Abstract. Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs, in short) with closed control regions are formulated and studied. Instead of using spike variation method as one may imagine, here we turn to treat the non-convexity of the control regions by borrowing some tools in set-valued analysis and adapting them into our stochastic control systems. A ...

Abstract We define a Skorohod type anticipative stochastic integral that extends the Itô integral not only with respect to the Wiener process, but also with respect to a wide class of stochastic processes satisfying certain homogeneity and smoothness conditions, without requirements relative to filtrations such as adaptedness. Using this integral, a change of variable formula that extends the c...

A combinatorial construction of the multiple stochastic integral is developed using sequences in Clifford (geometric) algebras. In particular, sequences of Berezin integrals in an ascending chain of geometric algebras converge in mean to the iterated stochastic integral. By embedding such chains within an infinite-dimensional Clifford algebra, an infinitedimensional analogue of the Berezin inte...

We construct the basis of a stochastic calculus for so-called Volterra processes, i.e., processes which are defined as the stochastic integral of a timedependent kernel with respect to a standard Brownian motion. For these processes which are natural generalization of fractional Brownian motion, we construct a stochastic integral and show some of its main properties: regularity with respect to ...

Almost stochastic dominance allows small violations of stochastic dominance rules to avoid situations where most decision makers prefer one alternative to another but stochastic dominance cannot rank them. However, the commonly used integral condition for almost seconddegree stochastic dominance does not map into the corresponding classes of increasing concave utility functions with bounded der...

We propose a method to construct the stochastic integral simultaneously under a non-dominated family of probability measures. Pathby-path, and without referring to a probability measure, we construct a sequence of Lebesgue-Stieltjes integrals whose medial limit coincides with the usual stochastic integral under essentially any probability measure such that the integrator is a semimartingale. Th...

We develop a stochastic calculus for the fractional Brownian motion with Hurst parameter H > 2 using the techniques of the Malliavin calclulus. We establish estimates in Lp, maximal inequalities and a continuity criterion for the stochastic integral. Finally, we derive an Itô’s formula for integral processes.

In a work of van Gaans (2005a) stochastic integrals are regarded as L2-curves. In Filipović and Tappe (2008) we have shown the connection to the usual Itô-integral for càdlàg-integrands. The goal of this note is to complete this result and to provide the full connection to the Itô-integral. We also sketch an application to stochastic partial differential equations.

A new computational method based on Haar wavelets is proposed for solving multidimensional stochastic Itô-Volterra integral equations. The block pulse functions and their relations to Haar wavelets are employed to derive a general procedure for forming stochastic operational matrix of Haar wavelets. Then, Haar wavelets basis along with their stochastic operational matrix are used to approximate...

We consider the solution u(x, t) to a stochastic heat equation. For fixed x, the process F (t) = u(x, t) has a nontrivial quartic variation. It follows that F is not a semimartingale, so a stochastic integral with respect to F cannot be defined in the classical Itô sense. We show that for sufficiently differentiable functions g(x, t), a stochastic integral ∫ g(F (t), t) dF (t) exists as a limit...