6 research outputs found
Continuous viscosity solutions to linear-quadratic stochastic control problems with singular terminal state constraint
This paper establishes the existence of a unique nonnegative continuous
viscosity solution to the HJB equation associated with a Markovian
linear-quadratic control problems with singular terminal state constraint and
possibly unbounded cost coefficients. The existence result is based on a novel
comparison principle for semi-continuous viscosity sub- and supersolutions for
PDEs with singular terminal value. Continuity of the viscosity solution is
enough to carry out the verification argument
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
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