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This paper extends the valuation and optimal surrender framework for variable annuities with guaranteed minimum benefits in a Lévy equity market environment by incorporating a stochastic interest rate described by the Hull-White model. This approach frames a more dynamic and realistic financial setting compared to previous literature. We exploit a robust valuation mechanism employing a hybrid numerical method that merges tree methods for interest rate modeling with finite difference techniques for the underlying asset price. This method is particularly effective for addressing the complexities of variable annuities, where periodic fees and mortality risks are significant factors. Our findings reveal the influence of stochastic interest rates on the strategic decision-making process concerning the surrender of these financial instruments. Through comprehensive numerical experiments, and by comparing our results with those obtained through the Longstaff-Schwartz Monte Carlo method, we illustrate how our refined model can guide insurers in designing contracts that equitably balance the interests of both parties. This is particularly relevant in discouraging premature surrenders while adapting to the realistic fluctuations of financial markets. Lastly, a comparative statics analysis with varying interest rate parameters underscores the impact of interest rates on the cost of the optimal surrender strategy, emphasizing the importance of accurately modeling stochastic interest rates. [ABSTRACT FROM AUTHOR]

We consider a continuous-time two-dimensional risk model, in which the claims from the two lines of insurance businesses satisfy an extensive asymptotic independence structure and the stochastic return is driven by a geometric Lévy process. Under a mild technical condition regarding the Laplace exponent of the Lévy process, we obtain explicit asymptotic expansions for both finite-time and infinite-time ruin probabilities when the claim sizes have regularly varying distributions. [ABSTRACT FROM AUTHOR]

This paper studies the Cramér–Lundberg asymptotics of the ruin probability for a model in which the reserve level process is described by a spectrally-positive light-tailed Markov additive process. By applying a change-of-measure technique in combination with elements from Wiener-Hopf theory, the exact asymptotics of the ruin probability are expressed in terms of the model primitives. In addition a simulation algorithm of generalized Siegmund type is presented, under which the returned estimate of the ruin probability has bounded relative error. Numerical experiments show that, compared to direct estimation, this algorithm greatly reduces the number of runs required to achieve an estimate with a given accuracy. The experiments also reveal that our asymptotic results provide a good approximation of the ruin probability even for relatively small initial surplus levels. [ABSTRACT FROM AUTHOR]

This paper endeavours to directly establish the observability inequality of backward stochastic heat equations involving a Lévy process for measurable sets. As an immediate application, we attain the approximate controllability of forward stochastic heat equations featuring a Lévy process. Subsequently, we embark upon the formulation of a nuanced optimization problem, encompassing both the determination of an optimal actuator location and its associated minimum norm control. This formulation is elegantly cast as a two-person zero-sum game problem. We then proceed to articulate a set of conditions–both sufficient and necessary–required for optimizing the solution through the prism of a Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem. [ABSTRACT FROM AUTHOR]

By partly invoking the refined basic coupling for the Lévy process, we establish quantitative estimates of distributions associated with two Lévy driven SDEs with different drifts, which are partially dissipative. This quantitative result is powerful not only to investigate the exponentially contractive property of distributions corresponding to partially dissipative McKean-Vlasov SDEs with jumps but also to handle the long time behavior of time-inhomogeneous SDEs with jumps. Our approach also works very well for two Lévy driven SDEs with different drift terms, which satisfy monotone and Lyapunov conditions. As important applications of the result in this aspect, under monotone and Lyapunov conditions, we obtain succinctly the exponential decay and the locally exponential contractivity as well as the exponential ergodicity for McKean-Vlasov SDEs driven by Lévy noises. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we investigate the trajectory fitting estimation for the integrated Ornstein-Uhlenbeck process driven by the Lévy process based on continuous observations. The strong consistency and asymptotic distributions of the estimator are obtained by the strong law of large numbers and the inner clock property of the α-stable stochastic integral. [ABSTRACT FROM AUTHOR]

