## 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 $$B_t^{H_t}$$.

### MSC:

 60G22 Fractional processes, including fractional Brownian motion 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G15 Gaussian processes
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### References:

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