2025/11/12 by Yanbin Zhu, Xiaomeng Jiang, Zhu, Yanbin +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60F10 #60G22 #60H10 #FOS: Mathematics #Fractional Differential Equations Solutions #Probability (math.PR) #Stochastic processes and financial applications #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.2511.09300
openalex publication_date 2025/11/12 · openalex created_date 2025/11/14 · openalex updated_date 2026/07/28
In this paper, we derive the Onsager-Machlup functional for stochastic differential equations driven by time-varying fractional noise of the form X(t) = x0 + integral from 0 to t bs(X(s)) ds + integral from 0 to t sigmas dBH(s), where BH denotes fractional Brownian motion with Hurst parameter H. Our main results are established for H in (1/4, 1) by extending small ball probability estimates and the Girsanov theorem for fractional Brownian motion to the setting with time-dependent coefficients. Regarding the choice of norms, for 1/4 < H < 1/2 the analysis is valid under the supremum norm and Holder norms of order 0 < beta < H - 1/4. For 1/2 < H < 1 the analysis applies to Holder norms of order beta satisfying H - 1/2 < beta < H - 1/4. In the case H = 1/2, the admissible norms depend on the spatial regularity of the drift coefficient b: specifically, if b is n-times continuously differentiable, then Holder norms of order 0 < beta < 1/2 - 1/(2n) are permissible. To validate our theoretical findings, we perform numerical simulations for a classical double-well potential system, illustrating how time-varying fractional noise influences transition dynamics between metastable states.