Homogenization of time-fractional diffusion equations with periodic coefficients

We consider the initial boundary value problem for the time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial data $a(x)\in L^{2}(D)$ in a bounded domain $D\subset \mathbb{R}^d$ with a sufficiently smooth boundary. We analyze the homogenized solution under the assumption that the diffusion coefficient $\kappa^{\epsilon}(x)$ is smooth and periodic with the period $\epsilon>0$ being sufficiently small. We derive that its first order approximation has a convergence rate of $\mathcal{O}(\epsilon^{1/2})$ when the dimension $d\leq 2$ and $\mathcal{O}(\epsilon^{1/6})$ when $d=3$. Several numerical tests are presented to show the performance of the first order approximation

Essays on Volatility Risk and Security Returns

This dissertation studies the determinants of expected option returns and equilibrium determinants of variance risk and the variance risk premium. In the first essay, I analyze the relation between expected option returns and the volatility of the underlying securities. In the Black-Scholes-Merton and stochastic volatility models, the expected return from holding a call (put) option is a decreasing (increasing) function of the volatility of the underlying. These predictions are strongly supported by the data. In the cross-section of stock option returns, returns on call (put) option portfolios decrease (increase) with underlying stock volatility. This strong negative (positive) relation between call (put) option returns and volatility is not due to cross-sectional variation in expected stock returns. It holds in various option samples with different maturities and moneyness, and it is robust to alternative measures of underlying volatility and different weighting methods. Time-series evidence also supports the predictions from option pricing theory: Future returns on S&P 500 index call (put) options are negatively (positively) related to S&P 500 index volatility. In the second essay, I show that in many consumption-based general equilibrium models with Epstein-Zin-Weil preferences, the market variance risk premium is related to the leverage effect, defined as the conditional covariance between market returns and changes in the conditional market variance. The sign of the relation between the market variance risk premium and the market leverage effect depends on the coefficient of relative risk aversion and the elasticity of intertemporal substitution. I find a statistically significant negative intertemporal relationship between the variance risk premium and the leverage effect for the S\&P 500 from 1996 to 2014. This implies an elasticity of intertemporal substitution less than one and a preference for the early resolution of uncertainty. Exploiting the relation between the variance risk premium and the leverage effect also allows me to characterize the historical behavior of the variance risk premium going back to 1926.Finance, Department o

OsCAF1, a CRM Domain Containing Protein, Influences Chloroplast Development

The chloroplast RNA splicing and ribosome maturation (CRM) domain proteins are involved in the splicing of chloroplast gene introns. Numerous CRM domain proteins have been reported to play key roles in chloroplast development in several plant species. However, the functions of CRM domain proteins in chloroplast development in rice remain poorly understood. In the study, we generated oscaf1 albino mutants, which eventually died at the seedling stage, through the editing of OsCAF1 with two CRM domains using CRISPR/Cas9 technology. The mesophyll cells in oscaf1 mutant had decreased chloroplast numbers and damaged chloroplast structures. OsCAF1 was located in the chloroplast, and transcripts revealed high levels in green tissues. In addition, the OsCAF1 promoted the splicing of group IIA and group IIB introns, unlike orthologous proteins of AtCAF1 and ZmCAF1, which only affected the splicing of subgroup IIB introns. We also observed that the C-terminal of OsCAF1 interacts with OsCRS2, and OsCAF1–OsCRS2 complex may participate in the splicing of group IIA and group IIB introns in rice chloroplasts. OsCAF1 regulates chloroplast development by influencing the splicing of group II introns