We study the least squares estimation of drift parameters for a class of stochastic differential equations driven by small a-stable noises, observed at n regularly spaced time points ti = i/n, i = 1,...,n on [0, 1]. U...We study the least squares estimation of drift parameters for a class of stochastic differential equations driven by small a-stable noises, observed at n regularly spaced time points ti = i/n, i = 1,...,n on [0, 1]. Under some regularity conditions, we obtain the consistency and the rate of convergence of the least squares estimator (LSE) when a small dispersion parameter ε→0 and n →∞ simultaneously. The asymptotic distribution of the LSE in our setting is shown to be stable, which is completely different from the classical cases where asymptotic distributions are normal.展开更多
We introduce anticipating quadrant and symmetric integrals in the plane, and establish the associated chain rules which are the same as the deterministic ones. In particular, we deduce the relation between quadrant in...We introduce anticipating quadrant and symmetric integrals in the plane, and establish the associated chain rules which are the same as the deterministic ones. In particular, we deduce the relation between quadrant integrals, symmetric integral, and Skorohod integral with respect to two-parameter Wiener processes.展开更多
In this paper we obtain the uniform bounds on the rate of convergence in the central limit theorem (CLT) for a class of two-parameter martingale difference sequences under certain conditions.
We study the problem of parameter estimation for mean-reverting α-stable motion, dXt = (a0 - θ0Xt)dt + dZt, observed at discrete time instants. A least squares estimator is obtained and its asymptotics is discuss...We study the problem of parameter estimation for mean-reverting α-stable motion, dXt = (a0 - θ0Xt)dt + dZt, observed at discrete time instants. A least squares estimator is obtained and its asymptotics is discussed in the singular case (a0, θ0) = (0, 0). If a0 = 0, then the mean-reverting α-stable motion becomes Ornstein-Uhlenbeck process and is studied in [7] in the ergodic case θ0 〉 0. For the Ornstein-Uhlenbeck process, asymptotics of the least squares estimators for the singular case (θ0 = 0) and for ergodic case (θ0 〉 0) are completely different.展开更多
In this paper, we consider the approximation problem of stochastic differential equation with respect to two-parameter Wiener processes and deal with the relation between the solution of stochastic differential equati...In this paper, we consider the approximation problem of stochastic differential equation with respect to two-parameter Wiener processes and deal with the relation between the solution of stochastic differential equation dX(x) = sigma(X(x)dB(x) + 1/4 sigmasigma' (X(z))dz and the solution of ordinary differential equation dX(n)(z) = sigma(X(n)(z))dB(z)n, where B(z)n is a special sequence of approximation to B(x).展开更多
In this paper, we consider the approximation problem of stochastic integral with respect to two-parameter Wiener process. We first introduce a kind of symmetric integral and prove it obeys the chain rule. Then we appl...In this paper, we consider the approximation problem of stochastic integral with respect to two-parameter Wiener process. We first introduce a kind of symmetric integral and prove it obeys the chain rule. Then we apply an integral formula of bounded variation functions with two variables to show the approximation theorem of stochastic integral in the plane. In particular, we prove that the symmetric stochastic integral is stable when the limit is taken in the sense of L~2convergence.展开更多
Let (Ω,(?), P) be a complete probability space with a family of sub-σ-fields {(?)_z}_z∈R_+~2 which satisfies the usual conditions. Yeh considered the existence and uniqueness of strong solutions of the following no...Let (Ω,(?), P) be a complete probability space with a family of sub-σ-fields {(?)_z}_z∈R_+~2 which satisfies the usual conditions. Yeh considered the existence and uniqueness of strong solutions of the following non-Markovian stochastic differential equations (SDE)展开更多
基金supported by FAU Start-up funding at the C. E. Schmidt Collegeof Science
文摘We study the least squares estimation of drift parameters for a class of stochastic differential equations driven by small a-stable noises, observed at n regularly spaced time points ti = i/n, i = 1,...,n on [0, 1]. Under some regularity conditions, we obtain the consistency and the rate of convergence of the least squares estimator (LSE) when a small dispersion parameter ε→0 and n →∞ simultaneously. The asymptotic distribution of the LSE in our setting is shown to be stable, which is completely different from the classical cases where asymptotic distributions are normal.
文摘We introduce anticipating quadrant and symmetric integrals in the plane, and establish the associated chain rules which are the same as the deterministic ones. In particular, we deduce the relation between quadrant integrals, symmetric integral, and Skorohod integral with respect to two-parameter Wiener processes.
文摘In this paper we obtain the uniform bounds on the rate of convergence in the central limit theorem (CLT) for a class of two-parameter martingale difference sequences under certain conditions.
基金Hu is supported by the National Science Foundation under Grant No.DMS0504783Long is supported by FAU Start-up funding at the C. E. Schmidt College of Science
文摘We study the problem of parameter estimation for mean-reverting α-stable motion, dXt = (a0 - θ0Xt)dt + dZt, observed at discrete time instants. A least squares estimator is obtained and its asymptotics is discussed in the singular case (a0, θ0) = (0, 0). If a0 = 0, then the mean-reverting α-stable motion becomes Ornstein-Uhlenbeck process and is studied in [7] in the ergodic case θ0 〉 0. For the Ornstein-Uhlenbeck process, asymptotics of the least squares estimators for the singular case (θ0 = 0) and for ergodic case (θ0 〉 0) are completely different.
文摘In this paper, we consider the approximation problem of stochastic differential equation with respect to two-parameter Wiener processes and deal with the relation between the solution of stochastic differential equation dX(x) = sigma(X(x)dB(x) + 1/4 sigmasigma' (X(z))dz and the solution of ordinary differential equation dX(n)(z) = sigma(X(n)(z))dB(z)n, where B(z)n is a special sequence of approximation to B(x).
基金Work supported by National Natural Science Foundation of China.
文摘In this paper, we consider the approximation problem of stochastic integral with respect to two-parameter Wiener process. We first introduce a kind of symmetric integral and prove it obeys the chain rule. Then we apply an integral formula of bounded variation functions with two variables to show the approximation theorem of stochastic integral in the plane. In particular, we prove that the symmetric stochastic integral is stable when the limit is taken in the sense of L~2convergence.
文摘Let (Ω,(?), P) be a complete probability space with a family of sub-σ-fields {(?)_z}_z∈R_+~2 which satisfies the usual conditions. Yeh considered the existence and uniqueness of strong solutions of the following non-Markovian stochastic differential equations (SDE)