Abstract Lie symmetries play a central role in the analysis and solution of differential equations, offering systematic techniques for order reduction and integration. However, many differential equations do not admit classical Lie point symmetries, limiting the applicability of this approach. To address this limitation, various generalisations have been developed, among which the notion of $$\lambda$$ -symmetries (also known as $$C^{\infty }$$ -symmetries) has proven particularly effective. These $$\lambda$$ -symmetries extend the symmetry framework, enabling the reduction of order and the derivation of first integrals even in the absence of classical symmetries. In this paper, we investigate the Painlevé–Ince equation within the context of $$\lambda$$ -symmetries for the first time. We identify the conditions under which the equation admits such symmetries and employ them to obtain a novel reduction of order. This analysis yields new insights into the structural properties and integrability of the equation.

Abstract From the era of Black–Scholes, European options have dwelt on pay-offs that are a linear function of the asset price. In this paper, we will look at a special case of exotic options - power options - whose payoffs are nonlinear functions of the underlying asset price. Exotic options are derivatives which have features that makes them more complex than commonly traded products - thus finding their fair value is not an always easy task. Previous analyses of the power option partial differential equation (PDE) have only obtained closed form solutions either by guessing solutions, similarity methods or the martingale approach [8, 9, 25]. Using Lie symmetry analysis we obtain an optimal system of the Lie point symmetries of the power option PDE and demonstrate an algorithmic method for finding solutions to the equation. In addition, we find a new analytical solution to the asymmetric type of the power option.

This paper considered the problem of estimation of population variance of the study character in two-occasion (successive) sampling in absence and presence of non-response and measurement error. Logarithmic type estimators have been developed to reduce the nuisance effect of non-response in sample surveys. The proposed logarithmic based estimators aim to enhance the precision and validity of inferences drawn from successive sampling surveys, where non-sampling errors can significantly affect the quality of the data and the resulting population parameter estimates. The expressions for the biases and mean squared errors of the proposed estimators were derived to quantify their statistical properties and performance. The empirical results through simulation studies demonstrated that the proposed logarithmic-type estimators outperform the traditional variance estimator obtained through linear combination of sample variance and conventional ratio estimator in terms of relative absolute bias and mean squared error, especially when non-response and measurement errors are substantial.