Search

Value at risk (VaR) is a quantitative measure used to evaluate the risk linked to the potential loss of investment or capital. Estimation of the VaR entails the quantification of prospective losses in a portfolio of investments, using a certain likelihood, under normal market conditions within a specific time period. The objective of this article is to construct a model and estimate the VaR for a diversified portfolio consisting of multiple cash commodity positions driven by standard Brownian motions and jump processes. Subsequently, a thorough analytical estimation of the VaR is conducted for the proposed model. The results are then applied to two distinct commodities—corn and soybean—enabling a comprehensive comparison of the VaR values in the presence and absence of jumps. [ABSTRACT FROM AUTHOR]

This study develops a novel periodic regime-switching model (the PRS model) to improve the forecasting of stock market volatility by accounting for the information from non-trading and trading periods, including regular trading and after-hour trading. Empirical analysis of the Taiwan Futures Exchange (TAIFEX) demonstrates the significant improvements of the PRS model in both in-sample and out-of-sample periods. Our results also show that the introduction of after-hour trading sessions has provided valuable information for volatility forecasting in subsequent regular trading sessions, emphasizing the importance of considering diverse information flows across different trading and non-trading times. The PRS model effectively captures the dynamics of non-trading and trading sessions and the influence of unusual news arrivals and jumps on market volatility, contributing to investment and risk management strategies. [ABSTRACT FROM AUTHOR]

Martingales with jumps on Riemannian manifolds and harmonic maps with respect to Markov processes are discussed in this paper. Discontinuous martingales on manifolds were introduced in Picard (Séminaire de Probabilités de Strasbourg 25:196–219, 1991). We obtain results about the convergence of martingales with finite quadratic variations on Riemannian submanifolds of higher-dimensional Euclidean space as t → ∞ and as t → 0 . Furthermore, we apply the result about martingales with jumps on submanifolds to harmonic maps with respect to Markov processes such as fractional harmonic maps. [ABSTRACT FROM AUTHOR]

This paper studies a dynamic principal–agent setting in which the principal needs to dynamically schedule an agent to work or be suspended. When the agent is directed to work and exert effort, the arrival rate of a Poisson process is increased, which increases the principal's payoff. Suspension, on the other hand, serves as a threat to the agent by delaying future payments. A key feature of our setting is a switching cost whenever the suspension stops and the work starts again. We formulate the problem as an optimal control model with switching and fully characterize the optimal control policies/contract structures under different parameter settings. Our analysis shows that, when the switching cost is not too high, the optimal contract demonstrates a generalized control-band structure. The length of each suspension episode, on the other hand, is fixed. Overall, the optimal contract is easy to describe, compute, and implement. This paper was accepted by Ilia Tsetlin, behavioral economics and decision analysis. Funding: The work of P. Cao is financially supported by the National Natural Science Foundation of China [Grants 72122019 and 71771202]. The work of F. Tian is financially supported by the Hong Kong Research Grants Council (RGC) funding (RGC Early Career Scheme 2022/23) [Grant 27500822]. Supplemental Material: The e-companion and data are available at https://doi.org/10.1287/mnsc.2023.4845. [ABSTRACT FROM AUTHOR]

This paper deals with optimal control of systems driven by stochastic differential equations (SDEs), with controlled jumps, where the control variable has two components, the first being absolutely continuous and the second singular. We study the corresponding relaxed-singular problem, in which the first part of the admissible control is a measure-valued process and the state variable is governed by a SDE driven by a relaxed Poisson random measure, whose compensator is a product measure. We establish a stochastic maximum principle for this type of relaxed control problems extending existing results. The proofs are based on Ekeland’s variational principle and stability properties of the state process and adjoint variable with respect to the control variable. [ABSTRACT FROM AUTHOR]

Many variables show a tendency to increase over time in line with their nonstationary nature. It is notable, however, that the original time series can be transformed into a sequence of jumps measured by time distances between the successive maxima and present the resulting series as the compound Poisson process, which has powerful consequences discussed in the paper. Firstly, the jump-generating process is stationary, unlike the one generating the original data. Secondly, the dynamics of a variable can be determined using solely the properties of the derived stationary counterpart. Thirdly, using this framework for prediction offers substantial advantages. The proposed methodology allows forecasting the number of periods necessary for a process to achieve the desired level and decomposing the path leading to that level into jumps of different size. It also gives a unique insight into the shape of the trajectory over the prediction horizon, which the traditional approach to the forecasting of nonstationary time series is incapable of providing. [ABSTRACT FROM AUTHOR]

