Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLarge deviations for multidimensional SDEs with reflection
129
0
0.0
(
0
)
Authors
Zongxia Liang
Ask ChatGPT about the research
No Arabic abstract
The large deviations principles are established for a class of multidimensional degenerate stochastic differential equations with reflecting boundary conditions. The results include two cases where the initial conditions are adapted and anticipated.
rate research
Read More
Consider the projection of an $n$-dimensional random vector onto a random $k_n$-dimensional basis, $k_n leq n$, drawn uniformly from the Haar measure on the Stiefel manifold of orthonormal $k_n$-frames in $mathbb{R}^n$, in three different asymptotic regimes as $n rightarrow infty$: constant ($k_n=k$), sublinear ($k_n rightarrow infty$ but $k_n/n rightarrow 0$) and linear $k_n/n rightarrow lambda$ with $0 < lambda le 1$). When the sequence of random vectors satisfies a certain asymptotic thin shell condition, we establish annealed large deviation principles (LDPs) for the corresponding sequence of random projections in the constant regime, and for the sequence of empirical measures of the coordinates of the random projections in the sublinear and linear regimes. We also establish LDPs for certain scaled $ell_q$ norms of the random projections in these different regimes. Moreover, we verify our assumptions for various sequences of random vectors of interest, including those distributed according to Gibbs measures with superquadratic interaction potential, or the uniform measure on suitably scaled $ell_p^n$ balls, for $p in [1,infty)$, and generalized Orlicz balls defined via a superquadratic function. Our results complement the central limit theorem for convex sets and related results which are known to hold under a thin shell condition. These results also substantially extend existing large deviation results for random projections, which are first, restricted to the setting of measures on $ell_p^n$ balls, and secondly, limited to univariate LDPs (i.e., in $mathbb{R}$) involving either the norm of a $k_n$-dimensional projection or the projection of $X^{(n)}$ onto a random one-dimensional subspace. Random projections of high-dimensional random vectors are of interest in a range of fields including asymptotic convex geometry and high-dimensional statistics.
In this paper, the strong solutions $ (X, L)$ of multidimensional stochastic differential equations with reflecting boundary and possible anticipating initial random variables is established. The key is to obtain some substitution formula for Stratonovich integrals via a uniform convergence of the corresponding Riemann sums and to prove continuity of functionals of $ (X, L)$.
Let $X^{(delta)}$ be a Wishart process of dimension $delta$, with values in the set of positive matrices of size $m$. We are interested in the large deviations for a family of matrix-valued processes ${delta^{-1} X_t^{(delta)}, t leq 1 }$ as $delta$ tends to infinity. The process $X^{(delta)}$ is a solution of a stochastic differential equation with a degenerate diffusion coefficient. Our approach is based upon the introduction of exponential martingales. We give some applications to large deviations for functionals of the Wishart processes, for example the set of eigenvalues.
In this paper we prove the existence of strong solutions to a SDE with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<1/2. Here the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a local time variational calculus argument.
The Large Deviation Principle is established for stochastic models defined by past-dependent non linear recursions with small noise. In the Markov case we use the result to obtain an explicit expression for the asymptotics of exit time.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
