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is considered, where W is a standard Wiener process, , , and . It is proved that the process X is well-defined and continuous. The asymptotic properties of the variances and bounds for the variances of the increments of the process X are studied. It is also proved that the process X satisfies the single-point Hölder condition up to order at point 0, the “interval” Hölder condition up to order on the interval (where ), and the Hölder condition up to order on the entire interval .