We give an overview of heat kernels on ultra-metric spaces based on the results of [9] and [11]. In particular, we present estimates of the heat kernel of the Vladimirov operator in [ABSTRACT FROM AUTHOR]

Exchange rate is an important part of financial markets. Our analysis finds that the fluctuations of exchange rates have several obvious features, such as spikes, thick tails, fluctuation aggregations and asymmetry. Based on this, we build novel GARCH class model by introducing a jumping process to describe the dynamics of their fluctuations. Our empirical results show that the models with jump factors can better characterize the agglomeration and thick tail characteristics of these return fluctuations than the models without jump factors. In particular, the model with double exponential jumps can fully handle and capture the fluctuation characteristics of the returns. Our findings will be useful for individuals and governments to predict exchange rate fluctuations, provide reference for the effective management of exchange rate risk in China, and further improve the financial risk management mechanism.

Consider a symmetric Markovian jump process {Xt} on a metric measure space (M, d, μ). Chen, Kumagai, and Wang recently showed that two-sided heat kernel estimates and the parabolic Harnack inequality are both stable under bounded perturbations of the jumping measure, assuming (M, d, μ) satisfies the volume-doubling and reverse-volume-doubling conditions. These results do not apply if (M, d, μ) is a graph (or more generally, if M contains any atoms x such that μ(x) > 0) because it is impossible for reverse-volume-doubling to hold on a space with atoms. We generalize the results of Chen, Kumagai, and Wang to a larger class of metric measure spaces, including all infinite graphs with volume-doubling. Our main tool is the construction of an "auxiliary space" that smooths out the atoms. We show that many properties transfer from (M, d, μ) to the auxiliary space, and vice versa, including heat kernel estimates, the parabolic Harnack inequality, and their stable characterizations. [ABSTRACT FROM AUTHOR]

In this article, we study the relaxed control problem where the admissible controls are measure-valued processes and the state variable is governed by a G-stochastic differential equation (SDEs) driven by a relaxed Poisson measure where the compensator is a product measure. The control variable appears in the drift and in the jump term. We prove that every solution of our SDE associated to a relaxed control can be written as a limit of a sequence of solutions of SDEs associated to strict controls (stability results). In the end, we show the existence of our relaxed control. [ABSTRACT FROM AUTHOR]

We consider a stochastic differential equation (SDE) governed by a fractional Brownian motion (B t H) and a Poisson process (N t) associated with a stochastic process (A t) such that: d X t = μ X t d t + σ X t d B t H + A t X t − d N t , X 0 = x 0 > 0. The solution of this SDE is analyzed and properties of its trajectories are presented. Estimators of the model parameters are proposed when the observations are carried out in discrete time. Some convergence properties of these estimators are provided according to conditions concerning the value of the Hurst index and the nonequidistance of the observation dates. [ABSTRACT FROM AUTHOR]

In this paper, we establish an abstract framework for the approximation of the invariant probability measure for a Markov semigroup. Following Pagès and Panloup (2022) we use an Euler scheme with decreasing step (unadjusted Langevin algorithm). Under some contraction property with exponential rate and some regularization properties, we give an estimate of the error in total variation distance. This abstract framework covers the main results in Pagès and Panloup (2022) and Chen et al. (2023). As a specific application we study the convergence in total variation distance to the invariant measure for jump type equations. The main technical difficulty consists in proving the regularization properties — this is done under an ellipticity condition, using Malliavin calculus for jump processes. [ABSTRACT FROM AUTHOR]

Motivated by entropic optimal transport, time reversal of Markov jump processes in R n is investigated. Relying on an abstract integration by parts formula for the carré du champ of a Markov process recently obtained in Cattiaux et al. (2021), and using an entropic improvement strategy discovered by Föllmer (1985, 1986), we compute the semimartingale characteristics of the time reversed process for a wide class of jump processes in R n with possibly unbounded variation sample paths and singular intensities of jump. [ABSTRACT FROM AUTHOR]

In this comment, we first use a counterexample to demonstrate that the optimal contract structure proposed in section 4 of the paper [Sun P, Tian F (2018) Optimal contract to induce continued effort. Management Sci. 64(9):4193–4217] can be wrong when the two players' discount rates are different. We then specify correct optimal contract structures, which involve generalizing the contract space to allow random termination. Numerical study with a wide range of model parameters illustrates that such a random termination only occurs sparingly in optimal contracts. Moreover, the suboptimality gap, measured by the relative improvement of the optimal contract over the best contract without random termination, is extremely small. This paper was accepted by Manel Baucells, decision analysis. [ABSTRACT FROM AUTHOR]

We consider a stochastic differential equation (SDE) governed by a fractional Brownian motion (BtH) and a Poisson process (Nt) associated with a stochastic process (At) such that: dXt=μXtdt+σXtdBtH+AtXt−dNt,X0=x0>0. The solution of this SDE is analyzed and properties of its trajectories are presented. Estimators of the model parameters are proposed when the observations are carried out in discrete time. Some convergence properties of these estimators are provided according to conditions concerning the value of the Hurst index and the nonequidistance of the observation dates.