In this paper, we study McKean–Vlasov stochastic differential equations driven by Lévy processes. Firstly, under the non-Lipschitz condition which include classical Lipschitz conditions as special cases, we establish the existence and uniqueness for solutions of McKean–Vlasov stochastic differential equations using Carathéodory approximation. Then under certain averaging conditions, we establish a stochastic averaging principle for McKean–Vlasov stochastic differential equations driven by Lévy processes. We find that the solutions to stochastic systems concerned with Lévy noise can be approximated by solutions to averaged McKean–Vlasov stochastic differential equations driven by Lévy processes in the sense of convergence in p th moment. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior for multi-scale stochastic differential equations driven by Lévy processes. The optimal strong convergence order 1/2 is obtained by studying the regularity estimates for the solution of Poisson equation with polynomial growth coefficients, and the optimal weak convergence order 1 is got by using the technique of Kolmogorov equation. The main contribution is that the obtained results can be applied to a class of multi-scale stochastic differential equations with monotonicity coefficients, as well as the driven processes can be the general Lévy processes, which seems new in the existing literature. [ABSTRACT FROM AUTHOR]

A system of n$$ n $$ components is here considered, with component deterioration modeled by non decreasing time‐scaled Lévy processes. When a component fails, a sudden change in the time‐scaling functions of the surviving components is induced, which makes the components stochastically dependent. We compute the reliability function of coherent systems under this new dependence model. We next study the distribution of the ordered failure times, and establish some positive dependence properties. We also provide stochastic comparison results in the usual multivariate stochastic order between failure times of two dependence models with different parameters. Finally, some numerical experiments illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Univariate Lévy processes have become quite common in the reliability literature for modeling accumulative deterioration. In case of correlated deterioration indicators, several possibilities have been suggested for modeling their dependence. The point of this article is the study of three different dependence models: use of a regular copula, superposition of independent univariate Lévy processes and use of a Lévy copula. The three methods are first presented and analyzed. In this way, it is shown that the multivariate process constructed through an ordinary copula cannot have independent increments in general, that is, it is not a Lévy process. The impact of a wrong choice for the model is next explored, based on data simulated from one model and next adjusted to all three models. It is shown that a wrong model can lead to either overestimate or underestimate the reliability function, which could be problematic in an application context. [ABSTRACT FROM AUTHOR]

In this paper we give a solution to the quickest drift change detection problem for a multivariate Lévy process consisting of both continuous (Gaussian) and jump components in the Bayesian approach. We do it for a general 0-modified continuous prior distribution of the change point as well as for a random post-change drift parameter. Classically, our criterion of optimality is based on a probability of false alarm and an expected delay of the detection, which is then reformulated in terms of a posterior probability of the change point. We find a generator of the posterior probability, which in case of general prior distribution is inhomogeneous in time. The main solving technique uses the optimal stopping theory and is based on solving a certain free-boundary problem. We also construct a Generelized Shiryaev-Roberts statistic, which can be used for applications. The paper is supplemented by two examples, one of which is further used to analyze Polish life tables (after proper calibration) and detect the drift change in the correlated force of mortality of men and women jointly. [ABSTRACT FROM AUTHOR]

Hybrid energy harvesting systems are broadly applied in various fields due to the advantage of improving energy harvesting efficiency. In actual environment, there are many complex phenomena exhibiting jump, flights, rare transition features, and intermittent features, which can be described by systems subjected to non-Gaussian Lévy process. Sometimes, it is difficult to build mathematical models of complex hybrid energy harvesting systems precisely, especially for those driven by non-Gaussian Lévy noise. With the development of simulation capabilities and observing techniques recently, massive noise measurement data or simulating data can be feasibly obtained and there are many existing techniques devoted to discovering governing laws from abundant data. In this paper, we aim to extract the system equations from observed data influenced by non-Gaussian Lévy noise via using a data-driven method. The expressions of drift term, diffusion term and Lévy term can be approximated with the help of Fokker-Planck equation and non-local Kramers-Moyal formulae, and the coefficients of the expressions are learned by utilizing a sparse regression approach in the least square sense. Three examples are given to demonstrate the effectiveness of the method. Results show that the approach can well be applied to not only hybrid energy harvesting systems under Gaussian Brownian process but also the systems subjected to non-Gaussian Lévy process. Additionally, the relations between the demarcation parameter and an indicator denoted as Ratio for different time steps are analyzed, and results demonstrate that the indicator can be regarded as the criterion of selecting the appropriate demarcation parameter. [ABSTRACT FROM AUTHOR]

