arXiv subjects
Publications and source records attributed to Zhou Zhou.
We consider multi-player stopping games in continuous time. Unlike Dynkin games, in our games the payoff of each player is revealed after all the players stop. Moreover, each player can adjust her own stopping strategy by observing other players' behaviors. Assuming the continuity of the payoff functions in time, we show that there always exists an $\eps$-Nash equilibrium in pure stopping strategies for any $\eps>0$.
We consider two-player non-zero-sum stopping games in discrete time. Unlike Dynkin games, in our games the payoff of each player is revealed after both players stop. Moreover, each player can adjust her own stopping strategy according to the other player's action. In the first part of the paper, we consider the game where players act simultaneously at each stage. We show that there exists a Nash equilibrium in mixed stopping strategies. In the second part, we assume that one player has to act first at each stage. In this case, we show the existence of a Nash equilibrium in pure stopping strategies.
On a filtered probability space $(Ω,\mathcal{F}, (\mathcal{F}_t)_{t\in[0,\infty]}, \mathbb{P})$, we consider the two-player non-zero-sum stopping game $u^i := \mathbb{E}[U^i(ρ,τ)],\ i=1,2$, where the first player choose a stopping strategy $ρ$ to maximize $u^1$ and the second player chose a stopping strategy $τ$ to maximize $u^2$. Unlike the Dynkin game, here we assume that $U(s,t)$ is $\mathcal{F}_{s\vee t}$-measurable. Assuming the continuity of $U^i$ in $(s,t)$, we show that there exists an $ε$-Nash equilibrium for any $ε>0$.
We consider a zero-sum continuous time stopping game in which the pay-off is revealed in the maximum of the two stopping times instead of the minimum, which is the case in Dynkin games.
We consider the optimal stopping problem $v^{(\eps)}:=\sup_{τ\in\mathcal{T}_{0,T}}\mathbb{E}B_{(τ-\eps)^+}$ posed by Shiryaev at the International Conference on Advanced Stochastic Optimization Problems organized by the Steklov Institute of Mathematics in September 2012. Here $T>0$ is a fixed time horizon, $(B_t)_{0\leq t\leq T}$ is the Brownian motion, $\eps\in[0,T]$ is a constant, and $\mathcal{T}_{\eps,T}$ is the set of stopping times taking values in $[\eps,T]$. The solution of this problem is characterized by a path dependent reflected backward stochastic differential equations, from which the continuity of $\eps \to v^{(\eps)}$ follows. For large enough $\eps$, we obtain an explicit expression for $v^{(\eps)}$ and for small $\eps$ we have lower and upper bounds. The main result of the paper is the asymptotics of $v^{(\eps)}$ as $\eps\searrow 0$. As a byproduct, we also obtain Lévy's modulus of continuity result in the $L^1$ sense.
We consider as given a discrete time financial market with a risky asset and options written on that asset and determine both the sub- and super-hedging prices of an American option in the model independent framework of ArXiv:1305.6008. We obtain the duality of results for the sub- and super-hedging prices. For the sub-hedging prices we discuss whether the sup and inf in the dual representation can be exchanged (a counter example shows that this is not true in general). For the super-hedging prices we discuss several alternative definitions and argue why our choice is more reasonable. Then assuming that the path space is compact, we construct a discretization of the path space and demonstrate the convergence of the hedging prices at the optimal rate. The latter result would be useful for numerical computation of the hedging prices. Our results generalize those of ArXiv:1304.3574 to the case when static positions in (finitely many) European options can be used in the hedging portfolio.
An important assumption in the work on testing for structural breaks in time series consists in the fact that the model is formulated such that the stochastic process under the null hypothesis of "no change-point" is stationary. This assumption is crucial to derive (asymptotic) critical values for the corresponding testing procedures using an elegant and powerful mathematical theory, but it might be not very realistic from a practical point of view. This paper develops change point analysis under less restrictive assumptions and deals with the problem of detecting change points in the marginal variance and correlation structures of a non-stationary time series. A CUSUM approach is proposed, which is used to test the "classical" hypothesis of the form $H_0: θ_1=θ_2$ vs. $H_1: θ_1 \not =θ_2$, where $θ_1$ and $θ_2$ denote second order parameters of the process before and after a change point. The asymptotic distribution of the CUSUM test statistic is derived under the null hypothesis. This distribution depends in a complicated way on the dependency structure of the nonlinear non-stationary time series and a bootstrap approach is developed to generate critical values. The results are then extended to test the hypothesis of a {\it non relevant change point}, i.e. $H_0: | θ_1-θ_2 | \leq δ$, which reflects the fact that inference should not be changed, if the difference between the parameters before and after the change-point is small. In contrast to previous work, our approach does neither require the mean to be constant nor - in the case of testing for lag $k$-correlation - that the mean, variance and fourth order joint cumulants are constant under the null hypothesis. In particular, we allow that the variance has a change point at a different location than the auto-covariance.
