Lipschitz continuity in the Hurst parameter of functionals of stochastic differential equations driven by a fractional Brownian motion
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2016
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| _version_ | 1866912005505417216 |
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| author | Richard, Alexandre Talay, Denis |
| author_facet | Richard, Alexandre Talay, Denis |
| contents | Sensitivity analysis w.r.t. the long-range/memory noise parameter for probability distributions of functionals of solutions to stochastic differential equations is an important stochastic modeling issue in many applications.
In this paper we consider solutions $\{X^H_t\}_{t\in \mathbb{R}_+}$ to stochastic differential equations driven by fractional Brownian motions. We develop two innovative sensitivity analyses when the Hurst parameter $H$ of the noise tends to the critical Brownian parameter $H=\tfrac{1}{2}$ from above or from below. First, we examine expected smooth functions of $X^H$ at a fixed time horizon $T$. Second, we examine Laplace transforms of functionals which are irregular with regard to Malliavin calculus, namely, first passage times of $X^H$ at a given threshold.
In both cases we exhibit the Lipschitz continuity w.r.t. $H$ around the value $\tfrac{1}{2}$. Therefore, our results show that the Markov Brownian model is a good proxy model as long as the Hurst parameter remains close to $\tfrac{1}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1605_03475 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Lipschitz continuity in the Hurst parameter of functionals of stochastic differential equations driven by a fractional Brownian motion Richard, Alexandre Talay, Denis Probability Sensitivity analysis w.r.t. the long-range/memory noise parameter for probability distributions of functionals of solutions to stochastic differential equations is an important stochastic modeling issue in many applications. In this paper we consider solutions $\{X^H_t\}_{t\in \mathbb{R}_+}$ to stochastic differential equations driven by fractional Brownian motions. We develop two innovative sensitivity analyses when the Hurst parameter $H$ of the noise tends to the critical Brownian parameter $H=\tfrac{1}{2}$ from above or from below. First, we examine expected smooth functions of $X^H$ at a fixed time horizon $T$. Second, we examine Laplace transforms of functionals which are irregular with regard to Malliavin calculus, namely, first passage times of $X^H$ at a given threshold. In both cases we exhibit the Lipschitz continuity w.r.t. $H$ around the value $\tfrac{1}{2}$. Therefore, our results show that the Markov Brownian model is a good proxy model as long as the Hurst parameter remains close to $\tfrac{1}{2}$. |
| title | Lipschitz continuity in the Hurst parameter of functionals of stochastic differential equations driven by a fractional Brownian motion |
| topic | Probability |
| url | https://arxiv.org/abs/1605.03475 |