On ill-posedness of nonlinear stochastic wave equations driven by rough noise

2021/12/16 by Aurélien Deya, Deya, Aurélien
Economics, Econometrics and Finance · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications

paper · doi:10.48550/arxiv.2112.08758

openalex publication_date 2021/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We highlight a fundamental ill-posedness issue for nonlinear stochastic wave equations driven by a fractional noise. Namely, if the noise becomes too rough (i.e., the sum of its Hurst indexes becomes too small), then there is essentially no hope to provide an interpretation of the model, whether directly or through a Wick-type renormalization procedure. This phenomenon can be compared with the situation of a general SDE driven by a two-dimensional fractional noise of index H≤ (1)/(4). Our results clarify and extend previous similar properties exhibited in [3] or in [14].

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