In this paper, we study general mean-field reflected backward stochastic differential equations with locally monotone coefficients. With the help of choosing the suitable approximation sequence, we obtain the existence and uniqueness of solution to general mean-field reflected backward stochastic differential equations.
We generalize a strong form of the three-dimensional Gaussian product inequality studied by Herry et al. (2024), who resolved the case of any triple of even positive integers. We extend the result to any triple consisting of a pair of positive real numbers and an even positive integer. Our result includes all existing results on the three-dimensional Gaussian product inequality conjecture.
We obtain a Stein characterisation of the distribution of the product of two correlated normal random variables with non-zero means, and more generally the distribution of the sum of independent copies of such random variables. Our Stein characterisation is shown to naturally generalise a number of other Stein characterisations in the literature. From our Stein characterisation we derive recursive formulas for the moments of the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, which allows for efficient computation of higher order moments.
Distribution continuity is established for extremes of Gaussian processes with bounded sample paths and positive variance or continuous sample paths over compact domain, and for multiparameter Brownian sheets. These results provide probabilistic support for global inference on unknown functions.
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