Jump-diffusion Stochastic Process Research Articles

Stochastic transmission in epidemiological models.

Recent empirical evidence suggests that the transmission coefficient in susceptible-exposed-infected-removed-like (SEIR-like) models evolves with time, presenting random patterns, and some stylized facts, such as mean-reversion and jumps. To address such observations we propose the use of jump-diffusion stochastic processes to parameterize the transmission coefficient in an SEIR-like model that accounts for death and time-dependent parameters. We provide a detailed theoretical analysis of the proposed model proving the existence and uniqueness of solutions as well as studying its asymptotic behavior. We also compare the proposed model with some variations possibly including jumps. The forecast performance of the considered models, using reported COVID-19 infections from New York City, is then tested in different scenarios. Despite the simplicity of the epidemiological model, by considering stochastic transmission, the forecasted scenarios were fairly accurate.

Solutions to the jump-diffusion linear stochastic differential equations

The jump-diffusion stochastic process is one of the most common forms in reality (such as wave propagation, noise propagation, turbulent flow, etc.), and researchers often refer to them in models of random processes such as Wiener process, Levy process, Ito-Hermite process, in research of G. D. Nunno, B. Oksendal, F. B. Hanson, etc. In our research, we have reviewed and solved three problems: (1) Jump-diffusion process (also known as the Ito-Levy process); (2) Solve the differential equation jump-diffusion random linear, in the case of one-dimensional; (3) Calculate the Wiener-Ito integral to the random Ito-Hermite process. The main method for dealing with the problems in our presentation is the Ito random-integrable mathematical operations for the continuous random process associated with the arbitrary differential jump by the Poisson random measure. This study aims to analyse the basic properties of jump-diffusion process that are solutions to the jump-diffusion linear stochastic differential equations: dX(t) = [a (t)X (t􀀀)+A(t)]dt + [b (t)X (t􀀀 ∫ )+B(t)]dW (t) + R0 [g (t; z)X (t􀀀)+G(t; z)] ¯N (dt;dz) with a set of stochastic continuous functions fa;b ;g ;A;B;Gg and assuming that the compensated Poisson process ¯N (t; z) is independent of the Wiener process W(t). Derived from the Ito-Hermite formulas for the Ito-Hermite process and for the Ito-Levy process class we presented the results for the differential and multiple stochastic integration for the Ito- Hermite process. We also provided a separation method to solve jump-diffusion linear differential equations.

Data-driven inference for stationary jump-diffusion processes with application to membrane voltage fluctuations in pyramidal neurons

The emergent activity of biological systems can often be represented as low-dimensional, Langevin-type stochastic differential equations. In certain systems, however, large and abrupt events occur and violate the assumptions of this approach. We address this situation here by providing a novel method that reconstructs a jump-diffusion stochastic process based solely on the statistics of the original data. Our method assumes that these data are stationary, that diffusive noise is additive, and that jumps are Poisson. We use threshold-crossing of the increments to detect jumps in the time series. This is followed by an iterative scheme that compensates for the presence of diffusive fluctuations that are falsely detected as jumps. Our approach is based on probabilistic calculations associated with these fluctuations and on the use of the Fokker–Planck and the differential Chapman–Kolmogorov equations. After some validation cases, we apply this method to recordings of membrane noise in pyramidal neurons of the electrosensory lateral line lobe of weakly electric fish. These recordings display large, jump-like depolarization events that occur at random times, the biophysics of which is unknown. We find that some pyramidal cells increase their jump rate and noise intensity as the membrane potential approaches spike threshold, while their drift function and jump amplitude distribution remain unchanged. As our method is fully data-driven, it provides a valuable means to further investigate the functional role of these jump-like events without relying on unconstrained biophysical models.

The use of real option in condition-based maintenance scheduling for wind turbines with production and deterioration uncertainties

Preventive maintenance planning is an important problem for the handling of energy production systems with high down time costs. Throughout the last decade different maintenance strategies have been developed and optimized in order to minimize operational and maintenance costs whilst conserving and improving the system reliability and productivity. Preventive maintenance strategies are usually based on the monitoring and the prediction of the system behavior and its deterioration process. However, some industrial systems may be operating under a dynamic environment and/or variable working conditions. In this case both the deterioration and the production processes may not be deterministic and incorporate different types of uncertainties. In this paper, we consider the case of a preventive maintenance strategy for a production system subject to uncertainty. For this system, a decision-making procedure for condition-based maintenance planning is proposed. In order to consider uncertainty in production and deterioration processes, these latter are modeled by non-monotonic stochastic processes. The modeling of deterioration processes by means of jump-diffusion stochastic processes has been proposed in our previous work. In this paper, a decision-making approach for preventive maintenance strategies is proposed. Knowing the remaining useful life of a system, a simulation-based real options analysis is used in order to determine the best date to maintain. Considering a case study of a wind turbine with PHM structure, the decision-making approach is described and tested through an empirical example.

