9 research outputs found
Regularity of the Optimal Stopping Problem for Jump Diffusions
The value function of an optimal stopping problem for jump diffusions is
known to be a generalized solution of a variational inequality. Assuming that
the diffusion component of the process is nondegenerate and a mild assumption
on the singularity of the L\'{e}vy measure, this paper shows that the value
function of this optimal stopping problem on an unbounded domain with
finite/infinite variation jumps is in with . As a consequence, the smooth-fit property holds.Comment: To Appear in the SIAM Journal on Control and Optimizatio
A Bank Salvage Model by Impulse Stochastic Controls
The present paper is devoted to the study of a bank salvage model with a finite time horizon that is subjected to stochastic impulse controls. In our model, the bank\u2019s default time is a completely inaccessible random quantity generating its own filtration, then reflecting the unpredictability of the event itself. In this framework the main goal is to minimize the total cost of the central controller, which can inject capitals to save the bank from default. We address the latter task, showing that the corresponding quasi-variational inequality (QVI) admits a unique viscosity solution\u2014Lipschitz continuous in space and H\uf6lder continuous in time. Furthermore, under mild assumptions on the dynamics the smooth-fit W(1,2),ploc property is achieved for any 1<+ 1e
Error estimates of penalty schemes for quasi-variational inequalities arising from impulse control problems
This paper proposes penalty schemes for a class of weakly coupled systems of
Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) arising from
stochastic hybrid control problems of regime-switching models with both
continuous and impulse controls. We show that the solutions of the penalized
equations converge monotonically to those of the HJBQVIs. We further establish
that the schemes are half-order accurate for HJBQVIs with Lipschitz
coefficients, and first-order accurate for equations with more regular
coefficients. Moreover, we construct the action regions and optimal impulse
controls based on the error estimates and the penalized solutions. The penalty
schemes and convergence results are then extended to HJBQVIs with possibly
negative impulse costs. We also demonstrate the convergence of monotone
discretizations of the penalized equations, and establish that policy iteration
applied to the discrete equation is monotonically convergent with an arbitrary
initial guess in an infinite dimensional setting. Numerical examples for
infinite-horizon optimal switching problems are presented to illustrate the
effectiveness of the penalty schemes over the conventional direct control
scheme.Comment: Accepted for publication in SIAM Journal on Control and Optimizatio
Impulse Control in Finance: Numerical Methods and Viscosity Solutions
The goal of this thesis is to provide efficient and provably convergent
numerical methods for solving partial differential equations (PDEs) coming from
impulse control problems motivated by finance. Impulses, which are controlled
jumps in a stochastic process, are used to model realistic features in
financial problems which cannot be captured by ordinary stochastic controls.
The dynamic programming equations associated with impulse control problems
are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than
in certain special cases, the numerical schemes that come from the
discretization of HJBQVIs take the form of complicated nonlinear matrix
equations also known as Bellman problems. We prove that a policy iteration
algorithm can be used to compute their solutions. In order to do so, we employ
the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a
byproduct of our analysis, we obtain some new results regarding a particular
family of Markov decision processes which can be thought of as impulse control
problems on a discrete state space and the relationship between w.c.d.d.
matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to
directly use the seminal result of Barles and Souganidis (concerning the
convergence of monotone, stable, and consistent numerical schemes to the
viscosity solution) to prove the convergence of our schemes. We address this
issue by extending the work of Barles and Souganidis to nonlocal PDEs in a
manner general enough to apply to HJBQVIs. We apply our schemes to compute the
solutions of various classical problems from finance concerning optimal control
of the exchange rate, optimal consumption with fixed and proportional
transaction costs, and guaranteed minimum withdrawal benefits in variable
annuities
Impulse Control in Finance: Numerical Methods and Viscosity Solutions
The goal of this thesis is to provide efficient and provably convergent
numerical methods for solving partial differential equations (PDEs) coming from
impulse control problems motivated by finance. Impulses, which are controlled
jumps in a stochastic process, are used to model realistic features in
financial problems which cannot be captured by ordinary stochastic controls.
The dynamic programming equations associated with impulse control problems
are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than
in certain special cases, the numerical schemes that come from the
discretization of HJBQVIs take the form of complicated nonlinear matrix
equations also known as Bellman problems. We prove that a policy iteration
algorithm can be used to compute their solutions. In order to do so, we employ
the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a
byproduct of our analysis, we obtain some new results regarding a particular
family of Markov decision processes which can be thought of as impulse control
problems on a discrete state space and the relationship between w.c.d.d.
matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to
directly use the seminal result of Barles and Souganidis (concerning the
convergence of monotone, stable, and consistent numerical schemes to the
viscosity solution) to prove the convergence of our schemes. We address this
issue by extending the work of Barles and Souganidis to nonlocal PDEs in a
manner general enough to apply to HJBQVIs. We apply our schemes to compute the
solutions of various classical problems from finance concerning optimal control
of the exchange rate, optimal consumption with fixed and proportional
transaction costs, and guaranteed minimum withdrawal benefits in variable
annuities