear map for stochastic processes X = (X_n, 𝔉_n) _n⩾1 and

()

for all stopping times τ ∈ R, then

()

where the constant C is from the definition of quasilinearity of the operator T.

Proof. Consider the following:

()

The theorem is proved.

Lemma 10. Let a > 1 and R = {k} _k∈ℕ be stopping times. Then for 0 < q < ∞

()

The proof is similar to the proof of the Lemma 4.

4. Interpolation Properties of the Spaces N_p,q(F)

Theorem 11. Let 1 < p₀ < p₁ < ∞, 1 ⩽ q₀, q₁, q ⩽ ∞, 0 < θ < 1, (1/p) = ((1 − θ)/p₀)+(θ/p₁), and R = {k} _k∈ℕ be the stopping times. Then for any stochastic process X = (X_n, 𝔉_n) _n⩾1,

()

where the constant c depends only on parameters p_i, q_i, θ, i = 0,1.

If X = (X_n, 𝔉_n) _n⩾1 ∈ W(F), then

()

where the constant c also depends only on parametres p_i, q_i, θ, i = 0,1.

Proof. Let X = Y + Z be any representation of a process X, where , . To prove the first statement of the theorem, we use the following inequality: .

For any a > 1 we have the following:

()

By putting

, we get

. Therefore, using (22) and Lemma 8, we have the following:

()

Let us prove the second statement of the theorem. Let X = (X_n, 𝔉_n) _n⩾1 ∈ W(F). By using Lemmas 8 and 3 we have the following:

()

Let k = [2^nγ], then taking into account that

, n ⩽ −1, we obtain the following:

()

Thus, we obtain the following:

()

Further, we have the following:

()

By using Lemma 8 for (1/γ) = (1/p₀)−(1/p₁), (46), and (47), we have the following:

()

By applying Minkowski’s inequality to (45) and using Lemma 4, estimates (48), we obtain

()

Corollary 12. Let 0 < p₀ < p₁ < ∞, 0 < q₀ < q₁ < ∞, 0 < θ < 1, 1 ⩽ s ⩽ ∞, (1/q) = ((1 − θ)/q₀)+(θ/q₁), (1/p) = ((1 − θ)/p₀)+(θ/p₁), X ∈ W(F), and be a quasilinear transform. If for any k ∈ ℕ ∪ {0} the following conditions hold:

()

then

()

where C > 0 depends only on p₀, p₁, q₀, q₁, and θ.

Taking into account that the measurable function may be considered as a martingale, by corollary we may receive Marcinkiewicz-Calderon interpolation theorem (see [13]).

Corollary 13 (Marcinkiewicz-Calderon theorem). Let 0 < p₀ < p₁ < ∞, 0 < q₀ < q₁ ⩽ ∞, θ ∈ (0,1), 1/p = (1 − θ)/p₀ + θ/p₁, and 1/q = (1 − θ)/q₀ + θ/q₁. If T is a quasilinear map and

()

then

()

Corollary 14. Let be a quasilinear transform such that for any p ∈ (a, b) and for any X ∈ N_p,1(F)∩W(F) the following weak inequality holds:

()

Then

()

for any p and s such that p ∈ (a, b), 1 ⩽ s ⩽ ∞.

5. Boundedness of Some Operators in Class N_p,q(F)

Let

be a stochastic sequence and V = (V_n, 𝔊_n−1) be a predicted sequence (𝔊₋₁ = 𝔊₀). A stochastic sequence V · Y = ((V·Y)_n, 𝔊_n) such that

()

where ΔY_i = Y_i − Y_i−1 is called the transform of Y with respect to V. If Y is a martingale then we say that V · Y is the martingale transform.

Theorem 15. Let 0 < q < p < ∞, 1 ⩽ τ ⩽ ∞ and 1/r = 1/q − 1/p. Let V · Y be a martingale transform of a martingale Y by predicted sequence V = (V_n, 𝔉_n−1). If

()

then

()

where a constant c depends only on parametres p, q, and τ.

Proof. Let V · Y be a martingale transform of a martingale Y by predicted sequence V = (V_n, 𝔉_n−1); that is,

()

where Y₀ = 0, ΔY_k = Y_k − Y_k−1. By Abel’s transform

, we get the following:

()

Taking into account that ΔV_k, Y_k are measurable functions with respect to the algebra 𝔊_k, we have

and

()

Hence the weak inequality is proved as follows:

()

for 1 < q < p < ∞.

Let 0 < q < p < ∞, (1/r) = (1/q)−(1/p), 0 < q₀ < q < q₁ < ∞ and 0 < p₀ < p < p₁ < ∞. Let a pair of numbers (p₀, p₁) and (q₀, q₁) satisfy the following condition:

()

Then from that is proved above it follows that

()

for 0 < p < q < ∞.

Taking into account that for any stopping time k ∈ N processes Y^k and Y − Y^k are martingales, it is possible to apply Theorem 9. Then

()

where

()

Note that there exists θ ∈ (0,1) such that (1/p_θ) = (1/p). Then it follows from (63) that (1/q_θ) = (1/q).

Theorem 16. Let 0 < p < ∞, 1 ⩽ q ⩽ ∞ and X = (X_n, 𝔊_n) _n⩾1 ∈ W(G) and τ(ω) be the Markov time and let X^τ = (X_n∧τ, 𝔊_n) _n⩾1 be a stopped process. Then

()

Proof. Denote

()

Let us show that

If 2^s ⩽ n < 2^s+1, then

()

Now, using Corollary 12 we get the statement of the Theorem 16.

