Narrow Results

We aim at detecting stress in newborns by observing heart rate variability (HRV). The HRV features nonlinearities. Fractal dynamics is a usual way to model them and the Hurst exponent summarizes the fractal information. In our framework, we have observations of short duration, for which usual estimators of the Hurst exponent, like detrended fluctuation analysis (DFA), are not adapted. Moreover, we observe that the Hurst exponent does not vary much between stress and rest phases, but its decomposition in memory and underlying properties of the probability distribution leads to satisfactory diagnostic tools. This decomposition of the Hurst exponent is in addition embedded in a mean-reverting model. The resulting model is a mean-reverting fractional Lévy stable motion (FLSM). We estimate it and use its parameters as diagnostic tools of neonatal stress. Indeed, the value of the speed of reversion parameter is a significant indicator of stress. The evolution of both parameters in which the Hurst exponent is decomposed provides us with significant indicators as well. On the contrary, the Hurst exponent itself does not bear useful information.

This paper studies the optimal VIX futures trading problems under a regime-switching model. We consider the VIX as mean reversion dynamics with dependence on the regime that switches among a finite number of states. For the trading strategies, we analyze the timings and sequences of the investor’s market participation, which leads to several corresponding coupled system of variational inequalities. The numerical approach is developed to solve these optimal double stopping problems by using projected-successive-over-relaxation (PSOR) method with Crank–Nicolson scheme. We illustrate the optimal boundaries via numerical examples of two-state Markov chain model. In particular, we examine the impacts of transaction costs and regime-switching timings on the VIX futures trading strategies.

We investigate the effect of using an optimized exit rule on pairs trading. For every asset pair, we optimize the positions so that resulting intraday portfolio value is best fitted to an Ornstein–Uhlenbeck (OU) process through maximum likelihood estimation. Using various asset pairs, we examine the risks and returns of pairs trading strategies with and without an optimize exit rule. We provide empirical evidence that using an optimized exit rule improves the profitability of the trades and reduces turnovers.

We study the optimal timing strategies for trading a mean-reverting price process with a finite deadline to enter and a separate finite deadline to exit the market. The price process is modeled by a diffusion with an affine drift that encapsulates a number of well-known models, including the Ornstein–Uhlenbeck (OU) model, Cox–Ingersoll–Ross (CIR) model, Jacobi model, and inhomogeneous geometric Brownian motion (IGBM) model. We analyze three types of trading strategies: (i) the long–short (long to open, short to close) strategy; (ii) the short–long (short to open, long to close) strategy, and (iii) the chooser strategy whereby the trader has the added flexibility to enter the market by taking either a long or short position, and subsequently close the position. For each strategy, we solve an optimal double stopping problem with sequential deadlines, and determine the optimal timing of trades. Our solution methodology utilizes the local time-space calculus of [Peskir (2005) A change-of-variable formula with local time on curves, Journal of Theoretical Probability18, 499–535] to derive nonlinear integral equations of Volterra-type that uniquely characterize the trading boundaries. The numerical implementation of the integral equations provides examples of the optimal trading boundaries.

The heart beat RR intervals extracted from the electrocardiogram recorded during the stress test show a non stationary profile consisting of a decreasing trend during the exercise phase, an increasing trend during the recovery and a global minimum (acme). In addition this time series exhibits a time-varying variance. We decompose the series into a deterministic trend and random fluctuation. The trend is obtained as an exponential fit of the data; the fluctuation is modeled as a mean reverting process driven by the trend, in which the random innovation has a time-varying variance. Data analysis, performed on ambulatory recorded electrocardiograms of 10 healthy subjects, shows that the model describes correctly the data series on a scale of at least 300 beats.

Consider a partially informed trader who does not observe the true drift of a financial asset. Under Gaussian price dynamics with stochastic unobserved drift, including cases of mean-reversion and momentum dynamics, we take a filtering approach to solve explicitly for trading strategies which maximize expected logarithmic, exponential and power utility. We prove that the optimal strategies depend on current price and an exponentially weighted moving average (EMA) price, and in some cases current wealth, not on any other stochastic variables. We establish optimality over all price-history-dependent strategies satisfying integrability criteria, not just EMA-type strategies. Thus the condition that the optimal trading strategy reduces to a function of EMA and current price is not an assumption but rather a consequence of our analysis. We solve explicitly for the optimal parameters of the EMA-type strategies and verify optimality rigorously.