In this paper we consider the following optimal stopping problem V A ω (s) = sup τ ∈ T E s [ e - ∫ 0 τ ω (S w) d w g (S τ) ] , where the process S t is a jump-diffusion process, T is a family of stopping times while g and ω are fixed payoff function and discount function, respectively. In a financial market context, if g (s) = (K - s) + or g (s) = (s - K) + and E is the expectation taken with respect to a martingale measure, V A ω (s) describes the price of a perpetual American option with a discount rate depending on the value of the asset process S t . If ω is a constant, the above problem produces the standard case of pricing perpetual American options. In the first part of this paper we find sufficient conditions for the convexity of the value function V A ω (s) . This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of V A ω (s) . We also present a put-call symmetry for American options with asset-dependent discounting. In the case when S t is a spectrally negative geometric Lévy process we give exact expressions using the so-called omega scale functions introduced in [30]. We show that the analysed value function satisfies the HJB equation and we give sufficient conditions for the smooth fit property as well. Finally, we present a few examples for which we obtain the analytical form of the value function V A ω (s) . [ABSTRACT FROM AUTHOR]

Generalised hyperbolic (GH) processes are a class of stochastic processes that are used to model the dynamics of a wide range of complex systems that exhibit heavy-tailed behavior, including systems in finance, economics, biology, and physics. In this paper, we present novel simulation methods based on subordination with a generalised inverse Gaussian (GIG) process and using a generalised shot-noise representation that involves random thinning of infinite series of decreasing jump sizes. Compared with our previous work on GIG processes, we provide tighter bounds for the construction of rejection sampling ratios, leading to improved acceptance probabilities in simulation. Furthermore, we derive methods for the adaptive determination of the number of points required in the associated random series using concentration inequalities. Residual small jumps are then approximated using an appropriately scaled Brownian motion term with drift. Finally the rejection sampling steps are made significantly more computationally efficient through the use of squeezing functions based on lower and upper bounds on the Lévy density. Experimental results are presented illustrating the strong performance under various parameter settings and comparing the marginal distribution of the GH paths with exact simulations of GH random variates. The new simulation methodology is made available to researchers through the publication of a Python code repository. [ABSTRACT FROM AUTHOR]

With view on global warming and the ongoing climate change, weather derivatives play an increasingly important role for many companies and financial investors, as they constitute useful hedging instruments against disadvantageous weather conditions. In this paper, we present a new temperature model based on generalized Langevin equations driven by Lévy processes. The proposed arithmetic approach captures numerous stylized facts of empirical temperature behavior like seasonal variations, time-dependent volatilities, memory effects, heavy tails and skewness. We further derive a representation for the related meteorological temperature forecast curve and infer the risk-neutral price dynamics of temperature derivatives like CAT, CDD and HDD futures. We finally deduce the minimal variance hedging portfolio in a specific temperature futures market by an application of a stochastic maximum principle and present several practical examples. [ABSTRACT FROM AUTHOR]

In this paper, we present an Lq(Lp)-regularity theory for parabolic equations of the form: ∂tu(t,x)=... represents anisotropic non-local operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: .... To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calderón-Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic non-local operators and parabolic equations with isotropic non-local operators. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic has triggered a groundbreaking reliance on mathematical modelling as an important tool for studying and managing the spread of the virus since its emergence. Public health preventive measures such as vaccination and therapeutics can effectively reduce or eradicate an infectious disease. This work investigates these two strategies for controlling the COVID-19 epidemic through a stochastic epidemiological modelling approach. The existence and uniqueness of a positive solution of the stochastic system is studied. A priori estimates of the vaccination and treatment controls are established. Sufficient and necessary conditions are obtained for the near-optimal control problem of the stochastic model using the maximum condition of the Hamiltonian function and the Ekeland principle. Finally, to support our theoretical results, numerical simulations for a combination of optimized vaccination and treatment strategies were presented to understand the challenges posed by COVID-19 in Brazil. [ABSTRACT FROM AUTHOR]

