Infinite Horizon Irregular Quadratic BSDE and Applications to Quadratic PDE and Epidemic Models with Singular Coefficients
Abstract
:1. Motivation and Introduction
2. Main Result
2.1. Notations
- is the space of -adapted processes Z in such that
- is the space of -adapted processes Z in satisfying
- is the set of -valued, continuous, and -adapted processes Y such that
- represents the Sobolev space consisting of functions u defined on for which both u and its generalized derivatives and belong to .
- is the collection of -valued -measurable random variables with
- denotes the -algebra of predictable sets on .
- For all is a stochastic process that is progressively measurable such that
- There exists two non-random functions and such that, for all, we have, -a.s.:
- .
- There exists a non-negative random coefficient and a measurable function f such that for every :
- (i)
- and for all most every x, .
- (ii)
- u is a bijective mapping the whole real line to itself.
- (iii)
- The inverse function is both continuously differentiable on and locally in the Sobolev space .
- (iv)
- For any pair of real numbers x and y, there exist positive constants m and M independent on x and y satisfying the following inequalitiesand
- (v)
- Moreover, if f is continuous, then both u and belong to the class .
- 1.
- The stochastic process is progressively measurable and satisfies the integrability condition
- 2.
- For fixed ω and s, the mapping is Lipschitz continuous for a bounded and integrable function f over the whole space .
2.2. Comparison Principle
3. Application to Quadratic Partial Differential Equations
- are uniformly Lipschitz, i.e., there exists a constant such that ,
- The coefficients and b are monotonic with respect to x, meaning that there exists a constant such that, for all and all :
- The functions and .
- The function is integrable, and is a continuous function such that, for some constant ,
- (i)
- In this section, f is assumed to be continuous only for making sense of the associated semi-linear QPDE, see [22] for more details.
- (ii)
4. Application to Epidemic Models
- is the constant migration rate of the susceptible population.
- represents the transmission rate between S and I at time t.
- denotes the rate at which immunity decreases for vaccinated individuals.
- represents the natural death rate of the S, I, V, and R compartments and .
- represents the proportional coefficient of vaccinated individuals among those susceptible at time t.
- signifies the rate at which recovered individuals lose their immunity and revert to being susceptible.
- is introduced to turn off/on the control in “summer”/“rest of the year”. See below for more details.
- for represents the noise for the suspected, infected, and recovered people, where these parameters can be estimated from the data of the history of the epidemic. The uncertainty comes from the fact that there are infected non-detected and also recovered non-detected. For the vaccination, the uncertainty comes from the imperfect type of vaccination that could be used.
- represents the constant recovery rate for the disease-infected individual.
- is the death rate due to disease.
- : Represents the government’s efforts in border protection (airports and ports to track) as well as awareness campaigns and guidance.
- : Represent the control over the number of vaccinated people
- A four-dimensional diffusion of the formHere, is a positive constant for simplicity.
- A QBSDE with measurable coefficients.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Eddahbi, M.; Kebiri, O.; Sene, A. Infinite Horizon Irregular Quadratic BSDE and Applications to Quadratic PDE and Epidemic Models with Singular Coefficients. Axioms 2023, 12, 1068. https://doi.org/10.3390/axioms12121068
Eddahbi M, Kebiri O, Sene A. Infinite Horizon Irregular Quadratic BSDE and Applications to Quadratic PDE and Epidemic Models with Singular Coefficients. Axioms. 2023; 12(12):1068. https://doi.org/10.3390/axioms12121068
Chicago/Turabian StyleEddahbi, Mhamed, Omar Kebiri, and Abou Sene. 2023. "Infinite Horizon Irregular Quadratic BSDE and Applications to Quadratic PDE and Epidemic Models with Singular Coefficients" Axioms 12, no. 12: 1068. https://doi.org/10.3390/axioms12121068
APA StyleEddahbi, M., Kebiri, O., & Sene, A. (2023). Infinite Horizon Irregular Quadratic BSDE and Applications to Quadratic PDE and Epidemic Models with Singular Coefficients. Axioms, 12(12), 1068. https://doi.org/10.3390/axioms12121068