3 bovina@pd.infn.it 4 camana@pd.infn.it 5 stella@pd.infn.it † Summary. A central problem of Quantitative Finance is that of formulating a probabilistic model of the time evolution of asset prices allowing reliable predictions on their future volatility. As in several natural phenomena, the predictions of such a model must be compared with the data of a single process realization in our records. In order to give statistical significance to such a comparison, assumptions of stationarity for some quantities extracted from the single historical time series, like the distribution of the returns over a given time interval, cannot be avoided. Such assumptions entail the risk of masking or misrepresenting non-stationarities of the underlying process, and of giving an incorrect account of its correlations. Here we overcome this difficulty by showing that five years of daily Euro/US-Dollar trading records in the about three hours following the New York market opening, provide a rich enough ensemble of histories. The statistics of this ensemble allows to propose and test an adequate model of the stochastic process driving the exchange rate. This turns out to be a non-Markovian, self-similar process with non-stationary returns. The empirical ensemble correlators are in agreement with the predictions of this model, which is constructed on the basis of the time-inhomogeneous, anomalous scaling obeyed by the return distribution. †

A complete procedure for identifying the area of convergence of blood drops originated from a single static source is presented. Both for bloodstains lying on a horizontal and on a vertical plane a complete study is developed, based on error analysis and on an opportunely defined joint probability density for the orientation of the horizontal projections of the trajectories of the drops. The method generates a probabilistic map for the area of convergence, directly linking the angles of impact, and their uncertainties, to the projection on the ground of the point of origin. One of the objectives consists in providing a statistical definition of area of convergence, extending to this topic the mathematical accuracy of the calculation of the angle of impact in bloodstain pattern analysis (BPA).

Regarding the intraday sequence of high-frequency returns of the S&P index as daily realizations of a given stochastic process, we first demonstrate that the scaling properties of the aggregated return distribution can be employed to define a martingale stochastic model which consistently replicates conditional expectations of the S&P 500 high-frequency data in the morning of each trading day. Then, a more general formulation of the above scaling properties allows to extend the model to the afternoon trading session. We finally outline an application in which conditioned forecasting is used to implement a trend-following trading strategy capable of exploiting linear correlations present in the S&P data-set and absent in the model. Trading signals are model based and not derived from chartist criteria. In-sample and out-of-sample tests indicate that the model-based trading strategy performs better than a benchmark one established on an asymmetric GARCH process, and show the existence of small arbitrage opportunities. We remark that in the absence of linear correlations the trading profit would vanish and discuss why the trading strategy is potentially interesting to hedge volatility risk for S&P index-based products.

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