We study a one-dimensional kinetic stochastic model driven by a Lévy process with a non-linear time-inhomogeneous drift. More precisely, the process (V,X) is considered, where X is the position of the particle and its velocity V is the solution to a stochastic differential equation with a drift of the form t-ßF(v). The driving process is a stable Lévy process of index a, the function F satisfies a homogeneity condition and ß is a real number. The behaviour in large time of the process (V,X) is analysed. A result concerning the moments estimates of the velocity process is one of main tools. [ABSTRACT FROM AUTHOR]

We deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication − a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication. [ABSTRACT FROM AUTHOR]

In a mortgage valuation model, the early termination (i.e., prepayment and default) hazard rates and the recovery rate can be specified as multivariate affine functions that include the correlated stochastic state variables. For good capturing of the distributions for state variables, we specify that the state variables follow Lévy models. Accordingly, the early termination hazard rates and the recovery rate also follow Lévy models. Three popular Lévy models, the normal, Variance Gamma (VG), and Negative Inverse Normal (NIG), were used to obtain the closed-form pricing formula for a mortgage. We also conduct numerical applications. Our results reveal the following findings: first, VG model is better than the normal and NIG models in fitting the actual distributions of the interest rate and the housing return rate. Thus, mortgage valuation using a VG model should be better than that using the other two models. Second, the mortgage value estimated by the normal model is the lowest among the three Lévy models. Third, a prepayment hazard rate with a deterministic value (e.g., using the PSA prepayment model) could result in an unreasonable mortgage duration. Fourth, the mortgage convexity calculated by the normal model is higher than that in the other two Lévy models. Our general pricing formula for a mortgage as described in this study can help market participants accurately value mortgages and effectively manage their risks. [ABSTRACT FROM AUTHOR]

Lévy processes are useful tools for analysis and modeling of jump‐diffusion processes. Such processes are commonly used in the financial and physical sciences. One approach to building new Lévy processes is through subordination, or a random time change. In this work, we discuss and examine a type of multiply subordinated Lévy process model that we term a deep variance gamma (DVG) process, including estimation and inspection methods for selecting the appropriate level of subordination given data. We perform an extensive simulation study to identify situations in which different subordination depths are identifiable and provide a rigorous theoretical result detailing the behavior of a DVG process as the levels of subordination tend to infinity. We test the model and estimation approach on a data set of intraday 1‐min cryptocurrency returns and show that our approach outperforms other state‐of‐the‐art subordinated Lévy process models. [ABSTRACT FROM AUTHOR]

The local time (LT) of a given stochastic process { X t : t ≥ 0 } is defined informally as L X ⁢ (t , x) = ∫ 0 t δ x ⁢ (X s) ⁢ d s , where δ x denotes the Dirac function; actually, it counts the duration of the process's stay at 푥 up to time 푡. Using an approximation approach, we study the existence and the regularity of the LT process for two kinds of stochastic processes. The first type is the stochastic process defined by the indefinite Wiener integral X t := ∫ 0 t f ⁢ (u) ⁢ d B u for a given deterministic function f ∈ L 2 ([ 0 , + ∞ [) , and secondly, for Lévy type processes, i.e. ones that are stationary and with independent increments. [ABSTRACT FROM AUTHOR]

This paper constructs a theoretical framework for multi-population mortality modeling via generalized linear models and Lévy stochastic perturbations driven by a common Brownian motion and idiosyncratic factors to capture the mortality shocks. By having Lévy stochastic perturbations, our model admits various jump types, which is increasingly important for capturing mortality shocks such as pandemics, particularly when they affect various populations differently. At the same time, the proposed model allows a novel dependence structure of multiple populations, which is essential when it comes to the development of multi-population or joint-life products in the context of mortality shocks. In our empirical investigations, the mortality experiences of male and female lives in the UK and Japan are used. Compared with pure Poisson-generalized linear models, the proposed multi-population model shows superiority in predicting future mortality rates. [ABSTRACT FROM AUTHOR]