Corollary 17. Let 0 < p < ∞, 1 ⩽ q ⩽ ∞ and a process X = (X_n, 𝔊_n) _n⩾1 be a nonnegative submartingale. Then the process is also submartingale and

()

Proof. It follows from Theorem 16 that

()

The reverse inequality is trivial.

6. Interpolation Properties of the Space , the Embedding Theorems

Theorem 18. Let 1 ⩽ p, q, q₀, q₁ ⩽ ∞, α₀ < α₁, 0 < θ < 1, α = (1 − θ)α₀ + θα₁, and R = {r} _r∈ℕ. Then

()

Proof. Using Lemmas 10 and 4 we have the following:

()

Putting r = [2^nγ], γ = α₁ − α₀ we get the following:

()

It follows from the definition that

for

for k⩾r; therefore,

()

Substituting these equalities in (74) and applying Lemmas 4 and 8, we get the following:

()

For the proof of reverse estimate we use the fact that for any r and k the following equality holds:

()

Then we have the following:

()

Substituting

and using Lemma 4, we have the following:

()

since

, the proof is complete.

Theorem 19. Let 1 < r ⩽ p < ∞, 1 ⩽ q ⩽ ∞ and α = (1/r)−(1/p) and let the filtration G = {𝔊_n} _n⩾1 be such that for every k = 1,2… and for all A ∈ 𝔊_k the following condition holds:

()

where the constant C > 0 does not depend on k. Then

()

Proof. Let us show that

()

According to the condition (80), for

we have that P(A)>(C/2^m). Therefore for α = (1/r)−(1/p) we get the following:

()

Thus, (82) is proved.

Now, let α₀ < α₁, 1 < p₀ < p₁ < ∞, and θ ∈ (0,1) such that

()

Then using interpolation Theorems 18 and 11 we obtain the following:

()

It follows from (82) that

. Hence,

, where α = (1/r)−(1/p). The proof is complete.

7. Spaces with Variable Approximation Properties by Haar System

In this paragraph we consider some applications of the introduced interpolation method to Besov type spaces with variable approximation properties.

Let Ω = [0,1] and let 𝔉 be a σ-algebra of Borel subsets of set Ω, P a linear Lebesgue measure on 𝔉, F = {𝔉_n} _n⩾1 the Haar filtration, and

a sequence of stopping times such that for any k⩾0 the following conditions hold: τ₀ ≡ 0, τ_k + 1 ⩽ τ_k+1 (a.p.) and

()

For a function f(x) ∈ L[0,1] we denote by {c_k(f)} _k⩾1 the Fourier coefficients by Haar functions system {H_k(x)} _k⩾1 ([14]). For the given stopping time τ_k(ω) we denote

()

which we call the Fourier-Haar partial sum of a function f, corresponding to the Markov time τ_k.

Let 1 < p < ∞, 0 < q ⩽ ∞, α ∈ ℝ. By

we denote the set of functions f ∈ L[0,1], for which

()

for 0 < q < ∞,

()

for q = ∞.

Conceptually, the introduced spaces are close to spaces with variable smoothness. Here we mention works of Leopold [15], Cobos and Fernandez [16], and Besov [17–20].

Lemma 20. Let 1 < p < ∞, f ∈ L_p[0,1], and let S(f, τ) be the Fourier-Haar partial sum with respect to the Markov time τ. Then

()

Proof. Denote

()

Let L_p,∞[0,1] be the Marcinkiewicz-Lorentz space. Using the equivalent norm of L_p,∞[0,1] spaces (see [6]) and martingale properties of Fourier-Haar partial sums we get the following:

()

Now, applying the interpolation theorem (see [12]), we obtain the statement of the lemma.

Lemma 21. Let 1 < p < ∞, 0 < q ⩽ ∞, α ∈ ℝ, and

()

Then

()

Proof is similar to the proof of Lemma 3.

Theorem 22. Let 1 ⩽ q₀, q₁, q ⩽ ∞, 0 < α₀ < α₁, 0 < θ < 1, and α = (1 − θ)α₀ + θα₁. Then

()

Proof. By using Lemma 21 we have the following:

()

By applying Lemma 8 we get the following:

()

Let us prove the reverse embedding. Let

, f = f₀ + f₁ be an arbitrary representation of a function,

and

. Then

()

Since the representation f = f₀ + f₁ is arbitrary, we have the following:

()

Hence, putting

we get the following:

()

The theorem is proved.

Acknowledgment

This research was partially supported by Ministry of Education and Science of the Republic of Kazakhstan (0112RK02176, 0112RK00608).

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Interpolation Methods for Stochastic Processes Spaces

Abstract

1. Introduction

2. Properties of the Spaces N_p,q(F) and

3. Interpolation Method for Stochastic Processes

4. Interpolation Properties of the Spaces N_p,q(F)

5. Boundedness of Some Operators in Class N_p,q(F)

6. Interpolation Properties of the Space , the Embedding Theorems

7. Spaces with Variable Approximation Properties by Haar System

Acknowledgment

References

References

Information

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Interpolation Methods for Stochastic Processes Spaces

Abstract

1. Introduction

2. Properties of the Spaces Np,q(F) and

3. Interpolation Method for Stochastic Processes

4. Interpolation Properties of the Spaces Np,q(F)

5. Boundedness of Some Operators in Class Np,q(F)

6. Interpolation Properties of the Space , the Embedding Theorems

7. Spaces with Variable Approximation Properties by Haar System

Acknowledgment

References

References

Related

Information

2. Properties of the Spaces N_p,q(F) and

4. Interpolation Properties of the Spaces N_p,q(F)

5. Boundedness of Some Operators in Class N_p,q(F)