Michele Giordano

We establish a framework for the study of backward stochastic Volterra integral equations (BSVIE) driven by time-changed Lévy noises. In fact we shall consider a random measure $\mu$ that can be decomposed as the sum of a conditional Gaussian measure and a conditional centered Poisson measure. In this paper we deal with two information flows: $F_t$ namely the smallest right continuous filtration to which $\mu$ is adapted, $G_t$, generated by $\mu$ and the entire history of the time change processes, and we shall consider the information $F$ as partial with respect to $G$. Given a controlled dynamic, we will consider the optimization problem of finding the supremum of a functional $J(u)$ for a suitable control set $A$ which we consider to be either $F$ or $G$ predictable. We prove both a sufficient and a necessary maximum principle for such performance functional, showing that in the $F$-predictable case we can find a solution by projecting the results obtained for the $G$-predictable case onto the $F$-predictable one. We shall make use of stochastic derivatives. We stress that we cannot use the classical Malliavin calculus as our integrators are not the Brownian motion nor the centered Poisson random measure. Indeed we could use a conditional form of such calculus as introduced by Yablonski, however we resolve by using the non-anticipating derivative introduced and for martingale random fields as integrators. The use of the non-anticipating derivative has also the advantage that we do not require more restrictive conditions on domains, since it is already well defined for all $L^2(P)$ random variables. When studying such problems, we come across a BSVIE driven by a noise $\mu$ as above: we prove existence and uniqueness results for such BSVIE and we compute an explicit solution in the linear case. Examples and applications will be presented.

We establish a framework for the study of backward stochastic Volterra integral equations (BSVIE) driven by time-changed Lévy noises. In fact we shall consider a random measure $\mu$ that can be decomposed as the sum of a conditional Gaussian measure and a conditional centered Poisson measure. In this paper we deal with two information flows: $F_t$ namely the smallest right continuous filtration to which $\mu$ is adapted, $G_t$, generated by $\mu$ and the entire history of the time change processes, and we shall consider the information $F$ as partial with respect to $G$. Given a controlled dynamic, we will consider the optimization problem of finding the supremum of a functional $J(u)$ for a suitable control set $A$ which we consider to be either $F$ or $G$ predictable. We prove both a sufficient and a necessary maximum principle for such performance functional, showing that in the $F$-predictable case we can find a solution by projecting the results obtained for the $G$-predictable case onto the $F$-predictable one. We shall make use of stochastic derivatives. We stress that we cannot use the classical Malliavin calculus as our integrators are not the Brownian motion nor the centered Poisson random measure. Indeed we could use a conditional form of such calculus as introduced by Yablonski, however we resolve by using the non-anticipating derivative introduced and for martingale random fields as integrators. The use of the non-anticipating derivative has also the advantage that we do not require more restrictive conditions on domains, since it is already well defined for all $L^2(P)$ random variables. When studying such problems, we come across a BSVIE driven by a noise $\mu$ as above: we prove existence and uniqueness results for such BSVIE and we compute an explicit solution in the linear case. Examples and applications will be presented.

We establish a framework for the study of backward stochastic Volterra integral equations (BSVIE) driven by time-changed Lévy noises. In fact we shall consider a random measure $\mu$ that can be decomposed as the sum of a conditional Gaussian measure and a conditional centered Poisson measure. In this paper we deal with two information flows: $F_t$ namely the smallest right continuous filtration to which $\mu$ is adapted, $G_t$, generated by $\mu$ and the entire history of the time change processes, and we shall consider the information $F$ as partial with respect to $G$. Given a controlled dynamic, we will consider the optimization problem of finding the supremum of a functional $J(u)$ for a suitable control set $A$ which we consider to be either $F$ or $G$ predictable. We prove both a sufficient and a necessary maximum principle for such performance functional, showing that in the $F$-predictable case we can find a solution by projecting the results obtained for the $G$-predictable case onto the $F$-predictable one. We shall make use of stochastic derivatives. We stress that we cannot use the classical Malliavin calculus as our integrators are not the Brownian motion nor the centered Poisson random measure. Indeed we could use a conditional form of such calculus as introduced by Yablonski, however we resolve by using the non-anticipating derivative introduced and for martingale random fields as integrators. The use of the non-anticipating derivative has also the advantage that we do not require more restrictive conditions on domains, since it is already well defined for all $L^2(P)$ random variables. When studying such problems, we come across a BSVIE driven by a noise $\mu$ as above: we prove existence and uniqueness results for such BSVIE and we compute an explicit solution in the linear case. Examples and applications will be presented.

We establish a framework for the study of backward stochastic Volterra integral equations (BSVIE) driven by time-changed Lévy noises. In fact we shall consider a random measure $\mu$ that can be decomposed as the sum of a conditional Gaussian measure and a conditional centered Poisson measure. In this paper we deal with two information flows: $F_t$ namely the smallest right continuous filtration to which $\mu$ is adapted, $G_t$, generated by $\mu$ and the entire history of the time change processes, and we shall consider the information $F$ as partial with respect to $G$. Given a controlled dynamic, we will consider the optimization problem of finding the supremum of a functional $J(u)$ for a suitable control set $A$ which we consider to be either $F$ or $G$ predictable. We prove both a sufficient and a necessary maximum principle for such performance functional, showing that in the $F$-predictable case we can find a solution by projecting the results obtained for the $G$-predictable case onto the $F$-predictable one. We shall make use of stochastic derivatives. We stress that we cannot use the classical Malliavin calculus as our integrators are not the Brownian motion nor the centered Poisson random measure. Indeed we could use a conditional form of such calculus as introduced by Yablonski, however we resolve by using the non-anticipating derivative introduced and for martingale random fields as integrators. The use of the non-anticipating derivative has also the advantage that we do not require more restrictive conditions on domains, since it is already well defined for all $L^2(P)$ random variables. When studying such problems, we come across a BSVIE driven by a noise $\mu$ as above: we prove existence and uniqueness results for such BSVIE and we compute an explicit solution in the linear case. Examples and applications will be presented.

We establish a framework for the study of backward stochastic Volterra integral equations (BSVIE) driven by time-changed Lévy noises. In fact we shall consider a random measure $\mu$ that can be decomposed as the sum of a conditional Gaussian measure and a conditional centered Poisson measure. In this paper we deal with two information flows: $F_t$ namely the smallest right continuous filtration to which $\mu$ is adapted, $G_t$, generated by $\mu$ and the entire history of the time change processes, and we shall consider the information $F$ as partial with respect to $G$. Given a controlled dynamic, we will consider the optimization problem of finding the supremum of a functional $J(u)$ for a suitable control set $A$ which we consider to be either $F$ or $G$ predictable. We prove both a sufficient and a necessary maximum principle for such performance functional, showing that in the $F$-predictable case we can find a solution by projecting the results obtained for the $G$-predictable case onto the $F$-predictable one. We shall make use of stochastic derivatives. We stress that we cannot use the classical Malliavin calculus as our integrators are not the Brownian motion nor the centered Poisson random measure. Indeed we could use a conditional form of such calculus as introduced by Yablonski, however we resolve by using the non-anticipating derivative introduced and for martingale random fields as integrators. The use of the non-anticipating derivative has also the advantage that we do not require more restrictive conditions on domains, since it is already well defined for all $L^2(P)$ random variables. When studying such problems, we come across a BSVIE driven by a noise $\mu$ as above: we prove existence and uniqueness results for such BSVIE and we compute an explicit solution in the linear case. Examples and applications will be presented.