We develop a novel Monte Carlo algorithm for the vector consisting of the supremum, the time at which the supremum is attained, and the position at a given (constant) time of an exponentially tempered Lévy process. The algorithm, based on the increments of the process without tempering, converges geometrically fast (as a function of the computational cost) for discontinuous and locally Lipschitz functions of the vector. We prove that the corresponding multilevel Monte Carlo estimator has optimal computational complexity (i.e. of order $\varepsilon^{-2}$ if the mean squared error is at most $\varepsilon^2$) and provide its central limit theorem (CLT). Using the CLT we construct confidence intervals for barrier option prices and various risk measures based on drawdown under the tempered stable (CGMY) model calibrated/estimated on real-world data. We provide non-asymptotic and asymptotic comparisons of our algorithm with existing approximations, leading to rule-of-thumb principles guiding users to the best method for a given set of parameters. We illustrate the performance of the algorithm with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we present a new bivariate model for the joint description of the Bitcoin prices and the media attention to Bitcoin. Our model is based on the class of the Lévy processes and is able to realistically reproduce the jump‐type dynamics of the considered time series. We focus on the low‐frequency setup, which is for the Lévy‐based models essentially more difficult than the high‐frequency case. We design a semiparametric estimation procedure for the statistical inference on the parameters and the Lévy measures of the considered processes. We show that the dynamics of the market attention can be effectively modelled by the Lévy processes with finite Lévy measures, and propose a data‐driven procedure for the description of the Bitcoin prices. [ABSTRACT FROM AUTHOR]

Summary: In this study, we demonstrate that engineered topological features can distinguish time series sampled from different stochastic processes with different noise characteristics, in both balanced and unbalanced sampling schemes. We compare our classification results against the results of the same classification on features coming from descriptive statistics and the wavelet transform. We conclude that machine learning models built on engineered topological features alone perform consistently better than those built on the standard statistical and wavelet features for time series classification tasks. We also apply dimension reduction techniques to our engineered features and compare the result of our classification models before and after dimensionality reduction. Finally, we also show that in our calculations of the engineered topological features, employing parallel computing methods does yield significant improvements in run time and memory footprint. [ABSTRACT FROM AUTHOR]

In this paper, we derive stochastic Leibniz formulas for Volterra integrals under enlarged filtrations. We investigate both pure-jump and Brownian Volterra processes under diverse initially enlarged filtration approaches. In these setups, we compare the ordinary with the stochastic (Doleans-Dade) exponential of a Volterra process and provide the corresponding martingale conditions. We also consider backward stochastic Volterra integral equations (BSVIEs) under enlarged filtrations and obtain the related solution formulas. We finally propose an anticipative Heath Jarrow Morton (HJM) forward rate model of Volterra-type and infer the associated bond price representation. In an introductory section, we compile various facts on deterministic and stochastic Leibniz formulas for parameter integrals. [ABSTRACT FROM AUTHOR]

In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by α -stable processes with α ∈ (1 , 2). The existence part is based on Euler's approximation as Hao et al. (2021), while, the uniqueness is based on the Schauder estimates in Besov spaces for nonlocal Fokker–Planck equations. For the existence, we only assume the drift being continuous in the density variable. For the weak uniqueness, the drift is assumed to be Lipschitz in the density variable, while for the strong uniqueness, we also need to assume the drift being β 0 -order Hölder continuous in the spatial variable, where β 0 ∈ (1 − α / 2 , 1). [ABSTRACT FROM AUTHOR]

In this paper, we extend the Fourier cosine expansion (COS) method to the pricing of {foreign exchange} target redemption note (FX-TARN), a popular exotic currency option. We take the FX spot rate and the cumulated positive cash flow as two state variables and factor the joint distribution by two marginals that can be approximated by Fourier cosine expansions. To recover the Fourier coefficients recursively, we approximate the two-dimensional integration by higher-order quadratures such as Gauss-Legendre or Clenshaw-Curtis quadrature for the integration over the spot rate. We derive the analytical formulas for the price under different knock-out types. We demonstrate that fast Fourier transform (FFT) can be employed to obtain the Fourier coefficients efficiently. We also evaluate the performance and accuracy of the method through a number of numerical experiments.

