Browsing by Author "Patidar, Kailash"

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    Data driven neural network approaches for pricing options
    (Elsevier B.V, 2025) Patidar, Kailash; Tarla, Divine; Nuugulu, Samuel
    This paper presents two data driven approaches, the purely data driven (PDD) and physics informed neural network (PINN) approach for solving asset pricing problems. The PDD approach relies purely on available data and does not require any governing partial differential equation (PDE) to solve a pricing problem. On the other hand, under the PINN approach, the pricing is done by solving a governing PDE. Both models are calibrated to observed market prices, and their implied volatilities are compared to those derived from market data and the classical Black–Scholes model. The absolute errors and maximum absolute errors metrics relative to observed implied volatilities and prices and the prices obtained from the classical Black–Scholes model were used in measuring the goodness-of-fit of the two proposed techniques. Several hyperparameter tuning techniques were employed to optimize the performance of the two methods. In addition, we analyze the probability density functions (PDFs) derived from each method and verify that they are valid by demonstrating positivity and proper normalization. Theoretical results, including propositions and theorems, are presented to establish conditions under which the PINN, trained using the Adam optimizer and initialized via the Xavier method, converges to an optimal solution, i.e., a set of trainable parameters that minimize the loss function. In further extensions, the PINN approach was applied to pricing European put options under a Heston stochastic volatility model (HSVM) model. While both methods exhibit competitive performance when calibrated, our empirical findings indicate that the PINN approach yields superior accuracy and stability.
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    Modeling the dynamics of an epidemic under vaccination in two interacting populations
    (Hindawi, 2012) Ahmed, Ibrahim H. I.; Witbooi, Peter J.; Patidar, Kailash
    Mathematical modeling of the numerical evolution of infectious diseases has become an important tool for disease control and eradication when possible. Much work has been done on the problem of how a given population is affected by another population when there is mutual interaction. The mere presence of migrant people poses a challenge to whatever health systems are in place in a particular region. Such epidemiological phenomena have been studied extensively, described by mathematical models with suggestions for intervention strategies. The epidemiological effect of migration within the population itself was modeled for sleeping sickness in a paper 1 by Chalvet-Monfray et al. In the case of malaria, there is for instance a study 2 by Tumwiine et al. on the effect of migrating people on a fixed population. The latter two diseases are vector borne. Diseases that propagate without a vector spread perhaps more easily when introduced into a new region. Various studies of models with immigration of infectives have been undertaken for tuberculosis, see for instance 3 by Zhou et al., or the work 4 of Jia et al., and for HIV, see the paper 5 of Naresh et al.
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    Optimal control strategies applied to a mathematical model of meningitis
    (New York Business Global, 2025) Patidar, Kailash; Mohamed, Sahar; Obaid, Hasim
    In this paper, we deal with the problem of optimal control for the transmission dynamics of the meningococcal meningitis. The problem is a mathematical model described by a system of nonlinear differential equations. Based on this, two controls are formulated and the resulting system is solved as an optimal control problem. Aiming to minimize the number of illnesses or deaths in the population, we used a control representing a vaccination and another one representing a treatment strategy. We prove that these controls are capable of reducing the number of carriers and infectious individuals. Numerical simulations are carried out to show how to perform the two strategies.