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Search Results (3)

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Keywords = regime switching in ordinary differential equations

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30 pages, 489 KiB  
Article
Pricing Variance Swaps under MRG Model with Regime-Switching: Discrete Observations Case
by Anqi Zou, Jiajie Wang and Chiye Wu
Mathematics 2023, 11(12), 2730; https://doi.org/10.3390/math11122730 - 16 Jun 2023
Cited by 1 | Viewed by 3717
Abstract
In this paper, we creatively price the discretely sampled variance swaps under the mean-reverting Gaussian model (MRG model in short) with regime-switching asymmetric double exponential jump diffusion. We extend the traditional MRG model by further considering the trend of the financial market as [...] Read more.
In this paper, we creatively price the discretely sampled variance swaps under the mean-reverting Gaussian model (MRG model in short) with regime-switching asymmetric double exponential jump diffusion. We extend the traditional MRG model by further considering the trend of the financial market as well as a sudden and unexpected event of the market. This new model is meaningful because it uses observable Markov chains that represent market states to adjust its parameters, which helps capture the movement of the market and fluctuations in asset prices. By utilizing the characteristic function and the conditional transition characteristic function, we obtain analytical solutions for pricing formulae. Note that this is our first effort to provide the analytical solution for the ordinary differential equations satisfied by the Feynman–Kac theorem. To achieve this, we have developed a new methodology in Proposition 2 that involves dividing the sampling interval into more detailed switching and non-switching intervals. One significant advantage of our closed-form solution is its high computational accuracy and efficiency. Subsequent semi-Monte Carlo simulations will provide specific validation results. Full article
(This article belongs to the Section E5: Financial Mathematics)
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23 pages, 1199 KiB  
Article
The Multi-Compartment SI(RD) Model with Regime Switching: An Application to COVID-19 Pandemic
by Manuel L. Esquível, Nadezhda P. Krasii, Gracinda R. Guerreiro and Paula Patrício
Symmetry 2021, 13(12), 2427; https://doi.org/10.3390/sym13122427 - 15 Dec 2021
Cited by 6 | Viewed by 2926
Abstract
We study—with existence and unicity results—a variant of the SIR model for an infectious disease incorporating both the possibility of a death outcome—in a short period of time—and a regime switch that can account for the mitigation measures used to control the spreading [...] Read more.
We study—with existence and unicity results—a variant of the SIR model for an infectious disease incorporating both the possibility of a death outcome—in a short period of time—and a regime switch that can account for the mitigation measures used to control the spreading of the infections, such as a total lockdown. This model is parametrised by three parameters: the basic reproduction number, the mortality rate of the infected, and the duration of the disease. We discuss a particular example of application to Portuguese COVID-19 data in two short periods just after the start of the epidemic in 4 March 2020, with the first two cases dated that day. We propose a simple and effective method for the estimation of the main parameters of the disease, namely, the basic reproduction number and the mortality rate of the infected. We correct these estimated values to take into account the asymptomatic non-diagnosed members of the population. We compare the outcome of the model in the cases of the existence, or not, of a regime switch, and under three different scenarios, with a remarkable agreement between model and data deaths in the case of our basis scenario. In a final short remark, we deal with the existence of symmetries for the proposed model. Full article
(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)
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25 pages, 730 KiB  
Article
Optimal Surplus-Dependent Reinsurance under Regime-Switching in a Brownian Risk Model
by Julia Eisenberg, Lukas Fabrykowski and Maren Diane Schmeck
Risks 2021, 9(4), 73; https://doi.org/10.3390/risks9040073 - 13 Apr 2021
Cited by 3 | Viewed by 3053
Abstract
In this paper, we consider a company that wishes to determine the optimal reinsurance strategy minimising the total expected discounted amount of capital injections needed to prevent the ruin. The company’s surplus process is assumed to follow a Brownian motion with drift, and [...] Read more.
In this paper, we consider a company that wishes to determine the optimal reinsurance strategy minimising the total expected discounted amount of capital injections needed to prevent the ruin. The company’s surplus process is assumed to follow a Brownian motion with drift, and the reinsurance price is modelled by a continuous-time Markov chain with two states. The presence of regime-switching substantially complicates the optimal reinsurance problem, as the surplus-independent strategies turn out to be suboptimal. We develop a recursive approach that allows to represent a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation and the corresponding reinsurance strategy as the unique limits of the sequence of solutions to ordinary differential equations and their first- and second-order derivatives. Via Ito’s formula, we prove the constructed function to be the value function. Two examples illustrate the recursive procedure along with a numerical approach yielding the direct solution to the HJB equation. Full article
(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)
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