Rate of convergence of uniform transport processes to a Brownian sheet

1 Introduction

Let W = { W ( s , t ) : ( s , t ) ∈ [ 0 , 1 ] 2 } be a Brownian sheet, i.e., a zero mean real continuous Gaussian process with covariance function E [ W ( s 1 , t 1 ) W ( s 2 , t 2 ) ] = ( s 1 ∧ s 2 ) ( t 1 ∧ t 2 ) for any ( s 1 , t 1 ) , ( s 2 , t 2 ) ∈ [0, 1] 2 . The aim of this study is to obtain the rate of convergence of strong approximations of the Brownian sheet. This result is not only interesting from a purely mathematical point of view but also of great interest to provide sound approximation strategies to solutions of stochastic partial differential equations which arise in many fields such as physics, biology or finance.

The study of the approximations of the Brownian sheet by uniform transport processes or processes constructed from a Poisson process begins with the proof of the weak convergence. Bardina and Jolis [1] prove that the process

where { N ( x , y ) , x ≥ 0 , y ≥ 0 } is a Poisson process in the plane, converges in law to a Brownian sheet when n goes to infinity, and Bardina et al. [2] extended this result to the d-parameter Wiener processes.

On the other hand, Bardina et al. [3] constructed a family of processes, starting from a set of independent standard Poisson processes, that has realizations that convergence almost surely to a Brownian sheet. Our purpose is to give the rate of convergence of such approximations. As far as we know, our work is the first rate of convergence for the multiparameter case for this family of approximations.

There exist several literature studies about strong convergence of uniform transport processes and the study of the corresponding rate of convergence. In the seminal paper of Griego et al. [4], the authors presented realizations of a sequence of uniform transform processes that converges almost surely to the standard Brownian motion, uniformly on the unit time interval. In [5], Gorostiza and Griego extended the result of [4] to the case of diffusions. Again Gorostiza and Griego [6] and Csörgo and Horváth [7] obtained the rate of convergence of the approximation sequence. More recently, Garzón et al. [8] defined a sequence of processes that converges strongly to fractional Brownian motion uniformly on bounded intervals, for any Hurst parameter H ∈ ( 0 , 1 ) and computed the rate of convergence. In [9,10], the same authors deal with subfractional Brownian motion and fractional stochastic differential equations. Bardina et al. [11] proved the strong convergence to a complex Brownian motion and obtained the corresponding rate of convergence.

The structure of the article is as follows. In Section 2, we recall the approximations that converge almost surely to the Brownian sheet and we present our theorem. In Section 3, we give the proof of our result. It is based on a combination of the properties of the Brownian sheet and the use of the rate of convergence for the Brownian motion given by Gorostiza and Griego in [6].

2 Approximations and main result

Let us recall the approximation processes introduced in [3]. For any n and λ > 0 , consider the partition of the unit square [ 0 , 1 ] 2 in disjoint rectangles

where ⌊ x ⌋ denotes the greatest integer less than or equal to x.

If W = { W ( s , t ) : ( s , t ) ∈ [ 0 , 1 ] 2 } is a Brownian sheet on the unit square, let W k denote its restriction to each of the above-defined rectangles k − 1 n λ , k n λ × [0 , 1] . That is,

for k ∈ { 1 , 2 , … , ⌊ n λ ⌋ } . Thus, for any l ∈ { 1 , 2 , … , ⌊ n λ ⌋ } and t ∈ [ 0 , 1 ]

Moreover, putting W ˜ k ( t ) ≔ n λ 2 W k ( t ) , we obtain a family

of independent standard Brownian motions defined in [ 0 , 1 ] .

From the paper of Griego et al. [4], it is known that there exist realizations of uniform transport processes that converge strongly and uniformly on bounded time intervals to Brownian motions. So, we can get an approximation sequence { W ˜ ( n ) k ; n ≥ 1 } for each one of the standard Brownian motions W ˜ k ; k ∈ { 1 , 2 , … , ⌊ n λ ⌋ } . We can state Theorem 1 in [6] for such an approximation sequence for any k.

