This study investigates valuation of guaranteed minimum maturity benefits (GMMB) in variable annuity contract in the case where the guarantees can be surrendered at any time prior to the maturity. In the event of the option being exercised early, early surrender charges will be applied. We model the underlying mutual fund dynamics under regime-switching volatility. The valuation problem can be reduced to an American option pricing problem, which is essentially an optimal stopping problem. Then, we obtain the pricing partial differential equation by a standard Markovian argument. A detailed discussion shows that the solution of the problem involves an optimal surrender boundary. The properties of the optimal surrender boundary are given. The regime-switching Volterra-type integral equation of the optimal surrender boundary is derived by probabilistic methods. Furthermore, a sensitivity analysis is performed for the optimal surrender decision. In the end, we adopt the trinomial tree method to determine the optimal strategy.

In this paper, we propose an innovative approach to determine the approximate solution of the coupled time-fractional Keller-Segel (K-S) model. We use the fractional complex transform (FCT) to switch the model into its differential partner, and then, the homotopy perturbation method (HPM) is introduced to tackle its nonlinear elements using He’s polynomials. This two-scale theory helps to define the physical meaning of the FCT for the solution of the K-S model. Some examples are illustrated to show that the proposed scheme presents the significant results. The considerable findings show that this strategy does not require any assumptions and also reduces the massive computations without imposing any constraints. This technique is also suitable in functional studies of fractal calculus due to its powerful and robust support for nonlinear problems.

The purpose of this paper is to investigate the approximate solution of the casting-mould heterogeneous system with Caputo derivative under the homotopy idea. The symmetry design of the system contains the integer partial differential equations and the fractional-order partial differential equations. We apply Yang transform homotopy perturbation method (T-HPM) to find the approximate solution of temperature distribution in the casting-mould heterogeneous system. The T-HPM is a combined form of Yang transform (T) and the homotopy perturbation method (HPM) using He’s polynomials. Some examples are provided to demonstrate the superiority of the suggested technique. The significant findings reveal that T-HPM minimizes the enormous without imposing any assumptions. Due to its powerful and robust support for nonlinear problems, this approach presents a remarkable appearance in the functional studies of fractal calculus.

In this analysis, we develop a new approach to investigate the semianalytical solution of the delay differential equations. Mohand transform coupled with the homotopy perturbation method is called Mohand homotopy perturbation transform method (MHPTM) and performs the solution results in the form of series. The beauty of this approach is that it does not need to compute the values of the Lagrange multiplier as in the variational iteration method, and also, there is no need to implement the convolution theorem as in the Laplace transform. The main purpose of this scheme is to reduce the less computational work and the error analysis of the problems than others studied in the literature. Some illustrated examples are interpreted to confirm the accuracy of the newly developed scheme.

The time-fractional Emden-Fowler model is an interesting mathematical model because it is widely used and important in mathematics and applied sciences. The current investigation compiles the Elzaki transform homotopy perturbation method (ET-HPM) and the Laplace transform homotopy perturbation method (LT-HPM) for the analytical solution of the considered model. In their developments, we show that fractional orders are simple to compute, and the resulting series lead to accurate results after a few iterations. We consider the fractional orders in the context of the Caputo form. The primary aim of these approaches is to compare the analytical results of the models under consideration. These schemes do not impose any constraints on the parameters or make any assumptions that could complicate the process of obtaining the analytical results. We show that the derived outcomes are in convergent sequence with a rapid iterative formula. The iterative series and the graphical illustrations in various fractional orders confirm the efficiency of the proposed approaches. The derived results are also compared to the precise solutions, which show that our suggested approaches are simple, effective, and accurate.

When the market environment changes, we extend the self‐exciting price impact model and further analysis of investors’ liquidation behaviour. It is assumed that the model is accompanied by an exponential decay factor when the temporary impact and its coefficient are linear and nonlinear. Using the optimal control method, we obtain that the optimal liquidation behaviours satisfy the second‐order nonlinear ODEs with variable coefficients in the case of linear and nonlinear temporary impact. Next, we solve the ODEs and get the form of the investors’ optimal liquidation behaviour in four cases. Furthermore, we prove the decreasing properties of the optimal liquidation behaviour under the linear temporary impact. Through numerical simulation, we further explain the influence of the changed parameters ρ, a, b, x, and α on the investors’ liquidation strategy X_t in twelve scenarios. Some interesting properties have been found.