On weak solutions of stochastic differential equations with sharp drift coefficients

We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of stochastic differential equations (SDEs for short). To be more precise, let $b: [0,T]\times{\mathbb R}^d\rightarrow{\mathbb R}^d$ be Borel measurable, where $T>0$ is arbitrarily fixed. Consider $$X_t=x+\int_0^tb(s,X_s)ds+W_t,\quad t\in[0,T], \, x\in\mathbb{R}^d,$$where $\{W_t\}_{t\in[0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that b_1(T-\cdot)\in\cC_q^0((0,T];L^p(\mR^d)) with $2/q+d/p=1$ for $p,q\ge1$ and \|b_1(T-\cdot)\|_{\cC_q((0,T];L^p(\mR^d))} is sufficiently small, and that $b_2$ is bounded and Borel measurable, then there exits a unique weak solution to the above equation. Furthermore, we obtain the strong Feller property of the semi-group and existence of density associated with above SDE. Besides, we extend the classical partial differential equations (PDEs) results for L^q(0,T;L^p(\mR^d)) coefficients to L^\infty_q(0,T;L^p(\mR^d)) ones, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Lemma \ref{lem2.1})

On-Line Portfolio Selection for a Currency Exchange Market

The purpose of this paper is to study on-line portfolio selection strategies for cur- rency exchange markets and our focus is on the markets with presence of decre- ments. To this end, we first analyze the main factors arising in the decrements. Then we develop a cross rate scheme which enables us to establish an on-line portfolio se- lection strategy for the currency exchange markets with presence of decrements. Fi- nally, we prove the universality of our on-line portfolio selections

On a stochastic nonlocal conservation law in a bounded domain

In this paper, we are concerned with the Dirichlet boundary value problem for a multi-dimensional nonlocal conservation law involving a multiplicative stochastic perturbation in a bounded domain. Using the concept of measure-valued solutions and Kruzhkov’s semi-entropy formulations, we establish the existence and uniqueness of entropy solution to the boundary value problem

BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations

In this paper, we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises \bessdu_t(x)=\Delta^{\frac{\alpha}{2}} u_t(x)dt+g(t,x)d\eta_t,\ \ \ u_0=0,\ \ \ t\in(0,T], \, x\in G, \eessfor a random field $u:(t,x)\in [0,T]\times G\mapsto u(t,x)=:u_t(x)\in\mathbb{R}$, where $\Delta^{\frac{\alpha}{2}}:=-(-\Delta)^{\frac{\alpha}{2}}, \alpha\in(0,2]$, is the fractional Laplacian, $T\in(0,\infty)$ is arbitrarily fixed, $G\subset\mathbb{R}^d$ is a bounded domain,$g:[0,T]\times G\times\Omega\to\mathbb{R}$ is a joint measurable coefficient, and $\eta_t, t\in[0,\infty)$, is either a Brownian motion or a L\'evy process on a given filtered probability space $(\Omega,\mathcal{F},P;\{\mathcal{F}_t\}_{t\in[0,T]})$. To this end, we derive the BMO estimates and Morrey-Campanato estimates, respectively, for stochastic singular integral operators arising from the equations concerned. Then, by utilising the embedding theory between the Campanato space and the H\"older space, we establish the controllability of the norm of the space $C^{\theta,\theta/2}(\bar D)$, where $\theta\ge0,\bar D=[0,T]\times\bar G$. With all these in hand, we are able to show that the $q$-th order BMO quasi-norm of the $\frac{\alpha}{q_0}$-order derivative of the solution $u$ is controlled by the norm of $g$ under the condition that $\eta_t$ is a L\'evy process. Finally, we derive the Schauder estimate for the $p$-moments of the solution of the above stochastic fractional heat equations driven by L\'evy noise

Model selection and estimation in high dimensional regression models with group SCAD

In this paper, we study the oracle property of the group SCAD under high dimensional settings where the number of groups can grow at a certain polynomial rate. Numerical studies are presented to demonstrate the merit of the group SCAD

Heterogeneous stochastic scalar conservation laws with non-homogeneous Dirichlet boundary conditions

We introduce a notion of stochastic entropy solutions for heterogeneous scalar conservation laws with multiplicative noise on a bounded domain with non-homogeneous boundary condition. Using the concept of measure-valued solutions and Kruzhkov's semi-entropy formulations, we show the existence and uniqueness of stochastic entropy solutions. Moreover, we establish an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux function and the random source function