Iterated stochastic integrals of nonrandom integrands are constructed in the two-dimensional case. They are applied to the velocity fluctuations in a two-dimensional flow and mean kinetic energy of the velocity fluctuations is discussed.

In this paper we implement a combination of data-science and fuzzy theory to improve the classical Barndorff-Nielsen and Shephard model, and implement this to analyze the S&P 500 index. We preprocess the index data based on fuzzy theory. After that, S&P 500 stock index data for the past 10 years are analyzed, and a deterministic parameter is extracted using various machine and deep learning methods. The results show that the new model, where fuzzy parameters are incorporated, can incorporate the long-term dependence in the classical Barndorff-Nielsen and Shephard model. The modification is based on only a few changes compared to the classical model. At the same time, the resulting analysis effectively captures the stochastic dynamics of the stock index time series. [ABSTRACT FROM AUTHOR]

We present a Monte Carlo approach to pairs trading on mean-reverting spreads modelled by Lévy-driven Ornstein-Uhlenbeck processes. Specifically, we focus on using a variance gamma driving process, an infinite activity pure jump process to allow for more flexible models of the price spread than is available in the classical model. However, this generalization comes at the cost of not having analytic formulas, so we apply Monte Carlo methods to determine optimal trading levels and develop a variance reduction technique using control variates. Within this framework, we numerically examine how the optimal trading strategies are affected by the parameters of the model. In addition, we extend our method to bivariate spreads modelled using a weak variance alpha-gamma driving process, and explore the effect of correlation on these trades. [ABSTRACT FROM AUTHOR]

We consider a particle evolving in the quadratic potential and subject to a time-inhomogeneous frictional force and to a random force. The couple of its velocity and position is solution to a stochastic differential equation driven by a symmetric $ \alpha $-stable Lévy process with $ \alpha \in (1,2] $ and the frictional force is of the form $ t^{-\beta}\text{sgn}(v)|v|^\gamma $. We identify three regimes for the behavior in long-time of the couple velocity-position with a suitable rescaling, depending on the balance between the frictional force and the index of stability $ \alpha $ of the noise [ABSTRACT FROM AUTHOR]

In the probability theory, selfdecomposable or class L0distributions play an important role as they are limit distributions of normalized partial sums of sequences of independent, not necessarily identically distributed, random variables. The class L0 is quite large and includes many known classical distributions. For this note, the most important feature of the selfdecomposable variables are their random integral representation with respect to a Lévy process. From those random integral representations we get the equality of logarithms of some characteristic functions. These allow us to get formulas for some definite integrals; some of them were previously unknown, and some are rarely quoted in popular tables of integrals and series. [ABSTRACT FROM AUTHOR]

As per the projections of conventional credit risk structured model, the risky asset values tend to adhere to the geometric Brownian motion. On the contrary, the risky asset values remain a non-continuous and dynamic ones and jump based on the conditions. Is not possible to measure the real Knight Uncertainty risks in financial markets with the help of a single probability measure. In this background, the current research work analyzes a structural credit risk model that belongs to Levy market under Knight Uncertainty. With the help of Lévy-Laplace exponent, the authors developed a dynamic pricing model in this study and acquired the price intervals for default probability, stock value and the bond value of enterprise. To be specific, the study intended to establish explicit solutions for three value processes, discussed earlier, with an assumption that the jump process follows a log-normal distribution. At the end, the study also conducted numerical analysis to understand the crucial role played by Knight Uncertainty upon the pricing of default probability and the stock value of the enterprise. [ABSTRACT FROM AUTHOR]

We consider reflected backward stochastic differential equations driven by Teugels martingales associated with a Lévy process, in which the barrier process is optional with regulated trajectories (i.e., trajectories with left and right finite limits), which is assumed to be right upper semi-continuous. We prove the existence and uniqueness of such equations by using the predictable representations for Lévy processes due to Nualart and Schoutens, and some tools from the general theory of processes such as Mertens decomposition of optional strong supermartingales. We also discuss the case where the barrier is assumed to be completely irregular, and we establish an infinitesimal characterization of the solution in terms of a value process to an extension of the optimal stopping problem. [ABSTRACT FROM AUTHOR]

