Sample path properties of multidimensional integral with respect to stochastic measure

The integral with respect to a multidimensional stochastic measure, for which we assume only $\sigma$-additivity in probability, is studied. The continuity and differentiability of its realizations are established.


Introduction
The main purpose of this article is studying the regularity property of the integral of a deterministic function with respect to stochastic measure µ, where µ is defined on Borel subsets of [0, 1] d .In basic statements of the paper, for µ we assume only σadditivity in probability.We study the regularity with respect to a parameter and with respect to the upper limit of the integral.Case d = 1 was considered in [20,Section 2.3], but methods of [20] do not work for d ≥ 2. In our paper, we study the integral using the Fourier -Haar expansion of the integrand.
Sample path behaviour of various stochastic processes was studied in many works; we mention only some of them.The conditions for sample boundedness and continuity of α-stable processes were established in [22, Chapter 10].Hölder continuity of harmonizable operator scaling stable random field is given by Corollary 5.5 [5].In Theorem 4.1 from [18] the uniform modulus of continuity for some self-similar SαS random field was obtained, while the upper bound of modulus of continuity of stable random fields is given by Proposition 5.1 and Corollary 5.3 [6].These results, for example, imply Hölder continuity of mentioned fields.
Conditions of continuity of Gaussian random fields may be found in Theorems 3.3.3,3.4.1 and 3.4.3[1].Hölder continuity of centered Gaussian random fields under certain conditions is proved in [28,Theorem 4.2].Theorem 2.1 [29] gives an estimate of modulus of continuity of general random fields on metric space, these results are applied to harmonizable stable random fields.
In our article, we consider stochastic processes which are integrals with respect to a general stochastic measure.The definition of these measures, their properties and examples as well as information about Fourier -Haar series may be found in Section 2. The continuity and differentiability of sample paths of parameter-dependent integral are established in Section 3. The continuity of realizations of the integral as a function of the upper limit is studied in Section 4.

Preliminaries
2.1.Stochastic measures.In this subsection, we give basic information concerning stochastic measures in a general setting.In statements of Sections 3 and 4, this set function is defined on Borel subsets of [0, 1] d .
Let L 0 = L 0 (Ω, F , P) be the set of all real-valued random variables defined on the complete probability space (Ω, F , P). Convergence in L 0 means the convergence in probability.Let X be an arbitrary set and B a σ-algebra of subsets of X.
In other words, We do not assume the moment existence or martingale properties for SM.We can say that µ is an L 0 -valued measure.
We note the following examples of SMs in multidimensional spaces.All measures, if nothing else is mentioned, are considered on Borel σ-algebra of sets in [0, 1] d .
Example 1.Let µ be a random orthogonal measure -it is defined, for example, in [ Therefore, µ is an SM.
If q = 1 then the Hermite sheet coincides with the fractional Brownian sheet.
Theorem 10.1.1 of [15] states the sufficient conditions under which the product of one-dimensional SMs generates an SM.
SMs may be used for study of stochastic dynamical systems (see [3]).
For deterministic measurable functions f : X → R, an integral of the form X f dµ is studied in [15,Chapter 7], [20,Chapter 1].In particular, every bounded measurable f is integrable w. r. t. any µ.An analogue of the Lebesgue dominated convergence theorem holds for this integral (see [15,Proposition 7.1.1]or [20,Theorem 1.5]).
Below we will use the following statement for SMs defined on arbitrary σ-algebra B.
For functions f ∈ L p ([0, 1] d ), we consider the value where ω p denotes the L p -modulus of continuity, and |h| denotes the Euclidean norm in R d .Then and Let µ be an SM defined on the Borel σ-algebra of [0, 1] d .For In the sequel, we will refer to the following assumption.
Assumption A1.There exists a real-valued finite measure m on (X, B) with the following property: if a measurable function g : X → R satisfies X g 2 dm < +∞ then g is integrable w. r. t. µ on X.
For orthogonal measure, we can take its structural measure as m.For an α-stable random measure A1 holds for the control measure m (see (3.4.1) [22]).For an SM from Example 3 we can take measure m such that where u and v are d-dimensional vectors with the components u i , v i .
We have the following statement concerning the Besov regularity of SMs defined on Borel subsets of [0, 1] d .Theorem 1. (Theorem 5.1 [19]) Let the random function has continuous paths and Assumption A1 holds.

Haar functions.
In order to construct a version of the stochastic integral, we use the approximation of the integrand with Fourier-Haar series.We follow the definition of one-dimensional Haar functions ( Now let χ 1 = 1, and . In discontinuity points and at the endpoints of [0, 1] we define For g ∈ L 1 ([0, 1]), we define the Fourier -Haar coefficients and sums in a standard way: We will approximate g taking the Fourier -Haar sums for values and For functions f : [0, 1] d → R, we will use approximation by multivariate Fourier -Haar sums.
The multivariate Haar functions are defined with equalities where χ n s are one-dimensional Haar functions, and sums In the sequel, we denote the set of elements (n 1 , . . ., n d ) such that 1 We need the following statement on the uniform convergence of multivariate Fourier -Haar sums for continuous functions.For d = 1 this fact may be found, for example, in [13,Theorem III.2].A similar statement for multivariate periodic functions was proved in [2].
Proof.For bounded functions u : [0, 1] d → R we use the following notation of the uniform modulus of continuity From ( 5) and ( 6) (see also (III.15)[13]) it follows that for d = 1 holds For t = (t 1 , . . .t d ) we get We consider the operators where we take one-dimensional Fourier -Haar sums of the function u(x 1 , . . ., x d ) in coordinate x j .Then and it is easy to see that for integral m ≥ 1 We have where ≤ ω(P k,l (. ≤ ω(f , (2l + 1)2 −k ). Therefore, For continuous f the right hand side of (10) tends to 0 as k → ∞, that implies the statement of our lemma.

