We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itô's formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Lévy noise is established.

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This review examines the methods used to optimize the process parameters of laser cladding, including traditional optimization algorithms such as single-factor, regression analysis, response surface, and Taguchi, as well as intelligent system optimization algorithms such as neural network models, genetic algorithms, support vector machines, the new non-dominance ranking genetic algorithm II, and particle swarm algorithms. The advantages and disadvantages of various laser cladding process optimization methods are analyzed and summarized. Finally, the development trend of optimization methods in the field of laser cladding is summarized and predicted. It is believed that the result would serve as a foundation for future studies on the preparation of high-quality laser cladding coatings.