Empirical evidence shows that calibrating exponential Lévy models by options with different maturities leads to conflicting information. In other words, the stationarity implicitly assumed in the exponential Lévy model is not satisfied. An identifiable time-inhomogeneous Lévy model is proposed that does not assume stationarity and that can integrate option prices from different maturities and different strike prices without leading to conflicting information. In the time-inhomogeneous Lévy model, the convergence rates are derived, and confidence intervals are shown for the estimators of the volatility, the drift, the intensity and the Lévy density. Previously, confidence intervals have been constructed for time-homogeneous Lévy models in an idealized Gaussian white noise model. In the idealized Gaussian white noise model, it is assumed that the observations are Gaussian and given continuously across the strike prices. This simplifies the analysis significantly. The confidence intervals are constructed in a discrete observation setting for time-inhomogeneous Lévy models, and the only assumption on the errors is that they are sub-Gaussian. In particular, all bounded errors with arbitrary distributions are covered. Additional results on the convergence rates extend existing results from time-homogeneous to time-inhomogeneous Lévy models.

In the one-dimensional case, the self-similarity of the stochastic process {X_t, t ≥ 0}
is inferred by assessing the statistical significance of the minimal diameter of the rescaled distributions of X_t. By construction, for i.i.d. random variables, the one-dimensional minimal diameter reduces to the widely recognized Kolmogorov-Smirnov (KS) statistic. However, when independence is not strictly enforced, the distribution of the KS statistic undergoes significant alterations, contingent upon the intensity and direction of dependence. An illustrative instance is fractional Brownian motion (fBm), for which the critical values of the KS test cease to offer
reliable rejection thresholds, displaying a type I or type II error depending on whether the value of the Hurst exponent of the fBm is greater or less than 1/2. We observe that, in case of fBm, the KS statistic distributes as the absolute maximum of a fractional Brownian bridge (fBb) on the time interval [0,T] associated to the number of data of the rescaled empirical distribution. Consistently with this premise, employing the Maximum Likelihood Estimation (MLE) approach, we find that the Generalized Extreme Value (GEV) distribution captures the distributions of minimal diameters, providing more accurate critical values depending on the value of H and on the sample length.

We investigate the Local Asymptotic Property for fractional Brownian models based on discrete observations contaminated by a Gaussian noise. We consider both situations of low and high-frequency observations in a unified setup and we show that the convergence rate $n^{1/2} (nu_n Delta_n^{-H})^{-1/(2H+1)}$ is optimal for estimating the Hurst index $H$, where $nu_n$ is the noise intensity.

Simulation of bridge processes is a widely used tool for statistics for stochastic processes. Such processes arise when the original process is conditioned to be in a given state at a given time. A common tool to study bridge processes is Doob's h-transform. However, a problem with the transformation is that it relies on the, typically intractable, transition density of the process. We instead consider the technique of conditioning by guiding, which circumvents this problem by using the same transformation but with a different h-function whilst maintaining absolute continuity with respect to the true bridge process.

The talk will focus on conditioning manifold-valued semimartingales. We describe semimartingales on manifolds through the so-called "rolling without slipping" (Eels-Elworthy-Malliavin) construction: Mapping an R^d-valued semimartingale to the frame bundle of the manifold and then projectiong it back to the manifold.

In the talk I'll briefly discuss the construction of manifold-valued semimartingales and then move to the simulation of bridge processes through guiding.

The Compound Poisson Process (CPP) is a mathematical model used to describe phenomena in medicine, reliability, risk and catastrophe bonds. In this work, we developed four methods to offer the confidence intervals of the homogeneous CPP. We presented the exact method for constructing the confidence intervals of CPP by using the Renewal Reward Theorem and the properties of the parameter estimators. Using the delta method, the other two approaches of CPP's confidence intervals were estimated in both the original and logarithm scales. Furthermore, a special case of CPP with exponential distribution was discussed in detail.

We presented our four proposed methods with the numerical simulation and compared these methods to the referential method. The simulated data was synthesised by combining a Poisson process with several non-negative distributions, including discrete and continuous. The delta method in the logarithm scale achieved the highest coverage compared to the other methods. Furthermore, this method outperformed the other methods when the sample size was small. This method has been proven to be effective and was recommended to be used in practice to achieve a more precise confidence interval.