Title : Parameter estimation for misspecified diffusion with market microstructure noise
Abstract :
We consider statistical inference for stock prices modeled by diffusion processes with high-frequency observations. In particular, we study parametric inference under the existence of market microstructure noise and nonsynchronous observations. We first consider maximum-likelihood-type estimation for parametric diffusion processes with noisy, nonsynchronous observations, assuming that the true model is contained in the parametric family. We show asymptotic mixed normality of the estimator with the convergence rate $n^{-1/4}$. We also see local asymptotic normality of the statistical model when coefficients of the stochastic differential equation is deterministic, and show asymptotic efficiency of the estimator.
In practice for high-frequency financial data, it is not easy task to choose parametric family so that the true model is contained in the parametric family. The statistical model without this assumption is called 'misspecified model'. In this setting, the maximum-likelihood-type estimator cannot attain the optimal convergence rate $n^{-1/4}$ due to the asymptotic bias. We construct a new estimator which attains the optimal rate by using a bias correction, and show the asymptotic mixed normality.
Title : Sampling from random partitions via A-hypergeometric systems associated with monomial curves
Abstract :
A random partition given by sampling from Ferguson's Dirichlet process (1973) follows distribution of cycle lengths in cycle decomposition of random permutation. This is a uniform distribution with respect to cardinality of conjugacy class of symmetric group. In this talk, we discuss probability measures on Young diagrams. Schur symmetric polynomials are indexed by Young diagrams, and the generalizations, Macdonald symmetric polynomials, induce probability measures on Young diagrams. Diaconis and Lam (2012) discussed mixing of MCMC with random walks on Young diagrams induced by the probability measures. It is well known that a direct sampling from Dirichlet processes is achieved by the Blackwell-MacQueen urn scheme. However, such an urn scheme is not available for the measures induced by the Macdonald polynomials. In applications, a direct sampling is attractive. In this talk, we will see that a direct sampling is possible for a class of probability measures including the measures induced by the Macdonald polynomials. The key idea is use of sufficiency rendered by the length of partitions. The conditional measure has the normalizing constant, which is known as a A-hypergeometric polynomial, whose matrix A determines a monomial curve. Random walks on A-hypergeometric polynomials enable direct sequential sampling from the conditional measure.
Title : Perfect hedging under endogenous permanent market impacts
Abstract :
We model a nonlinear price curve quoted in a market as the utility indifference curve of a representative liquidity supplier. As the utility function, we adopt a g-expectation. In contrast to the standard framework of financial engineering, a trader is no longer a price taker as any trade has a permanent market impact via an effect on the supplier’s inventory. The P&L of a trading strategy is written as a nonlinear stochastic integral. Under this market impact model, we introduce a completeness condition under which any derivative can be perfectly replicated by a dynamic trading strategy. In the special case of a Markovian setting, the corresponding pricing and hedging can be done by solving a semilinear PDE.
Title : Equilibrium returns with transaction costs
Abstract :
We study how trading costs are reflected in equilibrium returns. To this end, we develop a tractable continuous-time risk-sharing model, where heterogeneous mean–variance investors trade subject to a quadratic transaction cost. The corresponding equilibrium is characterized as the unique solution of a system of coupled but linear forward–backward stochastic differential equations. Explicit solutions are obtained in a number of concrete settings. The sluggishness of the frictional portfolios makes the corresponding equilibrium returns mean-reverting. Compared to the frictionless case, expected returns are higher if the more risk-averse agents are net sellers or if the asset supply expands over time.
Title : Consistent model selection for ergodic processes
Abstract :
There are several studies of model selection for stochastic differential equations (SDEs), for example, the contrast-based information criterion and Schwarz type information criterion for ergodic diffusion processes. However, most of the existing theoretical literature have been developed for nested models. We will give the mathematical validity of Bayesian model comparison for possibly misspecified ergodic SDE models driven by a large class of Lévy processes, and propose the quasi-Bayesian information criterion (QBIC).
Title : Rate of estimation for the stationary law of hypoelliptic processes
Abstract :
We consider the problem of non parametric estimation of the stationary distribution $\pi$ for a bidimensional hypo-elliptic diffusion. We assume that the process is observed continuously up to time $T$ with $T \to \infty$, and we characterize the minmax rate of estimation for the pointwise estimation of $\pi(x_0,y_0)$. We obtain that the rate of estimation depends on the smoothness of $\pi$ and on $(x_0,y_0)$.