The progression of disease for an individual can be described mathematically as a stochastic process. The individual experiences a failure event when the disease path first reaches or crosses a critical disease level. This happening defines a failure event and a first hitting time or time-to-event, both of which are important in medical contexts. When the context involves explanatory variables then there is usually an interest in incorporating regression structures into the analysis and the methodology known as threshold regression comes into play. To date, most applications of threshold regression have been based on parametric families of stochastic processes. This paper presents a semiparametric form of threshold regression that requires the stochastic process to have only one key property, namely, stationary independent increments. As this property is frequently encountered in real applications, this model has potential for use in many fields. The mathematical underpinnings of this semiparametric approach for estimation and prediction are described. The basic data element required by the model is a pair of readings representing the observed change in time and the observed change in disease level, arising from either a failure event or survival of the individual to the end of the data record. An extension is presented for applications where the underlying disease process is unobservable but component covariate processes are available to construct a surrogate disease process. Threshold regression, used in combination with a data technique called Markov decomposition, allows the methods to handle longitudinal time-to-event data by uncoupling a longitudinal record into a sequence of single records. Computational aspects of the methods are straightforward. An array of simulation experiments that verify computational feasibility and statistical inference are reported in an online supplement. Case applications based on longitudinal observational data from The Osteoarthritis Initiative (OAI) study are presented to demonstrate the methodology and its practical use. [ABSTRACT FROM AUTHOR]

This paper models stochastic process of price time series of $ CSI $ $ 300 $ index in Chinese financial market, analyzes volatility characteristics of intraday high-frequency price data. In the new generalized Barndorff-Nielsen and Shephard model, the lag caused by asynchrony of market information and market microstructure noises are considered, and the problem of lack of long-term dependence is solved. To speed up the valuation process, several machine learning and deep learning algorithms are used to estimate parameter and evaluate forecast results. Tracking historical jumps of different magnitudes offers promising avenues for simulating dynamic price processes and predicting future jumps. Numerical results show that the deterministic component of stochastic volatility processes would always be captured over short and longer-term windows. Research finding could be suitable for influence investors and regulators interested in predicting market dynamics based on high-frequency realized volatility.

A stochastic SIRS system with $ \mathrm {L\acute{e}vy} $ process is formulated in this paper, and the model incorporates the saturated incidence and vaccination strategies. Due to the introduction of $ \mathrm {L\acute{e}vy} $ jump, the jump stochastic integral process is a discontinuous martingale. Then the Kunita's inequality is used to estimate the asymptotic pathwise of the solution for the proposed model, instead of Burkholder-Davis-Gundy inequality which is suitable for continuous martingales. The basic reproduction number $ R_{0}^{s} $ of the system is also derived, and the sufficient conditions are provided for the persistence and extinction of SIRS disease. In addition, the numerical simulations are carried out to illustrate the theoretical results. Theoretical and numerical results both show that $ \mathrm {L\acute{e}vy} $ process can suppress the outbreak of the disease.

This paper deals with the problem of existence and uniqueness of 핃 p -solutions for a backward stochastic differential equation in a filtration that supports Lévy processes with p ∈ (1 , 2) . However, we will focus on when the data satisfy the appropriate integrability conditions and when the coefficient is Lipschitz. [ABSTRACT FROM AUTHOR]

In this paper, we study de Finetti's optimal dividend problem with capital injection under the assumption that the dividend strategies are absolutely continuous. In many previous studies, the process before being controlled was assumed to be a spectrally one-sided Lévy process, however in this paper we use a Lévy process that may have both positive and negative jumps. In the main theorem, we show that a refraction–reflection strategy is an optimal strategy. We also mention the existence and uniqueness of solutions of the stochastic differential equations that define refracted Lévy processes. [ABSTRACT FROM AUTHOR]

