New weak and strong existence and weak and strong uniqueness results for multi-dimensional stochastic McKean-Vlasov equation are established under relaxed regularity conditions. Weak existence is a variation of Krylov's weak existence for Itô's SDEs under the nondegeneracy of diffusion and no more than a linear growth in the state variable; this part is designed to fill in the existing gap, as earlier such results for McKean-Vlasov equations were not written. Weak and strong uniqueness is established under the restricted assumption of diffusion depending only on time and the state variable, yet without any regularity of the drift with respect to the state variable and also under a linear growth condition on this drift; this part is based on the analysis of the total variation metric.

We show that a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand g can have any prescribed distribution, moreover, we give both necessary and sufficient conditions when random variables can be represented in this form. We also prove that any random variable is a value of such integral in some improper sense. We discuss some applications of these results, in particular, to fractional Black-Scholes model of financial market.

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Fractional Brownian motion can be represented as an integral of a deterministic kernel w.r.t. an ordinary Brownian motion either on infinite or compact interval. In previous literature fractional Lévy processes are defined by integrating the infinite interval kernel w.r.t. a general Lévy process. In this article we define fractional Lévy processes using the com pact interval representation.We prove that the fractional Lévy processes presented via different integral transformations have the same finite dimensional distributions if and only if they are fractional Brownian motions. Also, we present relations between different fractional Lévy processes and analyze the properties of such processes. A financial example is introduced as well.

The paper deals with the expected maxima of continuous Gaussian processes X = (X t ) t≥0 that are Hölder continuous in L 2 -norm and/or satisfy the opposite inequality for the L 2 -norms of their increments. Examples of such processes include the fractional Brownian motion and some of its "relatives" (of which several examples are given in the paper). We establish upper and lower bounds for E max 0≤t≤1 X t and investigate the rate of convergence to that quantity of its discrete approximation E max 0≤i≤n X i/n . Some further properties of these two maxima are established in the special case of the fractional Brownian motion.