• A sediment replenishment problem as a stochastic control problem is analyzed. • The observation is only allowed to be partial and the replenishment is impulsive. • The HJB and FP equations having unique singularities are presented. • These equations are solved analytically and verified under a simplified case. • Numerical schemes that can handle the singularities of the equations are developed. We formulate a new two-variable river environmental restoration problem based on jump stochastic differential equations (SDEs) governing the sediment storage and nuisance benthic algae population dynamics in a dam-downstream river. Controlling the dynamics is carried out through impulsive sediment replenishment with discrete and random observation/intervention to avoid sediment depletion and thick algae growth. We consider a cost-efficient management problem of the SDEs to achieve the objectives whose resolution reduces to solving a Hamilton-Jacobi-Bellman (HJB) equation. We also consider a Fokker-Planck (FP) equation governing the probability density function of the controlled dynamics. The HJB equation has a discontinuous solution, while the FP equation has a Dirac's delta along boundaries. We show that the value function, the optimized objective function, is governed by the HJB equation in the simplified case and further that a threshold-type control is optimal. We demonstrate that simple numerical schemes can handle these equations. Finally, we numerically analyze the optimal controls and the resulting probability density functions. [ABSTRACT FROM AUTHOR]

We study a US OEM that outsources its production to two contract manufacturers, a local manufacturer (e.g. in the US or Mexico) and a foreign manufacturer (e.g. in China). The local manufacturer is relatively reliable, but low margin. The foreign manufacturer offers high margin, but is subject to disruption risks. Both manufacturers experience some level of operational uncertainties, and the operational risks can be positively or negatively correlated. Disruption risks are modelled as a Poisson jump process at a random magnitude, and operational risks are modelled as correlated stochastic diffusion processes. We develop a stochastic dynamic programming formulation to characterise the OEM’s optimal capital allocation decision to contract manufacturers, and provide the necessary and sufficient conditions for each optimal decision. The objective of our study is to investigate how dual sourcing balances the risks and opportunities, when the OEM bears disruption risks and correlated operational risks. We find that the two manufacturers can be substitutes or complements to each other. Risk of disruption renders the unreliable foreign manufacturer less attractive, but has a moderating effect on the allocation to the local manufacturer. We also provide managerial implications of our study. [ABSTRACT FROM PUBLISHER]

Maintenance outsourcing is quite common in industries that rely on complex and critical equipment. Instead of investing in the maintenance facilities, firms outsource maintenance activities to specialized companies. However, it may be hard for firms (i.e., principal) to observe whether maintenance companies (i.e., agent) put sufficient resources into providing the best service, which gives rise to agency issues. In a dynamic environment in which an agent is responsible for both maintenance and repair of a critical machine, how the principal uses payments and termination to tackle agency issues is a challenging problem. In "Optimal Contract for Machine Repair and Maintenance," F. Tian, P. Sun, and I. Duenyas provide theoretical guidance on designing the optimal contract to induce efforts from an agent to efficiently operate a machine. Although they consider the very general contract forms, the optimal contracts demonstrate simple and intuitive structures, making them easy to describe and implement in practice. A principal hires an agent to repair a machine when it is down and maintain it when it is up and earns a revenue flow when the machine is up. Both the up- and downtimes follow exponential distributions. If the agent exerts effort, the downtime is shortened, and uptime is prolonged. Effort, however, is costly to the agent and unobservable to the principal. We study optimal dynamic contracts that always induce the agent to exert effort while maximizing the principal's profits. We formulate the contract design problem as a stochastic optimal control model with incentive constraints in continuous time over an infinite horizon. Although we consider the contract space that allows payments and potential contract termination time to take general forms, the optimal contracts demonstrate simple and intuitive structures, making them easy to describe and implement in practice. [ABSTRACT FROM AUTHOR]

