Parsiad Azimzadeh

Parsiad Azimzadeh, Peter A. Forsyth

This work is motivated by numerical solutions to Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) associated with combined stochastic and impulse control problems. In particular, we consider (i) direct control, (ii) penalized, and (iii) semi-Lagrangian discretization schemes applied to the HJBQVI problem. Scheme (i) takes the form of a Bellman problem involving an operator which is not necessarily contractive. We consider the well-posedness of the Bellman problem and give sufficient conditions for convergence of the corresponding policy iteration. To do so, we use weakly chained diagonally dominant matrices, which give a graph-theoretic characterization of weakly diagonally dominant M-matrices. We compare schemes (i)--(iii) under the following examples: (a) optimal control of the exchange rate, (b) optimal consumption with fixed and proportional transaction costs, and (c) pricing guaranteed minimum withdrawal benefits in variable annuities. We find that one should abstain from using scheme (i).

Parsiad Azimzadeh, Erhan Bayraktar

We introduce high order Bellman equations, extending classical Bellman equations to the tensor setting. We introduce weakly chained diagonally dominant (w.c.d.d.) tensors and show that a sufficient condition for the existence and uniqueness of a positive solution to a high order Bellman equation is that the tensors appearing in the equation are w.c.d.d. M-tensors. In this case, we give a policy iteration algorithm to compute this solution. We also prove that a weakly diagonally dominant Z-tensor with nonnegative diagonals is a strong M-tensor if and only if it is w.c.d.d. This last point is analogous to a corresponding result in the matrix setting and tightens a result from [L. Zhang, L. Qi, and G. Zhou. "M-tensors and some applications." SIAM Journal on Matrix Analysis and Applications (2014)]. We apply our results to obtain a provably convergent numerical scheme for an optimal control problem using an "optimize then discretize" approach which outperforms (in both computation time and accuracy) a classical "discretize then optimize" approach. To the best of our knowledge, a link between M-tensors and optimal control has not been previously established.

Parsiad Azimzadeh

We present a test for determining if a substochastic matrix is convergent. By establishing a duality between weakly chained diagonally dominant (w.c.d.d.) L-matrices and convergent substochastic matrices, we show that this test can be trivially extended to determine whether a weakly diagonally dominant (w.d.d.) matrix is a nonsingular M-matrix. The test's runtime is linear in the order of the input matrix if it is sparse and quadratic if it is dense. This is a partial strengthening of the cubic test in [J. M. Peña., A stable test to check if a matrix is a nonsingular M-matrix, Math. Comp., 247, 1385-1392, 2004]. As a by-product of our analysis, we prove that a nonsingular w.d.d. M-matrix is a w.c.d.d. L-matrix, a fact whose converse has been known since at least 1964. We point out that this strengthens some recent results on M-matrices in the literature.

Parsiad Azimzadeh

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.

Parsiad Azimzadeh, Tommy Carpenter

The blocking probability of a finite-source bufferless queue is a fixed point of the Engset formula, for which we prove existence and uniqueness. Numerically, the literature suggests a fixed point iteration. We show that such an iteration can fail to converge and is dominated by a simple Newton's method, for which we prove a global convergence result. The analysis yields a new Turán-type inequality involving hypergeometric functions, which is of independent interest.

Parsiad Azimzadeh, Erhan Bayraktar, George Labahn

In [Azimzadeh, P., and P. A. Forsyth. "Weakly chained matrices, policy iteration, and impulse control." SIAM J. Num. Anal. 54.3 (2016): 1341-1364], we outlined the theory and implementation of computational methods for implicit schemes for Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs). No convergence proofs were given therein. This work closes the gap by giving rigorous proofs of convergence. We do so by introducing the notion of nonlocal consistency and appealing to a Barles-Souganidis type analysis. Our results rely only on a well-known comparison principle and are independent of the specific form of the intervention operator.