On difficulties appearing in the study of stochastic Volterra equations
This paper is concerned with stochastic Volterra equations of the general form NEWLINE\[NEWLINE X(t)=X_{0}+\int_{0}^{t}a(t-\tau)AX(\tau)d\tau + \int_{0}^{t}\Psi(\tau)dW(\tau), \quad t \geq 0,NEWLINE\]NEWLINE where \(X(\cdot)\) is an \(H\)-valued stochastic process on a separable Hilbert space \(H\), \(A : D(A) \subset H \to H\) is a closed linear operator with dense domain, \(W\) is the cylindrical Wiener process, \(a \in L^{1}_{\mathrm{loc}}({\mathbb R}_{+};{\mathbb R})\) and \(\Psi\) is an appropriate predictable process. Such equations arise as mathematical models for a wide variety of phenomena in natural science or in economics.NEWLINENEWLINEThe mathematical theory for such equations is reviewed, especially those aspects related to the resolvent approach, which is a generalization of the semigroup approach. To this end, the theory of resolvents for deterministic Volterra equations is briefly mentioned and its main technical difficulties as compared to the semigroup approach are highlighted, especially in connection with the properties of the scalar kernel function \(a\). Then, the class of completely positive kernels is introduced, along with results due to the author and Lizama, and inspired by work of Clement and Nohel, that guarantee the validity of analogues of the celebrated Hille-Yosida theorem in the case of resolvent operators. The author then moves to the study of the stochastic case by expressing the solution as a stochastic convolution of the cylindrical Wiener process with respect to the resolvent family. Results of Clement and Da Prato, as well as of the author and Lizama on the properties of the stochastic convolution are stated, especially in connection with regularity of the process and its relation with the properties of \(\Psi\), the operator \(A\) and the kernel \(a\). These results are then used to consider the well posedness and the regularity of the solutions of the stochastic Volterra equation in question. The paper concludes with some interesting comments. The paper forms a nice overview of a very interesting and often technical field, and will be of use to newcomers and experts in the field alike.NEWLINENEWLINEFor the entire collection see [Zbl 1238.81004].
scientific article; zbMATH DE number 6042764