Parameter dependent integral
In this section, we study the properties of random function Note that parameter stochastic integral w.r.t.Brownian sheet was considered in [23], [24].
If f is continuous in x then Lemma 2 and analogue of the dominated convergence theorem ([20, Theorem 1.5]) imply that for each fixed z is the version of η(z).
In (12) we have integrals of simple functions, and each integral is equal to respective linear combination of values of µ.We fix the same version of µ for all these values.We will use notation e i = (0, . . ., 0, 1, 0, . . ., 0), where 1 stays on i-th place, e i ∈ R d .
for some constant C f > 0, which is independent of z, and all 0 ≤ r ≤ l, s 1 , . . ., s r ⊂ {1, . . ., d}.Moreover, let l-th derivatives be Hölder continuous with an exponent α > 0: If l + α > d/2, then the version (12) satisfies the inequality where C Proof.We have that For each n s ≥ 2, take j s such that 2 We rewrite the expression for c (d) n 1 ,...,n d (f ) in the following way: In equality (*) we opened the brackets in the product 1≤s≤d,n s ≥2 and used the change of the variables The last sum in ( 17) can be represented as the sum of at most 2 d−1 summands of the form Now we repeat the same thoughts l − 1 times and use the properties of a function f , which leads to the following inequality Further, we have the estimate For a sufficiently small β > 0 we have the following estimates for P 1 : Here we denoted the set of indexes i, for which n s = 1, as A. It is left to prove that for each fixed β > 0 where by C (d) we denote positive constant that depends only on d.The inequality (20) holds true, as is proved below.
Here in inequality (*) we used that for each set j 1 , . . ., j d , for each x, x s i2 −j s , exists exactly one set (n 1 , . . ., n d ) such that χ (d) n 1 ,...,n d (x) 0 and 2 j s + 1 ≤ n s ≤ 2 j s +1 .If we have coordinates x s = i2 −j s , we take into account that 1 {i2 −j s } (x s ) has the coefficient 1/2, as follows from (4).Now the statement of the lemma is a consequence of ( 19) and ( 20).
Remark 2. Assume that constants in inequalities ( 13) and ( 14) depend not only on f , but also on z.Then the series (12) converges a. s. for each fixed z ∈ Z and the following analogue of (15): This statement is proved similarly to Lemma 3. We just refer to C f ,z everywhere instead of C f .
The following statement gives the conditions of the continuity with respect to parameter z ∈ Z.
Theorem 2. Let Z be a metric space, and the function f for some constant C f > 0 (independent of z) and all 0 ≤ r ≤ l, s 1 , . . ., s r ⊂ {1, . . ., d}.Let l-th derivatives be Hölder continuous with the exponent α and If, in addition, l + α > d/2, then the random function η defined by (11) has a version (12) with continuous paths on Z a. s.
Proof.From ( 17) it follows that each c (d) n 1 ,...,n d is continuous function of variable z, and (16) 2 k (f , x) dµ(x) converges to η(z) uniformly a.s. as k → ∞, that implies the statement of our theorem for all 0 ≤ r ≤ l, s 1 , . . ., s r ⊂ {1, . . ., d}.Moreover, let l + 1-th derivatives be Hölder continuous with the exponent α > 0: If, in addition, l +α > d/2, then paths of a random function η, which is defined by (12), have bounded derivatives on Z: Proof.Lemma 3 implies that where the series in the right hand side converges uniformly.According to equalities and Remark 2, we can differentiate the series in (12) and obtain that which finishes the proof.

Integral as a function of upper limit
For continuous function f (x) : [0, 1] d → R and y = (y 1 , . . ., y d ) ∈ [0, 1] d , an we consider the random function is the version of ξ(y).
We refer to the following assumptions on SM µ.
In the following condition on the uniform modulus of continuity we take the continuous version of µ.
We introduce the following stochastic process: where W is a d-dimensional Wiener process.Theorem 3.1 in [11] implies the existence of rectangle Here Therefore, ω [t,T ] (B q , τ) ≤ C| ln τ| 1/2−γ .Now we are ready to formulate the main result of the section.
has a continuous paths.We will prove that the convergence in (24) is uniform in y ∈ [0, 1] d a.s., and this will imply the continuity of ξ.
Our aim is to show that for some random constant C(d) f ,µ (ω) < ∞ a. s. that may depends on d, f , µ.
Recall that paths of µ(x 1 , . . ., x d ) are continuous, therefore for any cut of the set Thus, we obtain Here A k1 (y) is the sum of terms with (n 1 , . . (here for n s = 1 we take only ε s = + and (∆ n s ) ε s = (0, 1)).Taking into account the definition of (∆ n s ) + and (∆ n s ) − in (3), we get that in A k1 (y) Here j ′ s are taken such that 2 j ′ s + 1 ≤ m s ≤ 2 j ′ s +1 , and j ′ s = j s + 1 for respective m s .From estimates in (19), (20) it follows that for some random constant C(d) µ (ω) that depends only of d and µ.(It is easy to see that all estimates in (19), (20) remain valid if we change χ (d) n 1 ,...,n d (x) to 2 (j     Here in ( * ) we used the fact that sum (−1) |B|+1 1≤j ′ i ≤k, i∈A 2 − i∈B j ′ i depends only on |A| and |B|.Therefore, where 0 < β < 1. Applying Assumption A3 we get the statement of the theorem.
Proof.The statement follows from Theorems 1 and 4.