The Existence And Uniqueness Of Solutions To Forward And Backward Stochastic Pantograph Equations

In 1951, K. lto pioneered the theory of Ito stochastic differential equations(SDEs). Since then, SDEs have been developed rapidly. SDE can describe some random phenomena in the real society. It is widely used in several subjects, such as engineering cybernetics and ecology. It plays an effective role in other branches of mathematical fields. In the real society, many phenomena depend on the past events. The development at time tmay depend on both the state at time t and the history. Stochastic delay differential equations (SDDEs) describe the phenomena. Stochastic pantograph differential equation is a very special delay differential equation. It arises in different fields of pure and applied mathematics such as dynamical systems, probability, quantum mechanics and electro dynamics.In this thesis, we are dedicated to study the forward and backward stochastic pantograph equations. We prove separately the existence and uniqueness of solutions to forward and backward stochastic pantograph equations:In second chapter, we study the following generalized stochastic pantograph differential equation: dx(t)= f(t, x(t), x(q(t))dt+g(t, x(t), x(q(t)))dw,, t∈[0, T]. x(0)= ζ(0). where 0≤q(t)≤qot,0< q0<1, q’(t)≥, N> 0. First, by Burkholder-Davis-Gundy in-equality, Holder inequality and Gronwall inequality, we prove the existence and uniqueness of a solution to the above equation which is under the Lipschitz condition. Then, by defining truncation function and stopping time, we prove the existence and uniqueness of a solution to the above equation which is under the local Lipschitz condition.In third chapter, we study the following generalized backward stochastic pantograph differential equation: where 0≤q(t)≤qot,0≤q0< 1, q’(t)≥1/N, N> 0. By constructing a Picard scheme and showing its convergence, we prove that, for a sufficiently small time horizon T or for a sufficiently small Lipschitz constant K, the above equation has a unique solution.