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A B S T R A C TThis paper aims at obtaining the analytical solutions of some boundary value problems garnished with stochastic volatility, price volatility risk and the risk premium. A set of functions is constructed which transforms the problem into a Laplace equation and a heat equation. The analytical solutions of these equations are obtained. Then existence of a unique solution is achieved which represents the behavior of volatile behavior of the system. Some numerical illustration of the models is obtained using the Maple software.

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This paper investigates the maximum interval of stability and convergence of solution of a forced Mathieu's equation, using a combination of Frobenius method and Eigenvalue approach. The results indicated that the equilibrium point was found to be unstable and maximum bounds were found on the derivative of the restoring force showing sharp condition for the existence of periodic solution. Furthermore, the solution to Mathieu's equation converges which extends and improves some results in literature.