We consider the fundamental theorem of asset pricing (FTAP) and hedging prices of options under non-dominated model uncertainty and portfolio constrains in discrete time. We first show that no arbitrage holds if and only if there exists some family of probability measures such that any admissible portfolio value process is a local super-martingale under these measures. We also get the non-dominated optional decomposition with constraints. From this decomposition, we get duality of the super-hedging prices of European options, as well as the sub- and super-hedging prices of American options. Finally, we get the FTAP and duality of super-hedging prices in a market where stocks are traded dynamically and options are traded statically.
We show that the results of ArXiv:1305.6008 on the Fundamental Theorem of Asset Pricing and the super-hedging theorem can be extended to the case in which the options available for static hedging (\emph{hedging options}) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of \emph{robust no-arbitrage} which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are \emph{non-redundant}. A key result is the closedness of the set of attainable claims, which requires a new proof in our setting.
We investigate the behavior of the Generalized Likelihood Ratio Test (GLRT) (Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193]) for time varying coefficient models where the regressors and errors are non-stationary time series and can be cross correlated. It is found that the GLRT retains the minimax rate of local alternative detection under weak dependence and non-stationarity. However, in general, the Wilks phenomenon as well as the classic residual bootstrap are sensitive to either conditional heteroscedasticity of the errors, non-stationarity or temporal dependence. An averaged test is suggested to alleviate the sensitivity of the test to the choice of bandwidth and is shown to be more powerful than tests based on a single bandwidth. An alternative wild bootstrap method is proposed and shown to be consistent when making inference of time varying coefficient models for non-stationary time series.
We investigate the behavior of Fourier transforms for a wide class of nonstationary nonlinear processes. Asymptotic central and noncentral limit theorems are established for a class of nondegenerate and degenerate weighted $V$-statistics through the angle of Fourier analysis. The established theory for $V$-statistics provides a unified treatment for many important time and spectral domain problems in the analysis of nonstationary time series, ranging from nonparametric estimation to the inference of periodograms and spectral densities.
We consider controller-stopper problems in which the controlled processes can have jumps. The global filtration is represented by the Brownian filtration, enlarged by the filtration generated by the jump process. We assume that there exists a conditional probability density function for the jump times and marks given the filtration of the Brownian motion and decompose the global controller-stopper problem into controller-stopper problems with respect to the Brownian filtration, which are determined by a backward induction. We apply our decomposition method to indifference pricing of American options under multiple default risk. The backward induction leads to a system of reflected backward stochastic differential equations (RBSDEs). We show that there exists a solution to this RBSDE system and that the solution provides a characterization of the value function.
We consider the robust utility maximization using a static holding in derivatives and a dynamic holding in the stock. There is no fixed model for the price of the stock but we consider a set of probability measures (models) which are not necessarily dominated by a fixed probability measure. By assuming that the set of physical probability measures is convex and weakly compact, we obtain the duality result and the existence of an optimizer.
We consider the pricing and hedging of exotic options in a model-independent set-up using \emph{shortfall risk and quantiles}. We assume that the marginal distributions at certain times are given. This is tantamount to calibrating the model to call options with discrete set of maturities but a continuum of strikes. In the case of pricing with shortfall risk, we prove that the minimum initial amount is equal to the super-hedging price plus the inverse of the utility at the given shortfall level. In the second result, we show that the quantile hedging problem is equivalent to super-hedging problems for knockout options. These results generalize the duality results of [5,6] to the model independent setting of [1].
The paper considers nonparametric specification tests of quantile curves for a general class of nonstationary processes. Using Bahadur representation and Gaussian approximation results for nonstationary time series, simultaneous confidence bands and integrated squared difference tests are proposed to test various parametric forms of the quantile curves with asymptotically correct type I error rates. A wild bootstrap procedure is implemented to alleviate the problem of slow convergence of the asymptotic results. In particular, our results can be used to test the trends of extremes of climate variables, an important problem in understanding climate change. Our methodology is applied to the analysis of the maximum speed of tropical cyclone winds. It was found that an inhomogeneous upward trend for cyclone wind speeds is pronounced at high quantile values. However, there is no trend in the mean lifetime-maximum wind speed. This example shows the effectiveness of the quantile regression technique.
We consider estimation of quantile curves for a general class of nonstationary processes. Consistency and central limit results are obtained for local linear quantile estimates under a mild short-range dependence condition. Our results are applied to environmental data sets. In particular, our results can be used to address the problem of whether climate variability has changed, an important problem raised by IPCC (Intergovernmental Panel on Climate Change) in 2001.