Integro-differential Equations of Fractional Order with Nonlocal Fractional Boundary Conditions Associated with Financial Asset Model

In this article, we discuss the existence of solutions for a boundaryvalue problem of integro-differential equations of fractional order with nonlocal fractional boundary conditions by means of some standard tools of fixed point theory. Our problem describes a more general form of fractional stochastic dynamic model for financial asset. An illustrative example is also presented. 1. Formulation and basic result Fractional calculus, regarded as a branch of mathematical analysis dealing with derivatives and integrals of arbitrary order, has been extensively developed and applied to a variety of problems appearing in sciences and engineering. It is worthwhile to mention that this branch of mathematics has played a crucial role in exploring various characteristics of engineering materials such as viscoelastic polymers, foams, gels, and animal tissues, and their engineering and scientific applications. For a recent detailed survey of the activities involving fractional calculus, we refer a recent paper by Machado, Kiryakova and Mainardi [16]. Some recent work on the topic can be found in [1, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 17] and references therein. The underlying dynamics of equity prices following a jump process or a Levy process provide a basis for modeling of financial assets. The CGMY, KoBoL and FMLS are examples of some interesting financial models involving the dynamics of stock prices. In [8], it is shown that the prices of financial derivatives are expressible in terms of fractional derivative. In [15], the author described the dynamics of a financial asset by the fractional stochastic differential equation of order μ (representing the dynamical memory effects in the market stochastic evolution) with fractional boundary conditions. In the present paper, we study a more general model associated with financial asset. 2000 Mathematics Subject Classification. 34A08, 34B10, 34B15.