199 research outputs found
Mild solutions of semilinear elliptic equations in Hilbert spaces
This paper extends the theory of regular solutions ( in a suitable
sense) for a class of semilinear elliptic equations in Hilbert spaces. The
notion of regularity is based on the concept of -derivative, which is
introduced and discussed. A result of existence and uniqueness of solutions is
stated and proved under the assumption that the transition semigroup associated
to the linear part of the equation has a smoothing property, that is, it maps
continuous functions into -differentiable ones. The validity of this
smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck
transition semigroup and for the case of invertible diffusion coefficient
covering cases not previously addressed by the literature. It is shown that the
results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite
horizon optimal stochastic control problems in infinite dimension and that, in
particular, they cover examples of optimal boundary control of the heat
equation that were not treatable with the approaches developed in the
literature up to now
Verification Theorems for Stochastic Optimal Control Problems via a Time Dependent Fukushima - Dirichlet Decomposition
This paper is devoted to present a method of proving verification theorems
for stochastic optimal control of finite dimensional diffusion processes
without control in the diffusion term. The value function is assumed to be
continuous in time and once differentiable in the space variable ()
instead of once differentiable in time and twice in space (), like in
the classical results. The results are obtained using a time dependent
Fukushima - Dirichlet decomposition proved in a companion paper by the same
authors using stochastic calculus via regularization. Applications, examples
and comparison with other similar results are also given.Comment: 34 pages. To appear: Stochastic Processes and Their Application
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
Weak Dirichlet processes with a stochastic control perspective
The motivation of this paper is to prove verification theorems for stochastic
optimal control of finite dimensional diffusion processes without control in
the diffusion term, in the case that the value function is assumed to be
continuous in time and once differentiable in the space variable ()
instead of once differentiable in time and twice in space (), like in
the classical results. For this purpose, the replacement tool of the It\^{o}
formula will be the Fukushima-Dirichlet decomposition for weak Dirichlet
processes. Given a fixed filtration, a weak Dirichlet process is the sum of a
local martingale plus an adapted process which is orthogonal, in the
sense of covariation, to any continuous local martingale. The mentioned
decomposition states that a function of a weak Dirichlet process with
finite quadratic variation is again a weak Dirichlet process. That result is
established in this paper and it is applied to the strong solution of a Cauchy
problem with final condition. Applications to the proof of verification
theorems will be addressed in a companion paper.Comment: 22 pages. To appear: Stochastic Processes and Their Application
Optimal investment models with vintage capital: Dynamic Programming approach
The Dynamic Programming approach for a family of optimal investment models with vintage capital is here developed. The problem falls into the class of infinite horizon optimal control problems of PDE's with age structure that have been studied in various papers (see e.g. [11, 12], [30, 32]) either in cases when explicit solutions can be found or using Maximum Principle techniques. The problem is rephrased into an infinite dimensional setting, it is proven that the value function is the unique regular solution of the associated stationary Hamilton-Jacobi-Bellman equation, and existence and uniqueness of optimal feedback controls is derived. It is then shown that the optimal path is the solution to the closed loop equation. Similar results were proven in the case of finite horizon in [26][27]. The case of infinite horizon is more challenging as a mathematical problem, and indeed more interesting from the point of view of optimal investment models with vintage capital, where what mainly matters is the behavior of optimal trajectories and controls in the long run. The study of infinite horizon is performed through a nontrivial limiting procedure from the corresponding finite horizon problemsOptimal investment, vintage capital, age-structured systems, optimal control, dynamic programming, Hamilton-Jacobi-Bellman equations, linear convex control, boundary control
On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects
We consider a class of dynamic advertising problems under uncertainty in the
presence of carryover and distributed forgetting effects, generalizing a
classical model of Nerlove and Arrow. In particular, we allow the dynamics of
the product goodwill to depend on its past values, as well as previous
advertising levels. Building on previous work of two of the authors, the
optimal advertising model is formulated as an infinite dimensional stochastic
control problem. We obtain (partial) regularity as well as approximation
results for the corresponding value function. Under specific structural
assumptions we study the effects of delays on the value function and optimal
strategy. In the absence of carryover effects, since the value function and the
optimal advertising policy can be characterized in terms of the solution of the
associated HJB equation, we obtain sharper characterizations of the optimal
policy.Comment: numerical example added; minor revision
Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation
We consider a utility maximization problem for an investment-consumption
portfolio when the current utility depends also on the wealth process. Such
kind of problems arise, e.g., in portfolio optimization with random horizon or
with random trading times. To overcome the difficulties of the problem we use
the dual approach. We define a dual problem and treat it by means of dynamic
programming, showing that the viscosity solutions of the associated
Hamilton-Jacobi-Bellman equation belong to a suitable class of smooth
functions. This allows to define a smooth solution of the primal
Hamilton-Jacobi-Bellman equation, proving that this solution is indeed unique
in a suitable class and coincides with the value function of the primal
problem. Some financial applications of the results are provided
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