Abstract
Let \(\left\{ {{\text{W}}_{{\text{st}}} \;:{\text{s,t}} \in \left[ {0,1} \right]} \right\}\) be the Brownian sheet. We define the regularized process W{skst/ε} as the convolution of Wst and \(\varphi _\varepsilon \left( {{\text{s,t}}} \right) = \frac{1} {{\varepsilon ^2 }}\varphi \left( {\frac{s} {\varepsilon }} \right)\varphi \left( {\frac{t} {\varepsilon }} \right)\) where ϕ is a function satisfying some conditions. For ω fixed we prove that
almost surely, where λ is the Lebesgue measure in R2, Φ is the standard Gaussian distribution and ‖ · ‖2 is the usual norm in L2([− 1, 1], dx). These results are generalized to two parameter martingales M given by stochastic integrals of the Cairoli & Walsh type. Finally, as a consequence of our method we also obtain similar results for the normalized double increment of the processes W and M. These results constitute a generalisation of those obtained by Wschebor for Brownian stochastic integrals.
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León, J.R., Rondón, O. (2005). On the Increments of the Brownian Sheet. In: Baeza-Yates, R., Glaz, J., Gzyl, H., Hüsler, J., Palacios, J.L. (eds) Recent Advances in Applied Probability. Springer, Boston, MA. https://doi.org/10.1007/0-387-23394-6_12
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DOI: https://doi.org/10.1007/0-387-23394-6_12
Publisher Name: Springer, Boston, MA
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