86 research outputs found
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
-H\"{o}lder continuous drift coefficient, if . 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
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 , with generator with
quadratic growth with respect to . 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
. 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 , with nonlinear term with
quadratic growth with respect to and final condition only bounded
and continuous, and to solve stochastic optimal control problems with quadratic
growth
Stochastic Control Problems with Unbounded Control Operators: solutions through generalized derivatives
This paper deals with a family of stochastic control problems in Hilbert
spaces which arises in typical applications (such as boundary control and
control of delay equations with delay in the control) and for which is
difficult to apply the dynamic programming approach due to the unboudedness of
the control operator and to the lack of regularity of the underlying transition
semigroup. We introduce a specific concept of partial derivative, designed for
this situation, and we develop a method to prove that the associated HJB
equation has a solution with enough regularity to find optimal controls in
feedback form
Ergodic Optimal Quadratic Control for an Affine Equation with Stochastic and Stationary Coefficients
We study ergodic quadratic optimal stochastic control problems for an affine
state equation with state and control dependent noise and with stochastic
coefficients. We assume stationarity of the coefficients and a finite cost
condition. We first treat the stationary case and we show that the optimal cost
corresponding to this ergodic control problem coincides with the one
corresponding to a suitable stationary control problem and we provide a full
characterization of the ergodic optimal cost and control
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