We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation $$d{X}_t^{\left(\alpha \right)}=-\frac{\alpha }{T-t}{X}_t^{\left(\alpha \right)}dt+d{B}_t,t\in [0,T)$$ , with the initial condition X 0(α) = 0, where α > 0, T ∈ (0, ∞), and (B t)t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X (α). As applications, we calculate the Laplace transform and the distribution function...

We consider a stochastic process $X_t^x$ which solves an equation $$\mathrm{d}{X}_t^x=A{X}_t^x\mathrm{d}t+\Phi \mathrm{d}{B}_t^H,X_0^x=x$$ where $A$ and $\Phi$ are real matrices and $B^H$ is a fractional Brownian motion with Hurst parameter $H\in (1/2,1)$. The Kolmogorov backward equation for the function $u(t,x)=𝔼f\left(X_t^x\right)$ is derived and exponential convergence of probability distributions of solutions to the limit measure is established.

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [E. Bacry et al. Comm. Math. Phys. 236 (2003) 449–475]. If M is a non degenerate multifractal measure with associated metric ρ(x,y) = M([x,y]) and structure function ζ, we show that we have the following relation between the (Euclidian) Hausdorff dimension dimH of a measurable set K and the Hausdorff dimension dimHρ with respect to ρ of the same set: ζ(dimHρ(K)) = dimH(K). Our results can...

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [E. Bacry et al. Comm. Math. Phys.236 (2003) 449–475]. If M is a non degenerate multifractal measure with associated metric ρ(x,y) = M([x,y]) and structure function ζ, we show that we have the following relation between the (Euclidian) Hausdorff dimension dimH of a measurable set K and the Hausdorff dimension dimHρ with respect to ρ of the same set: ζ(dimHρ(K)) = dimH(K). Our results can...