Then, the Brownian sheet is approximated by a process W n such that for any l ∈ { 1 , 2 , … , [ n λ ] } and t ∈ [ 0 , 1 ]

where

and { N k , k ≥ 1 } is a family of independent standard Poisson processes and { A k , k ≥ 1 } is a sequence of independent random variables with law Bernoulli 1 2 , independent of the Poisson processes. Using linear interpolation, define W n ( s , t ) on the whole unit square as follows:

for any ( s , t ) ∈ 0 , ⌊ n λ ⌋ n λ × [ 0 , 1 ] and W n ( s , t ) = W n ⌊ n λ ⌋ n λ , t for any ( s , t ) ∈ ⌊ n λ ⌋ n λ , 1 × [ 0 , 1 ] .

In the following theorem, we give our main result, the rate of convergence of these processes.

The first part of the theorem has been proved in Theorem 2.1 in [3]. In Section 3, we give the proof of the rate of convergence.

3 Proof

The proof uses some well-known properties about the Brownian sheet { W ( s , t ) : ( s , t ) ∈ [ 0 , 1 ] 2 } . Let us recall, for instance, that for fixed t > 0 , the process { t − 1 2 W ( s , t ) , s ≥ 0 } is a standard Brownian motion or that for fixed a > 0 , the process { a − 1 2 W ( a s , t ) , s ≥ 0 , t ≥ 0 } is also a Brownian sheet. We refer the reader to Khoshnevisan [12] for a detailed description of the Brownian sheet.

Furthermore, we will begin recalling technical results about submartingales (Theorem 1, p. 74 from Imkeller [13]) that we will apply to the submartingale e W .

We will apply this theorem to the process

where { B ( s , t ) : ( s , t ) ∈ [ 0 , 1 ] 2 } is a Brownian sheet. Considering the family of σ -algebras ℱ s , t = σ < B ( u , v ) ; ( u , v ) ∈ [ 0 , s ] × [ 0 , t ] > , M ( s , t ) is clearly nonnegative, integrable, ℱ s , t -measurable and it is a submartingale. Indeed, if 0 < s ′ < s and 0 < t ′ < t , then s ′ t ′ < s t and

On the other hand, ∥ M ( 1 , 1 ) ∥ ψ = ∥ exp ( B ( 1 , 1 ) ) ∥ ψ is a finite constant. Indeed, using that B ( 1 , 1 ) is a N ( 0 , 1 ) ,

and this quantity is bigger than 1 for μ sufficiently small and smaller than 1 for μ big enough. We have used that

Let us give now the proof of our theorem.

where K n = c n − β , with β ∈ 0 , λ 2 .

Fixed s ∈ [ 0 , 1 ] , n ≥ 1 , there exists k ∈ { 0 , … , ⌊ n λ ⌋ } such that

that is, k = ⌊ s n λ ⌋ . So, we can write

Let us study first P 2 . Using the definitions of W ( n ) k and W k , for any k, we get

Finally, from the rate of convergence from W ˜ ( n ) k to W ˜ k , for any k, in Theorem 2.1, we get

for any q > 0 .

Let us consider now P 3 . We can write, using the well-known properties of the Brownian sheet

We have considered that W ( l n λ + r , t ) = W ( 1 , t ) if l n λ + r > 1 . Let us define B ( s , t ) ≔ n λ 2 W s n λ , t , for any ( s , t ) ∈ [ 0 , 1 ] 2 . Clearly, B is a Brownian sheet and P 3 can be bounded as

So, using Theorem 3.1

since ∥ exp ( B ( 1 , 1 ) ) ∥ ψ is a finite positive constant that does not depend on n.

Finally, let us study P 1 . Using that we have defined the intermediate points using linear interpolation we have that for any s ∈ l n λ , l + 1 n λ we get that

and we can write that

where if l + 1 n λ > 1 we assume that W l + 1 n λ , t = W n l + 1 n λ , t = W (1, t ) .

Note that P 1,2 can be bounded as P 3 and on the other hand, P 1,1 and P 1,3 can be studied as P 2 . Putting together the bounds for P 1 , P 2 and P 3 we finish the proof.□