Market calibration under a long memory stochastic volatility model

ABSTRACTIn this article, we study a long memory stochastic volatility model (LSV), under which stock prices follow a jump-diffusion stochastic process and its stochastic volatility is driven by a continuous-time fractional process that attains a long memory. LSV model should take into account most of the observed market aspects and unlike many other approaches, the volatility clustering phenomenon is captured explicitly by the long memory parameter. Moreover, this property has been reported in realized volatility time-series across different asset classes and time periods. In the first part of the article, we derive an alternative formula for pricing European securities. The formula enables us to effectively price European options and to calibrate the model to a given option market. In the second part of the article, we provide an empirical review of the model calibration. For this purpose, a set of traded FTSE 100 index call options is used and the long memory volatility model is compared to a popular pricing approach – the Heston model. To test stability of calibrated parameters and to verify calibration results from previous data set, we utilize multiple data sets from NYSE option market on Apple Inc. stock.

COMPUTATIONAL ASPECTS OF MONTE-CARLO SIMULATIONS OF THE FIRST PASSAGE TIME FOR MULTIVARIATE TRANSFORMED BROWNIAN MOTIONS WITH JUMPS

Many problems in science, engineering, and finance require the information on the first passage time (FPT) of a stochastic process. Mathematically, such problems are often reduced to the evaluation of the probability density of the time for such a process to cross a certain level, a boundary, or to enter a certain region. While in other areas of applications the FPT problem can often be solved analytically, in finance we usually have to resort to the application of numerical procedures, in particular when we deal with jump-diffusion stochastic processes (JDP). In this paper, we propose a Monte-Carlo-based methodology for the solution of the FPT problem in the context of multivariate (and correlated) JDPs. The developed technique provides an efficient tool for a number of applications, including credit risk and option pricing. We demonstrate its applicability to the analysis of default rates and default correlations of several different, but correlated firms via a set of empirical data.

A Two-Country Discontinuous General Equilibrium Model

The aim of this paper is to develop a continuous time general equilibrium model for a two country Lucas type economy. The model assumes that the output in the two countries follows a jump-diffusion stochastic process. We obtain the results concerning the evaluation of financial assets, the determination of the exchange rate, of the interest rate, and of the risk premium in this two-country economy.

Asset Pricing in a Two-Country Discontinuous General Equilibrium Model

The aim of this paper is to develop a framework for asset pricing in a continuous time general equilibrium model for a two country Lucas type economy. The model assumes that the output in the two countries follows a jump-diffusion stochastic process characterized by constant growth rates and volatilities and by log-normal amplitude of the jumps. Using this specification we deduce the fundamental evaluation equations for financial assets as well as a formula for the price of exchange rate options in this economy.

Hedging Adverse Bioclimatic Conditions Employing a Short Condor Position

AbstractWeather derivatives are a relatively new form of financial security, providing firms with the ability to hedge the impact of weather related risks to their activities. Participants in the energy industry have employed standardized temperature contracts trading on organized exchanges since 1999, and the availability and use of non-standardized contracts designed for specialized weather related risks is growing dramatically. The primary goal of this paper is to consider the potential design and use of a weather contract to hedge the risks faced in viticulture as measured by bioclimatic indices. Specifically we examine the Winkler and Huglin bioclimatic indices over a 43 year period for the Niagara region of Ontario, Canada's largest wine producing region, and identify a mixed jump diffusion stochastic process for cumulative growing season index values. We then employ Monte Carlo simulation to derive a range of benchmark prices for a “short condor” contract employing the Huglin index as the underlying variable. The results show that valuable hedging opportunities can be provided by such contracts. (JEL Classification: G13, G32, Q14, Q51, Q54)

STOCHASTIC INTENSITY MODELING FOR STRUCTURED CREDIT EXOTICS

We propose a class of credit models where we model default intensity as a jump-diffusion stochastic process. We demonstrate how this class of models can be specialised to value multi-asset derivatives such as CDO and CDO2 in an efficient way. We also suggest how it can be adapted to the pricing of option on tranche and leverage tranche deals. We discuss how the model performs when calibrated to the market.