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**Maximum upper bounds for the moments of stochastic integrals and solutions of stochastic differential equations that have fractional Brownian motion with Hurst index \(H<1/2\). II.**
*(Ukrainian, English)*
Zbl 1199.60214

Teor. Jmovirn. Mat. Stat. 76, 53-69 (2007); translation in Theory Probab. Math. Stat. 76, 59-76 (2008).

The authors study properties of solutions of stochastic differential equations which have additive terms in the form of the Wiener integral with respect to fractional Brownian motion with Hurst parameter \(H<1/2\). Properties of this type of integrals are studied in the first part of the paper (see Yu. V. Kozachenko and Yu. S. Mishura [Teor. Jmovirn. Mat. Stat. 75, 45–56 (2006); translation in Theory Probab. Math. Stat. 75, 51–64 (2007; Zbl 1164.60378)] ).
Similar stochastic differential equations with an additive fractional Brownian motion are studied in a paper by D. Nualart and Y. Ouknine [Stochastic Processes Appl. 102, No. 1, 103–116 (2002; Zbl 1075.60536)]. In the indicated paper the existence and uniqueness of a strong solution to the differential equation
\[
X_t=x+\int_0^t b(s,X_s)\,ds+B^H_t, t\geq 0,
\]
is established, where \(B^H_t\) is a fractional Brownian motion with Hurst parameter \(H\in (0,1)\) and \(b(s,x)\) is a bounded Borel function with at most linear growth in \(x\).

In the article under review the authors deal with the stochastic differential equation \[ X_t= x+\int_0^tf(s)dB^H_s+\int_0^t b(s,X_s)\,ds, t\in[0,T],x\in\mathbb R. \] The Wiener integral \(I_t:=\int_0^tf(s)dB^H_s\) with respect to fractional Brownian motion \(B^H_t\) is defined as \[ \int_0^tf(s)dB^H_s=\int_{\mathbb R}(M_-^Hf1_{(0,1)})(s)dW_s, \] where \(W_t\) is a Wiener process. It is shown that solutions of these differential equations belong to the Orlicz space \(L_U(\Omega)\) of random variables generated by the \(N\)-function \(U(x)=\exp\{x^2\}-1\). This allows applying the theory of Orlicz spaces (see V. V. Buldygin and Yu. V. Kozachenko [Metric characterization of random variables and random processes. Transl. from the Russian by V. Zaiats, American Mathematical Society (2000; Zbl 0998.60503)]) when analyzing the behaviour of solutions of stochastic differential equations. Estimates for the norms of the solutions in the space \(L_U(\Omega)\) are obtained. It is proved that the supremum of a solution belongs to the same Orlicz space as the solution itself. An estimate for the distribution of the supremum of a solution is proposed. These estimates have the same rate of growth as in the case of Gaussian processes. The modulus of continuity of solutions is studied. It is shown that a solution of the equations belongs to certain Lipschitz space with probability one. Estimates for the distribution of the norm of the solution in this space are found.

In the article under review the authors deal with the stochastic differential equation \[ X_t= x+\int_0^tf(s)dB^H_s+\int_0^t b(s,X_s)\,ds, t\in[0,T],x\in\mathbb R. \] The Wiener integral \(I_t:=\int_0^tf(s)dB^H_s\) with respect to fractional Brownian motion \(B^H_t\) is defined as \[ \int_0^tf(s)dB^H_s=\int_{\mathbb R}(M_-^Hf1_{(0,1)})(s)dW_s, \] where \(W_t\) is a Wiener process. It is shown that solutions of these differential equations belong to the Orlicz space \(L_U(\Omega)\) of random variables generated by the \(N\)-function \(U(x)=\exp\{x^2\}-1\). This allows applying the theory of Orlicz spaces (see V. V. Buldygin and Yu. V. Kozachenko [Metric characterization of random variables and random processes. Transl. from the Russian by V. Zaiats, American Mathematical Society (2000; Zbl 0998.60503)]) when analyzing the behaviour of solutions of stochastic differential equations. Estimates for the norms of the solutions in the space \(L_U(\Omega)\) are obtained. It is proved that the supremum of a solution belongs to the same Orlicz space as the solution itself. An estimate for the distribution of the supremum of a solution is proposed. These estimates have the same rate of growth as in the case of Gaussian processes. The modulus of continuity of solutions is studied. It is shown that a solution of the equations belongs to certain Lipschitz space with probability one. Estimates for the distribution of the norm of the solution in this space are found.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

### MSC:

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

60H05 | Stochastic integrals |

60G15 | Gaussian processes |

60G22 | Fractional processes, including fractional Brownian motion |