In this paper we investigate the class of grey Brownian motions $B_{α,β}$ ($0<α<2$, $0<β\leq1$). We show that grey Brownian motion admits different representations in terms of certain known processes, such as fractional Brownian motion, multivariate elliptical distribution or as a subordination. The weak convergence of the increments of $B_{α,β}$ in $t$, $w$-variables are studied. Using the Berman criterium we show that $B_{α,β}$ admits a $λ$-square integrable local time $L^{B_{α,β}}(\cdot,I)$ almost surely ($λ$ Lebesgue measure). Moreover, we prove that this local time can be weak-approximated by the number of crossings $C^{B_{α,β}^{\varepsilon}}(x,I)$, of level $x$, of the convolution approximation $B_{α,β}^{\varepsilon}$ of grey Brownian motion.
We show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the Hölder norm, we also prove a large deviations estimate.
In this article, we show that the standard vector-valued generalization of a generalized grey Brownian motion (ggBm) has independent components if and only if it is a fractional Brownian motion. In order to extend ggBm with independent components, we introduce a vector-valued generalized grey Brownian motion (vggBm). The characteristic function of the corresponding measure is introduced as the product of the characteristic functions of the one-dimensional case. We show that for this measure, the Appell system and a calculus of generalized functions or distributions are accessible. We character…
Estimating dynamic correlation between a pair of time series is of importance in many applications. We present new estimators for the dynamic correlation between a pair of correlated Brownian motions and separately for dynamic correlation between a pair of correlated Geometric Brownian motions. We show that, as the sample size increases, all estimators presented in this paper converge in probability to the underlying true dynamic correlation.
In this paper, we obtain the existence and uniqueness theorem of $L^{p}$-solution for coupled forward-backward stochastic differential equations driven by G-Brownian motion (G-FBSDEs) with arbitrary $T$ under weakly coupling condition. Specially, the result for $p\in(1,2)$ is completely different from the one for $p\geq2$. Furthermore, by considering the dual linear FBSDE under a suitable reference probability, we establish the comparison theorem for G-FBSDEs under weakly coupling condition.
We study a simple model for the trajectory of a particle in a turbulent fluid, where a Brownian motion travels through a random Gaussian velocity field. We study the quenched law of the process and prove that in a weak environment setting, the fluctuations at the transition from weak to strong disorder are described by the KPZ equation. We conjecture the same is true without the assumption of a weak environment.
Let $D_N$ be the set of points around which a planar Brownian motion winds at least $N$ times. We prove that the random measure on the plane with density $2 πN 1_{D_N}$ with respect to the Lebesgue measure converges almost surely weakly, as $N$ tends to infinity, towards the occupation measure of the Brownian motion.
We first study the convergence of solutions of a system of F-KPP equations related to irreducible multitype branching Brownian motions with Heaviside-type initial conditions to traveling wave solutions. Then we apply this convergence result to prove that the extremal processes of irreducible multitype branching Brownian motions converges weakly to a cluster point process.
We study a modification of the fractional analogue of the Brownian meander, which is Brownian motion conditioned to be positive on the time interval ${[0,1]}$. More precisely, we determine the weak limit of a fractional Brownian motion which is penalized -- instead of being killed -- when leaving the positive half-axis. In the Brownian case, we give a representation of the limiting process in terms of an explicit SDE and compare it to the SDE fulfilled by the Brownian meander.
We prove a fluctuating limit theorem of a sequence of super-Brownian motions over $\mbb{R}$ with a single point catalyst. The weak convergence of the processes on the space of Schwarz distributions is established. The limiting process is an Ornstein-Uhlenbeck type process solving a Langevin type equation driven by a one-dimensional Brownian motion.
We give a new approach to prove the existence of a weak solution of \[dx_t = f(t,x_t)dt + g(t)dB^H_t\] where $B^H_t$ is a fractional Brownian motion with values in a separable Hilbert space for suitable functions $f$ and $g$. Our idea is to use the implicit function theorem and the scaling property of the fractional Brownian motion in order to obtain a weak solution for this equation.
We study the pointwise regularity of the Multifractional Brownian Motion and in particular, we get the existence of slow points. It shows that a non self-similar process can still enjoy this property. We also consider various extensions of our results in the aim of requesting a weaker regularity assumption for the Hurst function without altering the regularity of the process.
Sticky Brownian motion on the real line can be obtained as a weak solution of a system of stochastic differential equations. We find the conditional distribution of the process given the driving Brownian motion, both at an independent exponential time and at a fixed time t>0. As a classical problem, we find the distribution of the occupation times of a half-line, and at 0, which is the sticky point for the process.
In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE.
