Noise Sensitivity of Functionals of Fractional Brownian Motion Driven Stochastic Differential Equations: Results and Perspectives

Abstract

We present an innovating sensitivity analysis for stochastic differential equations: We study the sensitivity, when the Hurst parameter H of the driving fractional Brownian motion tends to the pure Brownian value, of probability distributions of smooth functionals of the trajectories of the solutions $\{X^H_t\}_{t\in \mathbb {R}_+}$ and of the Laplace transform of the first passage time of $X^H$ at a given threshold. Our technique requires to extend already known Gaussian estimates on the density of $X^H_t$ to estimates with constants which are uniform w.r.t. t in the whole half-line $\mathbb {R}_+-\{0\}$ and H when H tends to $\tfrac{1}{2}$.

Appendix: Tables

Table 1 Values of $\Delta _H = \mathbb {E}\big [e^{-\lambda \tau _{\frac{1}{2}}}\big ] - \mathbb {E}\left[ e^{-\lambda \tau _H}\right]$ when $H\rightarrow \tfrac{1}{2}$. Set of parameters: $T=20,\ N=2^{16}\ (\delta \approx 3.10^{-4}),\ M=10^5$

Table 2 Test case: Error estimation of our procedure in the Brownian case ($H=\tfrac{1}{2}$). Set of parameters: $T=20,\ N=2^{16}\ (\delta _0\approx 3.10^{-4}),\ M=10^5$ for the simple estimator $T=20,\ N=2^{15}\ (\delta _1\approx 6.10^{-4}),\ M=10^5$ for the bridge estimator

Table 3 Comparison of estimators in the fractional case ($H=0,54$). Set of parameters: $T=20,\ N=2^{16}\ (\delta _0\approx 3.10^{-4}),\ M=10^5$ for the simple estimator $T=20,\ N=2^{15}\ (\delta _1\approx 6.10^{-4}),\ M=10^5$ for the simple estimator $T=20,\ N=2^{15}\ (\delta _1\approx 6.10^{-4}),\ M=10^5$ for the bridge estimator