A note on the existence of solutions to Markovian superquadratic BSDEs with an unbounded terminal condition

In [Stochastc Process. Appl., 122(9):3173-3208], the author proved the existence and the uniqueness of solutions to Markovian superquadratic BSDEs with an unbounded terminal condition when the generator and the terminal condition are locally Lipschitz. In this paper, we prove that the existence result remains true for these BSDEs when the regularity assumptions on the terminal condition is weakened

Well-posedness of semilinear stochastic wave equations with H\"{o}lder continuous coefficients

We prove that semilinear stochastic abstract wave equations, including wave and plate equations, are well-posed in the strong sense with an $\alpha$-H\"{o}lder continuous drift coefficient, if $\alpha \in (2/3,1)$. The uniqueness may fail for the corresponding deterministic PDE and well-posedness is restored by adding an external random forcing of white noise type. This shows a kind of regularization by noise for the semilinear wave equation. To prove the result we introduce an approach based on backward stochastic differential equations. We also establish regularizing properties of the transition semigroup associated to the stochastic wave equation by using control theoretic results

Infinite Horizon and Ergodic Optimal Quadratic Control for an Affine Equation with Stochastic Coefficients

We study quadratic optimal stochastic control problems with control dependent noise state equation perturbed by an affine term and with stochastic coefficients. Both infinite horizon case and ergodic case are treated. To this purpose we introduce a Backward Stochastic Riccati Equation and a dual backward stochastic equation, both considered in the whole time line. Besides some stabilizability conditions we prove existence of a solution for the two previous equations defined as limit of suitable finite horizon approximating problems. This allows to perform the synthesis of the optimal control

Stochastic Optimal Control with Delay in the Control: solution through partial smoothing

Stochastic optimal control problems governed by delay equations with delay in the control are usually more difficult to study than the the ones when the delay appears only in the state. This is particularly true when we look at the associated Hamilton-Jacobi-Bellman (HJB) equation. Indeed, even in the simplified setting (introduced first by Vinter and Kwong for the deterministic case) the HJB equation is an infinite dimensional second order semilinear Partial Differential Equation (PDE) that does not satisfy the so-called "structure condition" which substantially means that "the noise enters the system with the control." The absence of such condition, together with the lack of smoothing properties which is a common feature of problems with delay, prevents the use of the known techniques (based on Backward Stochastic Differential Equations (BSDEs) or on the smoothing properties of the linear part) to prove the existence of regular solutions of this HJB equation and so no results on this direction have been proved till now. In this paper we provide a result on existence of regular solutions of such kind of HJB equations and we use it to solve completely the corresponding control problem finding optimal feedback controls also in the more difficult case of pointwise delay. The main tool used is a partial smoothing property that we prove for the transition semigroup associated to the uncontrolled problem. Such results holds for a specific class of equations and data which arises naturally in many applied problems

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

Lifting partial smoothing to solve HJB equations and stochastic control problems

We study a family of stochastic control problems arising in typical applications (such as boundary control and control of delay equations with delay in the control) with the ultimate aim of finding solutions of the associated HJB equations, regular enough to find optimal feedback controls. These problems are difficult to treat since the underlying transition semigroups do not possess good smoothing properties nor the so-called "structure condition" which typically allows to apply the backward equations approach. In the papers [14], [15], and, more recently, [16] we studied such problems developing new partial smoothing techniques which allowed us to obtain the required regularity in the case when the cost functional is independent of the state variable. This is a somehow strong restriction which is not verified in most applications. In this paper (which can be considered a continuation of the research of the above papers) we develop a new approach to overcome this restriction. We extend the partial smoothing result to a wider class of functions which depend on the whole trajectory of the underlying semigroup and we use this as a key tool to improve our regularity result for the HJB equation. The fact that such class depends on trajectories requires a nontrivial technical work as we have to lift the original transition semigroup to a space of trajectories, defining a new "high-level" environment where our problems can be solved.Comment: arXiv admin note: text overlap with arXiv:2107.0430