Stochastic Volterra Type Integral Equations And Multivalued Stochastic Evolution Equations With Non-Lipschitz Coefficients

In the study of non-linear differential and integral equations,one usually cannot obtain the explicit expression of solutions.Thus,one has to make some qualitative analysis and find the approximate solution by numerical calculations.As a first step, we must prove the existence and uniqueness for the corresponding equation so that the further discussion makes sense.In most of the well known works about the stochastic differential equations,the non-linear coefficients are always subject to global Lipschitz condition or locally Lipschitz together with linear growth condition.Undoubtedly,it is of great importance to establish the existence and uniqueness under weaker conditions.In this direction,a classical result was obtained in 1971 by Yamada and Watanabe under a kind of non-Lipschitz condition.In recent years,systematic research was made on stochastic differential equations with non-Lipschitz coefficients by many scholars from the following aspects:stochastic homeomorphic flows,large deviations, continuity modulus,Euler approximations,and so on.Compared with the study of stochastic differential equations,non-Lipschitz stochastic integral equations seems relatively lagging.In this thesis,we mainly study the following three type of equations under certain non-Lipschitz conditions:(forward) stochastic Volterra type integral equation,backward stochastic Volterra type integral equation(including two cases:with jump and in infinite dimensional space),multivalued stochastic evolution equation.We shall establish the existence and uniqueness of solutions to these equations,and give regularity results in some places.These contents will be the base of future research.Main tools and methods:(1) In most of cases,we shall use the classical Picard iteration in the theory of ordinary differential equations and stochastic ones.In particular, when we consider the backward stochastic equations,since there exist several solution processes,the iteration scheme will be used repeatedly:from simple to complex and step by step.(2) To derive the continuity of solution for stochastic Volterra equation, the Kolmogorov continuity criterion will play a key role.(3) Instead of Gronwall's inequality,the non-linear Bihari inequality is crutial in dealing with the non-Lipschitz equations.Moreover,we emphasize that the dominating functions in non-Lipschitz conditions always satisfy concavity and some non-integrability near zero.The concavity is for using Jensen's inequality.Because of the non-integrability near zero,we can apply the comparison theorem of ordinary differential equations together with Bihari's inequality to yield many results.(4) Burkholder-Davis-Gundy inequality and H(o｜¨)lder's inequality are important tools in our proof and will be frequently used.Main results:(1) For the stochastic Volterra type integral equation with singular kernels,it is the first time to be investigated under non-Lipschitz conditions. The existence-uniqueness and H(o｜¨)lder's continuity are established.Here,an extended Minkowski's inequality plays a key role.Subsequently,Bihari's inequality is generalized to the equation with fractional integral kernel.With the help of the new inequality,we further weaken the condition on equations with special integral kernels,and obtain the corresponding results.(2) Some work has been done to backward stochastic Volterra type equations under Lipschitz condition.However,it seems that their proofs are not complete.In this thesis, we avoid using the previous idea of backward translation plus segmentation recurrence by applying It(?)'s formula to a properly choosing function,and fill up the gap.Here we generalize the known results to the case with non-Lipschitz coefficients and jump(driven by Brownian motion and Poisson process).And then,for such equation just driven by Brownian motion,the continuity is obtained by rather subtle analysis.(3) It has important significance to consider backward stochastic Volterra type integral equation in infinite dimensional space.By a similar calculation as in the second part,we study this kind of equations driven by cylindrical Brownian motion under non-Lipschitz condition.(4) The theory of maximal monotone operator occupies an important position in the study of nonlinear partial differential equations.Several years ago,some interesting phenomenon was found about stochastic equations involving maximal monotone opera- tor.This stimulates scholars to consider it in the framework of evolutional triple.In the last part we prove the existence and uniqueness of solutions for multivalued stochastic evolution equations with non-Lipschitz coefficients.