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Brownian motion in one or more dimensions is extensively used as a stochastic process to model natural and engineering signals, as well as financial data. Most works dealing with multidimensional Brownian motion assume that each component is independent. In this article, we investigate a model of correlated Brownian motion in $\mathbb{R}^2$, where the individual components are not necessarily independent. We explore various statistical properties of the process under consideration, going beyond the conventional analysis of the second moment. Our particular focus lies on investigating the distribution of turning angles. This distribution reveals particularly interesting characteristics for processes with dependent components that are relevant to applications in diverse physical systems. Theoretical considerations are supported by numerical simulations and analysis of two real-world datasets: the financial data of the Dow Jones Industrial Average and the Standard and Poor's 500, and trajectories of polystyrene beads in water. Finally, we show that the model can be readily extended to trajectories with correlations that change over time., Comment: 13 pages, 10 figures

The Langevin equation is a common tool to model diffusion at a single-particle level. In non-homogeneous environments, such as aqueous two-phase systems or biological condensates with different diffusion coefficients in different phases, the solution to a Langevin equation is not unique unless the interpretation of stochastic integrals involved is selected. We analyze the diffusion of particles in such systems and evaluate the mean, the mean square displacement, and the distribution of particles, as well as the variance of the time-averaged mean-squared displacements. Our analytical results provide a method to choose the interpretation parameter from single particle tracking experiments.

The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due to which the motion changes along the trajectories. Such effects manifest themselves as spatiotemporal correlations. Despite the broad occurrence of heterogeneous complex systems in nature, their analysis is still quite poorly understood and tools to model them are largely missing. We contribute to tackling this problem by employing an integral representation of Mandelbrot's fractional Brownian motion that is compliant with varying motion parameters while maintaining long memory. Two types of switching fractional Brownian motion are analysed, with transitions arising from a Markovian stochastic process and scale-free intermittent processes. We obtain simple formulas for classical statistics of the processes, namely the mean squared displacement and the power spectral density. Further, a method to identify switching fractional Brownian motion based on the distribution of displacements is described. A validation of the model is given for experimental measurements of the motion of quantum dots in the cytoplasm of live mammalian cells that were obtained by single-particle tracking.

In this paper, we generalise the results presented in the literature for the ruin probability for the insurer--reinsurer model under a pro-rata reinsurance contract. We consider claim amounts that are described by a phase-type distribution that includes exponential, mixture of exponential, Erlang, and mixture of Erlang distributions. We derive the ruin probability formulas with the use of change-of-measure technique and present important special cases. We illustrate the usefulness of the introduced model by fitting it to the real-world loss data. With the use of statistical tests and graphical tools, we show that the mixture of Erlangs is well-fitted to the data and is superior to other considered distributions. This justifies the fact that the presented results can be useful in the context of risk assessment of co-operating insurance companies.

We propose a generalization of the widely used fractional Brownian motion (FBM), memory-multi-FBM (MMFBM), to describe viscoelastic or persistent anomalous diffusion with time-dependent memory exponent $\alpha(t)$ in a changing environment. In MMFBM the built-in, long-range memory is continuously modulated by $\alpha(t)$. We derive the essential statistical properties of MMFBM such as response function, mean-squared displacement (MSD), autocovariance function, and Gaussian distribution. In contrast to existing forms of FBM with time-varying memory exponents but reset memory structure, the instantaneous dynamic of MMFBM is influenced by the process history, e.g., we show that after a step-like change of $\alpha(t)$ the scaling exponent of the MSD after the $\alpha$-step may be determined by the value of $\alpha(t)$ before the change. MMFBM is a versatile and useful process for correlated physical systems with non-equilibrium initial conditions in a changing environment., Comment: 15 pages, 10 figures, RevTeX

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity, or the Hurst exponent. However, recent single particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that can not be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process which preserves the properties of fractional Brownian motion at a single trajectory level, however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical and statistical analysis of fractional Brownian motion with random Hurst exponent. The explicit formulas for probability density function, mean square displacement and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely two-point, uniform and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition which we demonstrate analytically and numerically., Comment: 16 pages, 10 figures

Fractional Brownian motion (FBM) is the only Gaussian self-similar process with stationary increments. Its increment process, called fractional Gaussian noise, is ergodic and exhibits a property of power-like decaying autocorrelation function (ACF) which leads to the notion of long memory. These properties have made FBM important in modelling real-world data recorded in different experiments ranging from biology to telecommunication. These experiments are often disturbed by a noise which source can be just the instrument error. In this paper we propose a rigorous statistical test based on the ACF for FBM with added white Gaussian noise. To this end we derive a distribution of the test statistic which is given explicitly by the generalized chi-squared distribution. This allows us to find critical regions for the test with a given significance level. We check the quality of the introduced test by studying its power and comparing with other tests existing in the literature. We also note that the introduced test procedure can be applied to an arbitrary Gaussian process., Comment: 25 pages, 6 figures

Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as "superstatistics" or "diffusing diffusivity". Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models. We start from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion., Comment: 28 pages, 9 figures, IOP LaTeX

Within the context of the banking-related literature on contingent convertible bonds, we comprehensively formalise the design and features of a relatively new type of insurance-linked security, called a contingent convertible catastrophe bond (CocoCat). We begin with a discussion of its design and compare its relative merits to catastrophe bonds and catastrophe-equity puts. Subsequently, we derive analytical valuation formulae for index-linked CocoCats under the assumption of independence between natural catastrophe and financial markets risks. We model natural catastrophe losses by a time-inhomogeneous compound Poisson process, with the interest-rate process governed by the Longstaff model. By using an exponential change of measure on the loss process, as well as a Girsanov-like transformation to synthetically remove the correlation between the share and interest-rate processes, we obtain these analytical formulae. Using selected parameter values in line with earlier research, we empirically analyse our valuation formulae for index-linked CocoCats. An analysis of the results reveals that the CocoCat prices are most sensitive to changing interest-rates, conversion fractions and the threshold levels defining the trigger times.