• Abstract: There are many statistical methods available for constructing prediction models in the presence of high-dimensional data. The key concept underlying the analysis of high-dimensional data is dimension reduction or regularization. Producing a useful forecasting model requires regularization; that is, the estimates must be constrained regarding the bias-variance trade-off. We will cover commonly used penalization methods and their theoretical properties. We will mainly focus on the so-called oracle-property both from a finite-sample and asymptotic point of view. We will extensively study the asymptotic properties of the adaptive Sparse Group Lasso estimator within the penalized M-estimtor framework for dependent variables. We prove that this sparsity based estimator recovers the true underlying sparse model and is asymptotically normally distributed. Then we will study its asymptotic properties in a double-asymptotic framework, where the number of parameters diverges with the sample size. We show by simulations that the adaptive SGL outperforms other oracle-like methods in terms of estimation precision and variable selection.

• Abstract: Consider a marked Hawkes process $L$ with the conditional intensity process $\lambda$. The goal in this talk is to study the Malliavin calculus for the process $L$ and its conditional intensity $\lambda$, and to apply it to attack the problems on the sensitivity analysis for their conditional laws, and the asymptotic behavior on the law of the marked Hawkes process. 

• Abstract: We explore how the timings of compensation and contract termination are jointly and optimally determined in a continuous-time principal-agent model under the discretionary termination policy of investors (the principal) when the agent has loss averse preferences. We show that the larger replacement cost of the agent induces investors to reward the agent later and to be more likely to fire the agent at the contract termination, but does not necessarily induce investors to terminate the contract relation later. We also indicate that the larger degree of loss aversion induces investors to reward the agent earlier and to be less likely to fire the agent at the contract termination if the reference point is sufficiently small or large; however, it does not necessarily induce investors to terminate the contract relation later. Our theoretical findings provide several new empirical implications of the backloaded compensation and forced turnover of managers. Our model also shows that the mandatory deferral regulation of incentive pay induces investors to terminate the contract relation earlier and results in the more frequent replacement of managers. The endogenous determination of the utility reference point does not affect our main results.

• Abstract:本研究では, 近年研究が進展中の非整数確率ボラティリティモデルに対して, 高頻度取引データに基づいた未知定数の推定問題を考える. 特に我々は, ボラティリティ変動の激しさやスケール則を支配するHurst指数の推定に興味があるが, 潜在的な因子であるボラティリティ過程が非セミマルチンゲールであり, また直接的な観測データが得られないことから, 理論的に精緻な推定手法はこれまで提案されていない. 講演者たちは, 対数価格過程の累積ボラティリティが取引価格から誤差なしで計測できる状況下で, 新たに推定手法を提案し, 推定誤差の漸近分布などの理論的性質を明らかにした. 本講演では, これまでに得られた研究成果の報告を行う.

• Abstract: 

In this talk, we propose novel methods for obtaining hard bounding functions, without recourse to sample path simulation, without truncating the naturally unbounded domain that arises in this problem, and without discretizing the time and state variables.

Unlike accurate approximate solutions via the existing discretization-based methods, our hard bounding functions are free from statistical error and act as pointwise 100% confidence intervals within which the unknown solution is guaranteed to exist.

The proposed approaches can be applied to a variety of problem settings, such as mixed boundary conditions, stochastic volatility, stochastic processes with jumps, regime-switching and obstacle problems.

Numerical results are presented throughout to support our theoretical developments and to illustrate the effectiveness of the proposed approaches.

This talk consist of two parts; (i) optimization and (ii) perturbation.

(i)

We propose a novel method for obtaining and tightening hard bounding functions for the expected value on stochastic differential equations with the help of the mathematical programming and the Dynkin formula.

In a single implementation of semi-definite programming, the proposed approach obtains explicit bounds in the form of piecewise polynomial functions, which bound the expectation over the whole domain both in time and state.

As a consequence, these global bounds store a continuum of bounding information in the form of a finite number of polynomial coefficients.

In this talk, we pay particular attention to the American style option pricing problem.

(ii)

It is often the case that expectations are easy to compute for a simple model, while small perturbations make the computation of expectation suddenly prohibitive.

We propose a novel method for measuring the impact of such small perturbations in expectations without significant computing effort.

Our hard bounding functions are deterministic in the form of Markov-type inequalities, parametrized by the perturbation parameter, so that the upper and lower bounds converge to each other when the perturbation tends to vanish.

The proposed method requires only well-developed numerical methods for boundary value problems for partial differential equations and elementary numerical integration of smooth functions.