A nonlinear Bismut-Elworthy formula for HJB equations with quadratic Hamiltonian in Banach spaces

We consider a Backward Stochastic Differential Equation (BSDE for short) in a Markovian framework for the pair of processes $(Y,Z)$, with generator with quadratic growth with respect to $Z$. The forward equation is an evolution equation in an abstract Banach space. We prove an analogue of the Bismut-Elworty formula when the diffusion operator has a pseudo-inverse not necessarily bounded and when the generator has quadratic growth with respect to $Z$. In particular, our model covers the case of the heat equation in space dimension greater than or equal to 2. We apply these results to solve semilinear Kolmogorov equations for the unknown $v$, with nonlinear term with quadratic growth with respect to $\nabla v$ and final condition only bounded and continuous, and to solve stochastic optimal control problems with quadratic growth

Thermoresponsive microgels based on strong polycations

The aim of this work is to synthetize a series of thermoresponsive microgels that have never been reported before, based on strong polycations, and study their properties such as the change in volume in response to a temperature stimulus. Polymer microgels are interesting materials for practical applications as drug delivery systems, in separation techniques and catalysis. The interest on these materials arises from their physical properties of colloids combined with gel properties. The microgels presented in this work can undergo phase transitions not only in water but also in DMF/water mixtures. A crosslinked polymer that displays cloud point behaviour when heated forms a temperature-sensitive gel network. Cloud point is the temperature above which an aqueous solution of a water-soluble polymer becomes turbid in the case of polymer with LCST (Lower Critical Solution Temperature) behaviour. Upon heating such a gel, the gel shrinkage is observed by expelling water over a temperature range. The transition is largely driven by the entropy gain associated with the release of water from the network, and the concomitant collapse of the polymer chains. In addition, the size of the microgels is tuneable by adding NaCl at different concentration. The synthesis is carried out as a normal radical polymerization always in the same conditions except for the solvent mixture. The homopolymer, synthetized for comparison, is polymerized with RAFT (Reversible Addition-Fragmentation chain-Transfer) method. Nuclear Magnetic Resonance (NMR) confirmed the structure of the microgels validating the synthetic method. The hydrodynamic radius of the microgels after the addition of salty solutions at different concentration is determined by Dynamic Light Scattering (DLS). The thermo-responsive properties are investigated in terms of polarity using fluorescence and turbidity measurements and in terms of changes in volume calculated from the hydrodynamic radius with DLS at different temperature. The microgels show a thermo-responsive behaviour in the temperature range between 10 °C and 90 °C. In fact, the raise in temperature causes an increase in volume and hydrophobicity. Finally, it is reported a trend that follows the NaCl concentration of salt solutions added to the microgels. These microgels can be used for a wide range of applications, amongst them, they are useful support for metal nanoparticles for catalytic purposes. Here, AuNPs are formed directly on the microgel and the formation is ascertained by DLS and TGA (Thermogravimetric Analysis). Then, they are tested to effectively work during a catalysis experiment

Path-dependent SDEs with jumps and irregular drift: well-posedness and Dirichlet properties

We discuss a concept of path-dependent SDE with distributional drift with possible jumps. We interpret it via a suitable martingale problem, for which we provide existence and uniqueness. The corresponding solutions are expected to be Dirichlet processes, nevertheless we give examples of solutions which do not fulfill this property. In the second part of the paper we indeed state and prove significant new results on the class of Dirichlet processes

Stochastic filtering of a pure jump process with predictable jumps and path-dependent local characteristics

The objective of this paper is to study the filtering problem for a system of partially observable processes $(X, Y)$, where $X$ is a non-Markovian pure-jump process representing the signal and $Y$ is a general jump-diffusion which provides observations. Our model covers the case where both processes are not necessarily quasi left-continuous, allowing them to jump at predictable stopping times. By introducing the Markovian version of the signal, we are able to compute an explicit equation for the filtering process via the innovations approach

Optimal control of Piecewise Deterministic Markov Processes: a BSDE representation of the value function

We consider an infinite-horizon discounted optimal control problem for piecewise deterministic Markov processes, where a piecewise open-loop control acts continuously on the jump dynamics and on the deterministic flow. For this class of control problems, the value function can in general be characterized as the unique viscosity solution to the corresponding Hamilton−Jacobi−Bellman equation. We prove that the value function can be represented by means of a backward stochastic differential equation (BSDE) on infinite horizon, driven by a random measure and with a sign constraint on its martingale part, for which we give existence and uniqueness results. This probabilistic representation is known as nonlinear Feynman−Kac formula. Finally we show that the constrained BSDE is related to an auxiliary dominated control problem, whose value function coincides with the value function of the original non-dominated control problem