In this paper, we study a Lévy risk process consisting of Brownian component together with premiums and claims that are phase type with many phases. Our aim is to approximate the probability of ruin without touching an upper barrier a. In line with this, the study demonstrates that the described Lévy risk process can essentially be replaced with a simpler risk process in which both premiums and claims are phase-type with just few phases. [ABSTRACT FROM AUTHOR]

Let J 1 > J 2 > ⋯ be the ranked jumps of a gamma process τ α on the time interval [ 0 , α ] , such that τ α = ∑ k = 1 ∞ J k . In this paper, we design an algorithm that samples from the random vector (J 1 , ⋯ , J N , ∑ k = N + 1 ∞ J k) . Our algorithm provides an analog to the well-established inverse Lévy measure (ILM) algorithm by replacing the numerical inversion of exponential integral with an acceptance-rejection step. This research is motivated by the construction of Dirichlet process prior in Bayesian nonparametric statistics. The prior assigns weight to each atom according to a GEM distribution, and the simulation algorithm enables us to sample from the N largest random weights of the prior. Then we extend the simulation algorithm to a generalised gamma process. The simulation problem of inhomogeneous processes will also be considered. Numerical implementations are provided to illustrate the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

We consider a Lévy process Y (t) that is not continuously observed, but rather inspected at Poisson( $\omega$) moments only, over an exponentially distributed time $T_\beta$ with parameter $\beta$. The focus lies on the analysis of the distribution of the running maximum at such inspection moments up to $T_\beta$ , denoted by $Y_{\beta,\omega}$. Our main result is a decomposition: we derive a remarkable distributional equality that contains $Y_{\beta,\omega}$ as well as the running maximum process $\bar Y(t)$ at the exponentially distributed times $T_\beta$ and $T_{\beta+\omega}$. Concretely, $\overline{Y}(T_\beta)$ can be written as the sum of two independent random variables that are distributed as $Y_{\beta,\omega}$ and $\overline{Y}(T_{\beta+\omega})$. The distribution of $Y_{\beta,\omega}$ can be identified more explicitly in the two special cases of a spectrally positive and a spectrally negative Lévy process. As an illustrative example of the potential of our results, we show how to determine the asymptotic behavior of the bankruptcy probability in the Cramér–Lundberg insurance risk model. [ABSTRACT FROM AUTHOR]

Growth-fragmentation processes describe the evolution of systems in which cells grow slowly and fragment suddenly. Despite originating as a way to describe biological phenomena, they have recently been found to describe the lengths of certain curves in statistical physics models. In this note, we describe a new growth-fragmentation process connected to random planar maps with faces of large degree, having as a key ingredient the ricocheted stable process recently discovered by Budd. The process has applications to the excursions of planar Brownian motion and Liouville quantum gravity. [ABSTRACT FROM AUTHOR]

We study the almost sure asymptotic behavior of the supremum of the local time for a transient sub-ballistic diffusion in a spectrally negative Lévy environment. More precisely, we provide the proper renormalizations for the extremely large values of the supremum of the local time. This is done by establishing a—so far unknown—connection between the latter and the exponential functional of a Lévy process conditioned to stay positive, which allows us to use the properties of such exponential functionals to characterize the sought behavior. It appears from our results that the renormalization of the extremely large values of the supremum of the local time is determined by the asymptotic behavior of the Laplace exponent of the Lévy environment and is surprisingly greater than the renormalization that was previously known for the recurrent case. Our results show moreover a rich variety of behaviors, which is a new phenomenon, as it does not occur in the discrete setting. [ABSTRACT FROM AUTHOR]

In this paper, we extend the popular Barndorff–Nielsen–Shephard stochastic volatility model to the case of a pure-jump Ornstein–Uhlenbeck equation with non-vanishing stochastic mean-reversion level. Based on this setup, we derive representations for the squared VIX process and related VIX futures prices. Having these results at hand, we introduce an initially enlarged filtration which models the view of a VIX market insider who has knowledge about the future behavior of the stochastic mean-reversion level of the squared volatility process available. In this enlarged filtration framework, we infer an explicit representation for the anticipative VIX process and obtain the associated time dynamics. We finally investigate the pricing of variance swaps under both backward- and forward-looking information. [ABSTRACT FROM AUTHOR]