Random perturbation to the geodesic equation

We study random "perturbation" to the geodesic equation. The geodesic equation is identified with a canonical differential equation on the orthonormal frame bundle driven by a horizontal vector field of norm $1$. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian motion scaled by ${\frac{8}{n(n-1)}}$ where $n$ is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion.Comment: Published at http://dx.doi.org/10.1214/14-AOP981 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Limits of Random Differential Equations on Manifolds

Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form $\sum_kY_k\alpha_k(z_t^\epsilon(\omega))$ where $Y_k$ are vector fields, $\epsilon$ is a positive number, $z_t^\epsilon$ is a ${1\over \epsilon} {\mathcal L}_0$ diffusion process taking values in possibly a different manifold, $\alpha_k$ are annihilators of $ker ({\mathcal L}_0^*)$. Under H\"ormander type conditions on ${\mathcal L}_0$ we prove that, as $\epsilon$ approaches zero, the stochastic processes $y_{t\over \epsilon}^\epsilon$ converge weakly and in the Wasserstein topologies. We describe this limit and give an upper bound for the rate of the convergence.Comment: 46 pages, To appear in Probability Theory and Related Fields In this version, we add a note in proof for the published versio

On the Semi-Classical Brownian Bridge Measure

We prove an integration by parts formula for the probability measure induced by the semi-classical Riemmanian Brownian bridge over a manifold with a pole

Coordination Failure in Technological Progress, Economic Growth and Volatility

Technological progress has long been posited to be crucial in a country's economic growth. This paper argues that coordination failure in a country's new technology investment can be one of the barriers in a country's capital accumulation and economic growth. The global game established by Morris and Shin(2000) is extended to a two-sector overlapping generations model where capital goods can be produced by two different technologies. The first is a conventional technology with constant returns, which are perfectly revealed to economic agents. The second is a new technology exhibiting increasing return to scale due to technological externalities, whose returns economic agents only have incomplete information about. Economic agents have to choose which technology to invest in. My model reveals that under certain circumstances coordination failure in the capital goods sector will occur and be manifested as under-investment in the new technology. In this way, I explain how coordination failure in a country's technology updating process leads to slower capital accumulation and economic growth. More interestingly, the model generates a positive correlation between economic growth and volatility through a new channel associated with coordination failure. Policy implications are discussed as well.Economic Growth, Technological externalities, Coordination Failure

Investment Complementarities, Coordination Failure and Systemic Bankruptcy

I argue that systemic bankruptcy of firms can originate from coordination failure in an economy with investment complementarities. This new explanation about the origin of systemic bankruptcy promotes better understanding of how financial fragility arises, and provides theoretical guidance for central banks to establish an "early warning system" to prevent the occurrence of financial crises. In a global game setup, investment decisions of firms are studied in the presence of uncertainty and investment complementarities. Uncertainty is twofold here: first, firms are uncertain about economic fundamentals; second, firms are also uncertain about other firms' investment decisions. I demonstrate that even small uncertainty about economic fundamentals can be magnified through the uncertainty about other firms' investment decisions and can lead to coordination failure, which may be manifested as systemic bankruptcy. Moreover, my model reveals that systemic bankruptcy tends to arise when economic fundamentals are in the middle range where coordination matters. High financial leverage of firms greatly increases the severity of systemic bankruptcy. Optimistic beliefs of firms and banks can alleviate coordination failure, but can also increase the severity of systemic bankruptcy once it happens.Systemic Bankruptcy, Financial Crises, Global Games