arXiv subjects
Publications and source records attributed to Yasutaka Shimizu.
We study the problem of parametric estimation for continuously observed stochastic processes driven by additive small fractional Brownian motion with Hurst index 0<H<1/2 and 1/2<H<1. Under some assumptions on the drift coefficient, we obtain the asymptotic normality and moment convergence of maximum likelihood estimator of the drift parameter .
We will consider the following stochastic differential equation (SDE): \begin{equation} X_t=X_0+\int_0^tb(X_s,θ_0)ds+σB_t,~~~t\in(0,T], \end{equation} where $\{B_t\}_{t\ge 0}$ is a fractional Brownian motion with Hurst index $H\in(1/2,1)$, $θ_0$ is a parameter that contains a bounded and open convex subset $Θ\subset\mathbb{R}^d$, $\{b(\cdot,θ),θ\inΘ\}$ is a family of drift coefficients with $b(\cdot,θ):\mathbb{R}\rightarrow\mathbb{R}$, and $σ\in\mathbb{R}$ is assumed to be the known diffusion coefficient.
The aim of this paper is to construct the confidence interval of the ultimate ruin probability under the insurance surplus driven by a Lévy process. Assuming a parametric family for the Lévy measures, we estimate the parameter from the surplus data and estimate the ruin probability via the delta method. However the asymptotic variance includes the derivative of the ruin probability with respect to the parameter, which is not generally given explicitly, and the confidence interval is not straightforward even if the ruin probability is well estimated. This paper gives the Cramér-type approximation for the derivative and gives an asymptotic confidence interval of ruin probability.
We consider an estimation problem of expected functionals of a general random element that values in a metric space. If the functional forms an explicit function of some unknown parameters, we can estimate it by plugging-in a suitable estimator in to the function, and we can find the asymptotic distribution. However, if the functional is implicit in the parameters, it causes a problem of specifying asymptotic distribution. This paper gives a general condition to specify the asymptotic distribution even if the functional is implicit in the parameters, and further investigates it in detail when the random elements are semimartingales.
The maximum composite likelihood estimator for parametric models of determinantal point processes (DPPs) is discussed. Since the joint intensities of these point processes are given by determinant of positive definite kernels, we have the explicit form of the joint intensities for every order. This fact enables us to consider the generalized maximum composite likelihood estimator for any order. This paper introduces the two step generalized composite likelihood estimator and shows the moment convergence of the estimator under a stationarity. Moreover, our results can yield information criteria for statistical model selection within DPPs.
Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent, but numerically unstable in the sense of large standard deviations under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data, and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivalent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However, in numerical study, it looked asymptotically normal in an example where filter was choosen suitably, and the noise was a Lévy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.
We study the problem of parameter estimation for discretely observed stochastic processes driven by additive small Lévy noises. We do not impose any moment condition on the driving Lévy process. Under certain regularity conditions on the drift function, we obtain consistency and rate of convergence of the least squares estimator (LSE) of the drift parameter when a small dispersion coefficient $\varepsilon \to 0$ and $n \to \infty$ simultaneously. The asymptotic distribution of the LSE in our general setting is shown to be the convolution of a normal distribution and a distribution related to the jump part of the Lévy process.