scholarly journals A Continuous-Time Ehrenfest Model with Catastrophes and Its Jump-Diffusion Approximation

2015 ◽  
Vol 161 (2) ◽  
pp. 326-345 ◽  
Author(s):  
Selvamuthu Dharmaraja ◽  
Antonio Di Crescenzo ◽  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Diffusion Approximation ◽  
Continuous Time ◽  
Jump Diffusion

scholarly journals Jump-Diffusion Models for Valuing the Future: Discounting under Extreme Situations

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1589
Author(s):  
Jaume Masoliver ◽  
Miquel Montero ◽  
Josep Perelló
Keyword(s):  
Diffusion Process ◽  
Discount Rate ◽  
Continuous Time ◽  
Environmental Planning ◽  
Dynamical Evolution ◽  
Jump Diffusion ◽  
Economic Evolution ◽  
Discount Function ◽  
Long Run ◽  
Future Discounting

We develop the process of discounting when underlying rates follow a jump-diffusion process, that is, when, in addition to diffusive behavior, rates suffer a series of finite discontinuities located at random Poissonian times. Jump amplitudes are also random and governed by an arbitrary density. Such a model may describe the economic evolution, specially when extreme situations occur (pandemics, global wars, etc.). When, between jumps, the dynamical evolution is governed by an Ornstein–Uhlenbeck diffusion process, we obtain exact and explicit expressions for the discount function and the long-run discount rate and show that the presence of discontinuities may drastically reduce the discount rate, a fact that has significant consequences for environmental planning. We also discuss as a specific example the case when rates are described by the continuous time random walk.

Effect of the Return Policy in a Continuous-Time Newsvendor Problem

2017 ◽  
Vol 34 (06) ◽  
pp. 1750031
Author(s):  
Weiwei Zhang ◽  
Zhongfei Li ◽  
Ke Fu ◽  
Fan Wang
Keyword(s):  
Diffusion Process ◽  
Continuous Time ◽  
Existence And Uniqueness ◽  
Stackelberg Game ◽  
Jump Diffusion ◽  
Newsvendor Problem ◽  
Stackelberg Equilibrium ◽  
Computational Results ◽  
Return Policy ◽  
Demand Rate

This paper studies the stochastic differential Stackelberg game in a continuous-time newsvendor problem with a return policy, in which one supplier sells products to one retailer and the two parties make the decisions sequentially to maximize their own expected profits. When the demand process is a general jump-diffusion process, we provide a general formula for the equilibrium if it exists. When the demand rate is an Ornstein–Uhlenbeck (O–U) process, we prove the existence and uniqueness of the equilibrium and find an explicit expression for the equilibrium. Computational results show that the return policy has significant impact on the Stackelberg equilibrium.

Portfolio Selection with Transaction Costs and Jump-Diffusion Asset Dynamics I: A Numerical Solution

2016 ◽  
Vol 06 (04) ◽  
pp. 1650018 ◽  
Author(s):  
Michal Czerwonko ◽  
Stylianos Perrakis
Keyword(s):  
Diffusion Process ◽  
Transaction Costs ◽  
Portfolio Selection ◽  
Discrete Time ◽  
Continuous Time ◽  
Jump Diffusion ◽  
Numerical Approach ◽  
Proportional Transaction Costs ◽  
Asset Portfolio ◽  
Time Formulation

We derive allocation rules under isoelastic utility for a mixed jump-diffusion process in a two-asset portfolio selection problem with finite horizon in the presence of proportional transaction costs. We adopt a discrete-time formulation, let the number of periods go to infinity, and show that it converges efficiently to the continuous-time solution for the cases where this solution is known. We then apply this discretization to derive numerically the boundaries of the region of no transactions. Our discrete-time numerical approach outperforms alternative continuous-time approximations of the problem.

scholarly journals Separation of timescales for the seed bank diffusion and its jump-diffusion limit

2021 ◽  
Vol 82 (6) ◽  
Author(s):  
Jochen Blath ◽  
Eugenio Buzzoni ◽  
Adrián González Casanova ◽  
Maite Wilke Berenguer
Keyword(s):  
Seed Bank ◽  
Continuous Time ◽  
Scaling Limit ◽  
Jump Diffusion ◽  
Diffusion Limit ◽  
Scaling Limits ◽  
Semi Group ◽  
Analytical Scheme ◽  
Convergence Results ◽  
Funct Anal

AbstractWe investigate scaling limits of the seed bank model when migration (to and from the seed bank) is ‘slow’ compared to reproduction. This is motivated by models for bacterial dormancy, where periods of dormancy can be orders of magnitude larger than reproductive times. Speeding up time, we encounter a separation of timescales phenomenon which leads to mathematically interesting observations, in particular providing a prototypical example where the scaling limit of a continuous diffusion will be a jump diffusion. For this situation, standard convergence results typically fail. While such a situation could in principle be attacked by the sophisticated analytical scheme of Kurtz (J Funct Anal 12:55–67, 1973), this will require significant technical efforts. Instead, in our situation, we are able to identify and explicitly characterise a well-defined limit via duality in a surprisingly non-technical way. Indeed, we show that moment duality is in a suitable sense stable under passage to the limit and allows a direct and intuitive identification of the limiting semi-group while at the same time providing a probabilistic interpretation of the model. We also obtain a general convergence strategy for continuous-time Markov chains in a separation of timescales regime, which is of independent interest.

scholarly journals Hybrid master equation for jump-diffusion approximation of biomolecular reaction networks

2019 ◽  
Vol 60 (2) ◽  
pp. 261-294
Author(s):  
Derya Altıntan ◽  
Heinz Koeppl
Keyword(s):  
Master Equation ◽  
Probability Density ◽  
Diffusion Approximation ◽  
Joint Probability ◽  
Planck Equation ◽  
Jump Diffusion ◽  
Reaction Rates ◽  
Conditional Probability Density ◽  
Multi Scale ◽  
Joint Probability Density

AbstractCellular reactions have a multi-scale nature in the sense that the abundance of molecular species and the magnitude of reaction rates can vary across orders of magnitude. This diversity naturally leads to hybrid models that combine continuous and discrete modeling regimes. In order to capture this multi-scale nature, we proposed jump-diffusion approximations in a previous study. The key idea was to partition reactions into fast and slow groups, and then to combine a Markov jump updating scheme for the slow group with a diffusion (Langevin) updating scheme for the fast group. In this study we show that the joint probability density function of the jump-diffusion approximation over the reaction counting process satisfies a hybrid master equation that combines terms from the chemical master equation and from the Fokker–Planck equation. Inspired by the method of conditional moments, we propose a efficient method to solve this master equation using the moments of reaction counters of the fast reactions given the reaction counters of the slow reactions. For each time point of interest, we then solve a set of maximum entropy problems in order to recover the conditional probability density from its moments. This finally allows us to reconstruct the complete joint probability density over all reaction counters and hence obtain an approximate solution of the hybrid master equation. Finally, we show the accuracy of the method applied to a simple multi-scale conversion process.

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