Financial services firms are subject to various types of risks. In particular, operational risk is difficult to assess and can be devastating, although it is often perceived by a firm's management as being more controllable than the cost of managing other types of risks. Understanding the management problems associated with operational risk is crucial to the performance of the firm. In "Operational Risk Management: A Stochastic Control Framework with Preventive and Corrective Controls," Xu, Zhu, and Pinedo introduce a general modeling framework for operational risk management for financial firms. They propose two types of controls and characterize the optimal control policies. They apply their model to a data set from a commercial bank, and through a proper investment strategy, one can achieve a significant performance improvement. In this paper, we propose a general modeling framework for operational risk management of financial firms. We consider operational risk events as shocks to a financial firm's value process and then study capital investments under preventive and corrective controls to mitigate risk losses. The optimal decisions are made in three scenarios: (i) preventive control only, (ii) corrective control only, and (iii) joint controls. We characterize the optimal control policies within a general modeling framework that comprises these three scenarios and then discuss an exponential risk reduction function. We conclude our work with an application of our model to a data set from a commercial bank. We find that, through a proper investment strategy, we can achieve a significant performance improvement, especially when the risk severity level is high. Moreover, with controls, the value of the firm tends to increase relative to the value of the firm without controls. Hence, the controls are essentially smoothing out the jump losses and increasing the value of the firm. At the bank we analyze we find that with a joint control strategy the bank can achieve profit increases from 7.45% to 11.62% when the risk reduction efficiencies of the two controls are high. In general, our modeling framework, which combines a typical operational risk process with stochastic control, may suggest a new research direction in operations management and operational risk management. [ABSTRACT FROM AUTHOR]

This study applies the real options valuation (ROV) method with a jump process to consider the uncertainty of multiple unexpected events (UEs) when building investment valuation models for natural tourist attractions (NTAs). Unlike previous studies, this investigation considers multiple UEs from a pre-events, micro-level perspective. In considering both homogeneous and heterogeneous UE scenarios, certain conclusions can be drawn, so long as the project value of an NTA is known. In assessing diverse scenarios, it can be generally predicted that the greater the UE effects, the higher the degree of uncertainty; also, the more extensive the damage caused by UEs and the more frequent their occurrence, the greater the expense of an investment opportunity, and the longer the wait until the optimal time for investment. In assessing both homogeneous and heterogeneous UE scenarios, this study finds that investments should be valued differently under each type of scenario. In determining the optimal time to invest in NTAs that face differing types of UEs, strategic delay is found to be advantageous in many cases. [ABSTRACT FROM AUTHOR]

We propose a new framework to value employee stock options (ESOs) that capture multiple exercises of different quantities over time. We also model the ESO holder's job termination risk and incorporate its impact on the payoffs of both vested and unvested ESOs. Numerical methods based on Fourier transform and finite differences are developed and implemented to solve the associated systems of PDEs. In addition, we introduce a new valuation method based on maturity randomization that yields analytic formulae for vested and unvested ESO costs. We examine the cost impact of job termination risk, exercise intensity and various contractual features. [ABSTRACT FROM AUTHOR]

We study nonlocal operators acting on functions in the Euclidean space. The operators under consideration generate anisotropic jump processes, e.g., a jump process that behaves like a stable process in each direction but with a different index of stability. Its generator is the sum of one-dimensional fractional Laplace operators with different orders of differentiability. We study such operators in the general framework of bounded measurable coefficients. We prove a weak Harnack inequality and Hölder regularity results for solutions to corresponding integro-differential equations. [ABSTRACT FROM AUTHOR]

We present a methodology to identify change-points in financial markets where the governing regime shifts from a constant rate-of-return, i.e. normal growth, to a superexponential growth described by a power-law hazard rate. The latter regime corresponds, in our view, to financial bubbles driven by herding behavior of market participants. Assuming that the time series of log-price returns of a financial index can be modeled by arithmetic Brownian motion, with an additional jump process with power-law hazard function to approximate the superexponential growth, we derive a threshold value of the hazard-function control parameter, allowing us to decide in which regime the market is more likely to be at any given time. An analysis of the Standard & Poors 500 index over the last 60 years provides evidence that the methodology has merit in identifying when a period of herding behavior begins, and, perhaps more importantly, when it ends. [ABSTRACT FROM AUTHOR]

Simple point processes are often characterized by their associated compensators or conditional intensities. For non-simple point processes, however, the conditional intensity and compensator do not uniquely determine the distribution of the process. Various ways of characterizing non-simple multivariate point processes are discussed here, some important classes of separable non-simple processes are investigated, and methods of simplification involving thinning, rescaling, and changing the mark space are presented.

