Minimization of the entropy for a mixture of standard and fractional Brownian motions

Approximate solution of the integral equations involving kernel with additional singularity

“…In order to write down the Radon-Nikodym derivative d Q dP , let us recall here the (weighted) Riemann-Liouville fractional integrals, see papers [9] and [12,14].…”

“…Namely, the paper [12] was devoted to the problem of optimization of small deviation for mixed fractional Brownian motion with trend for the case H ∈ (0, 1/2), whereas in the paper [13], the same problem was considered for the case H ∈ (1/2, 1). The paper [14] studied the problem of minimization of the entropy functional appearing under the distribution of the drift, and this problem was studied for H ∈ (0, 1/2). Now our goal is twofold: first, to present the existing results from [12,13,14] and second, to demonstrate how to reduce the problem of minimization of entropy functional in the case H ∈ (1/2, 1).…”

“…We present the problems which lead to the integral equations involving the kernels with additional singularity. These problems were discussed in detail in the papers [12,13,14]. More precisely, the paper [12] is devoted to the problem of optimization of small deviation for mixed fractional Brownian motion with trend for the case of the Hurst index H ∈ (0; 1/2); whereas in the paper [13], the same problem was considered for the case H ∈ (1/2; 1).…”

“…More precisely, the paper [12] is devoted to the problem of optimization of small deviation for mixed fractional Brownian motion with trend for the case of the Hurst index H ∈ (0; 1/2); whereas in the paper [13], the same problem was considered for the case H ∈ (1/2; 1). The paper [14] is devoted to the minimization of entropy in the system described by the mixed fractional Brownian motion with trend. Both cases, H ∈ (0; 1/2) and H ∈ (1/2; 1) were considered and as the result, the problem was reduced to the couple of the same integral equations as in the case where the minimization of small deviations was studied.…”

“…In Subsection 2.1 we describe the problems themselves and then, in Subsection 2.2, we explain how to reduce these optimization problems to the integral equation and describe the structure of the involved integral kernels. The representations of the kernels are new, in comparison with the papers [12,13,14], and they are much more convenient for the numerical solution. In Section 3, we provide approximation of the involved kernels K(t, s) by kernels with continuous numerators.…”