In this paper, we derive new closed-form valuations to European options under three-factor hybrid models that include stochastic interest rates and stochastic volatility and incorporate a nonzero covariance structure between factors. We make novel use of the empirically proven 3/2 stochastic volatility model with a time-dependent drift in which we are free to choose the moving reversion target. This model has been shown by many authors to empirically outperform other volatility models in maximising model fit. We also improve the valuation of European options under the Heston volatility and Cox, Ingersoll, Ross interest rate model, recently published in the literature, by replacing open-form infinite series with closed-form analytic expressions. For completeness, we also add a fuller covariance structure in this setting and detail closed-form valuations for options. The inclusion of nonzero covariances amongst the factors can significantly improve option pricing by allowing for a wider variety of market behaviour. The solutions are derived by firstly formulating the price of a European call option in terms of the corresponding characteristic function of the underlying price and then determining a partial differential equation for the characteristic function. By including empirically proven models into our analysis, the options formulae could provide more realistic prices for investors and practitioners.

Complementary exponential geometric distribution has many applications in survival and reliability analysis. Due to its importance, in this study, we are aiming to estimate the parameters of this model based on progressive type-II censored observations. To do this, we applied the stochastic expectation maximization method and Newton–Raphson techniques for obtaining the maximum likelihood estimates. We also considered the estimation based on Bayesian method using several approximate: MCMC samples, Lindely approximation and Metropolis–Hasting algorithm. In addition, we considered the shrinkage estimators based on Bayesian and maximum likelihood estimators. Then, the HPD intervals for the parameters are constructed based on the posterior samples from the Metropolis–Hasting algorithm. In the sequel, we obtained the performance of different estimators in terms of biases, estimated risks and Pitman closeness via Monte Carlo simulation study. This paper will be ended up with a real data set example for illustration of our purpose.

Let $pod_3(n)$ denote the number of 3-regular partitions with distinct odd parts (and even parts are unrestricted) of a non-negative integer n. In this paper, we present infinite families of Ramanujan-type congruences modulo 2 and 3 for $pod_3(n)$.

We give an exposition of a result of Borell (Commun Math Phys 86:143–147, 1982) that the probability function that Brownian motion hits the inner boundary before time t and before hitting the outer boundary is a space-time quasiconcave function.

In this paper, we consider weak horseshoe with bounded-gap-hitting times. For a flow $(M,\phi )$, it is shown that if the time one map $(M,\phi _1)$ has weak horseshoe with bounded-gap-hitting times, so is $(M,\phi _\tau )$ for all $\tau \ne 0$. In addition, we prove that for an affine homeomorphism of a compact metric abelian group, positive topological entropy is equivalent to weak horseshoe with bounded-gap-hitting times.

In this paper, we extend Fibonacci unimodal map to a wider class. We describe the combinatorial property of these maps by first return map and principal nest. We give the sufficient and necessary condition for the existence of this class of maps. Moreover, for maps with ’bounded combinatorics’, we prove that they have no absolutely continuous invariant probability measure when the critical order $\ell $ is sufficiently large; for maps with reluctantly recurrent critical point, we prove they have absolutely continuous invariant probability measure whenever the critical order $\ell >1$.