In models which allow for random jumps, statistical tests for jumps are typically non-standard and nuisance parameter-dependent. To handle these problems, we combine bounds and MonteCarlo (MC) simulation techniques to derive nuisance-parameter-free bounds and obtain level-exact p-values for a wide class of processes with random jumps and time varying heteroskedasticity. When identified nuisance parameters are absent under the null, we show that MC p-values are finite sample, level-exact. To illustrate this easy-to-implement approach, we analyze the spot prices of four commodities (aluminium, copper, gold and lead) and the closing prices of four technology stocks (Intel, Microsoft, Oracle and Sun). We find significant jumps in these time series. Our approach can easily be extended to other nuisance-parameter dependent tests. (C) 2003 Elsevier B.V. All rights reserved.

In this work, we will show strong convergence of the Multilevel Monte-Carlo (MLMC) algorithm with split-step backward Euler (SSBE) and backward (drift-implicit) Euler (BE) schemes for nonlinear jump-diffusion stochastic differential equations (SDEs) when the drift coefficient is globally one-sided Lipschitz and the test function is only locally Lipschitz. We also confirm these theoretical results by numerical experiments for the jump-diffusion processes. [ABSTRACT FROM AUTHOR]

The paper deals with homogenization of Lévy-type operators with rapidly oscillating coefficients. We consider cases of periodic and random statistically homogeneous micro-structures and show that in the limit we obtain a Lévy-operator. In the periodic case we study both symmetric and non-symmetric kernels whereas in the random case we only investigate symmetric kernels. We also address a nonlinear version of this homogenization problem. [ABSTRACT FROM AUTHOR]

We consider minimal variance hedging in a pure-jump multicurve interest rate model. In the first part, we derive arithmetic multifactor martingale representations for the spread, OIS, and LIBOR rate, which are bounded from below by a real-valued constant. In the second part, we investigate minimal variance hedging and provide a closed-form formula for the related minimal variance portfolio. We apply this result to several examples covering both replicable and nonreplicable claims. We conclude the paper with a consideration of delta hedging. [ABSTRACT FROM AUTHOR]

Abstract In this paper we propose an algorithm to numerically simulate Markov processes of jump type. While these processes can naturally be generated as solutions to jump-SDEs the algorithm we propose is instead based on the interlacing construction of the process. We show that one can construct in the sense of strong convergence a high order numerical scheme based on high order ODE solvers. This result is in sharp contrast to the well known difficulty of constructing high-order numerical schemes for diffusion processes. [ABSTRACT FROM AUTHOR]

State‐dependent jump linear systems (SJLSs) are a set of linear systems whose switching is determined by a finite state random process with state‐dependent transition rates. In this study, a SJLS is considered with state multiplicative noise and stochastic perturbations. In particular, the jump process is regarded to have dissimilar transition rates among sets of state evolution space. The aim of this study is to consider stochastic H∞ control of such systems using state‐feedback control input. Sufficient conditions for stochastic H∞ are obtained by solving linear matrix inequalities, which are validated by a simulation example. [ABSTRACT FROM AUTHOR]

Durotaxis, the phenomena where cells migrate up a gradient in substrate stiffness, remains poorly understood. It has been proposed that durotaxis results from the reinforcement of focal adhesions on stiff substrates. In this paper we formulate a mathematical model of single cell migration on elastic substrates with spatially varying stiffness. We develop a stochastic model where the cell moves by updating the position of its adhesion sites at random times, and the rate of updates is determined by the local stiffness of the substrate. We investigate two physiologically motivated mechanisms of stiffness sensing. From the stochastic model of single cell migration we derive a population level description in the form of a partial differential equation for the time evolution of the density of cells. The equation is an advection–diffusion equation, where the advective velocity is proportional to the stiffness gradient. The model shows quantitative agreement with experimental results in which cells tend to cluster when seeded on a matrix with periodically varying stiffness. [ABSTRACT FROM AUTHOR]

We consider the non-local operator of variable order as follows: Lf(x) = ... (f(x + z) -- f(x) -- (∇ f(x), z) 1{|z|≤1})n(x, z)/... dz, where α(x) ∈ [α0, α2] for any x ∈ Rd and some constants 0 < α0 ≤ α2 < 2 and n(x, z) is a positive Borel measurable function bounded from above and below. Under mild continuity conditions on α(x) and n(x, z), we establish the H¨older regularity for the associated semigroups. The proof is based on a new construction of the coupling by reflection for non-local operator L, and the results successfully apply to both stable-like processes in the sense of Bass and time-change of symmetric stable processes. [ABSTRACT FROM AUTHOR]