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**Approximation of multifractional Brownian motion by absolutely continuous processes.**
*(English.
Ukrainian original)*
Zbl 1237.60029

Theory Probab. Math. Stat. 82, 115-127 (2011); translation from Teor. Jmovirn. Mat. Stat. No. 82, 115-127.

Multifractional Brownian motion is a generalization of fractional Brownian motion \(B_t^H\) with time-dependent Hurst parameter \(H_t\); in this paper \(H_t \in (\frac12, 1)\) and is a Hölder continuous function of \(t\) with exponent \(\gamma\). The first main result of the paper is an approximation of \(B_t^{H_t}\) by the absolutely continuous average \(B_t^{H_t,\epsilon} := \frac{1}{\phi_t(\epsilon)} \int_t^{t+\phi_t(\epsilon)} B_s^{H_s} ds\), where the length of the interval \(\phi_t(\epsilon)\) satisfies certain regularity conditions. Then \(\| B^{H,\epsilon} - B^H\|_{1,\beta}\) tends to \(0\) in probability as \(\epsilon \to 0^+\), where the norm is a suitable Besov-type norm and \(\beta \in(0, H_{\min})\) is arbitrary; \(H_{\min} = \min\{\gamma, \min_t H_t\}\).

The second main result of the paper is an application of the above result to stochastic differential equations. If a SDE is driven by multifractional Brownian motion, then – under conditions that guarantee the existence and uniqueness of the solution – the approximate solution \(X_t^{\epsilon}\) obtained with the driving process \(B_t^{H_t,\epsilon}\), tends in probability, as \(\epsilon \to 0^+\), to the solution \(X_t\) obtained with the driving process \(B_t^{H_t}\).

The second main result of the paper is an application of the above result to stochastic differential equations. If a SDE is driven by multifractional Brownian motion, then – under conditions that guarantee the existence and uniqueness of the solution – the approximate solution \(X_t^{\epsilon}\) obtained with the driving process \(B_t^{H_t,\epsilon}\), tends in probability, as \(\epsilon \to 0^+\), to the solution \(X_t\) obtained with the driving process \(B_t^{H_t}\).

Reviewer: Tamas Szabados (Budapest)

### MSC:

60G22 | Fractional processes, including fractional Brownian motion |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60G15 | Gaussian processes |

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\textit{K. V. Ral'chenko}, Theory Probab. Math. Stat. 82, 115--127 (2011; Zbl 1237.60029); translation from Teor. Jmovirn. Mat. Stat. No. 82, 115--12

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### References:

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