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

Infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone-1

0 (A), and an MOI dose-response relationship 2 days after infection (B). Symbols : ▫, virulent strain AA100jm; ▪, avirulent strain mutant. Data are mean ± SD of three wells. * < 0.05.<p><b>Copyright information:</b></p><p>Taken from "infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone"</p><p>Respiratory Research 2008;9(1):39-39.</p><p>Published online 1 May 2008</p><p>PMCID:PMC2390540.</p><p></p

Infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone-4

Ed high-power fields, and TUNEL-positive cells were expressed as a ratio per total number of cells. Symbols: ▫, virulent strain AA100jm; ▪, avirulent strain mutant. Data are mean ± SD of three different experiments. * < 0.05.<p><b>Copyright information:</b></p><p>Taken from "infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone"</p><p>Respiratory Research 2008;9(1):39-39.</p><p>Published online 1 May 2008</p><p>PMCID:PMC2390540.</p><p></p

Infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone-13

infection. Symbols : ▫, virulent strain AA100jm; ▪, avirulent strain mutant. Data are mean ± SD of three wells. * < 0.05.<p><b>Copyright information:</b></p><p>Taken from "infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone"</p><p>Respiratory Research 2008;9(1):39-39.</p><p>Published online 1 May 2008</p><p>PMCID:PMC2390540.</p><p></p

Infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone-11

Cted cells were cultured further. The number of viable bacteria in each well was determined by the CFU counting method. Symbols : ▫, virulent strain AA100jm; ▪, avirulent strain mutant. Data are mean ± SD of four wells. * < 0.05.<p><b>Copyright information:</b></p><p>Taken from "infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone"</p><p>Respiratory Research 2008;9(1):39-39.</p><p>Published online 1 May 2008</p><p>PMCID:PMC2390540.</p><p></p

Infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone-8

TUNEL method was analyzed by flow cytometry. The nuclear DNA fragmentation of virulent strain AA100jm-infected cells without (A) and with methyl prednisolone (B), is shown with no-pretreatment/no-stimulation cells (control) (C), and 30 μM mitomycin C-exposed cells without (D) and with methyl prednisolone (E). FS indicates the cell sizes. m-P; methyl prednisolone.<p><b>Copyright information:</b></p><p>Taken from "infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone"</p><p>Respiratory Research 2008;9(1):39-39.</p><p>Published online 1 May 2008</p><p>PMCID:PMC2390540.</p><p></p

Infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone-12

(A), and an MOI dose-response relationship 2 days after infection (B). Symbols : ▫, virulent strain AA100jm; ▪, avirulent strain mutant. Data are mean ± SD of three wells. * < 0.05.<p><b>Copyright information:</b></p><p>Taken from "infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone"</p><p>Respiratory Research 2008;9(1):39-39.</p><p>Published online 1 May 2008</p><p>PMCID:PMC2390540.</p><p></p

Infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone-3

Entation of cells 2 days after exposure to 30 μM mitomycin C (I) and with no treatment (control) (J). Cells were observed with a confocal laser scanning microscopy (all 200 ×). The nuclear DNA fragmentation is shown in green (FITC staining), and A549 cell nuclei in red (PI staining).<p><b>Copyright information:</b></p><p>Taken from "infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone"</p><p>Respiratory Research 2008;9(1):39-39.</p><p>Published online 1 May 2008</p><p>PMCID:PMC2390540.</p><p></p

Infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone-7

Caspase 3) were detected.<p><b>Copyright information:</b></p><p>Taken from "infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone"</p><p>Respiratory Research 2008;9(1):39-39.</p><p>Published online 1 May 2008</p><p>PMCID:PMC2390540.</p><p></p

Infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone-9

Rically. The activities of caspase 3 (A), 8 (B), 9 (C), and 1 (D) are presented. Data are mean ± SD of four or six different experiments. * < 0.05. m-P; methyl prednisolone.<p><b>Copyright information:</b></p><p>Taken from "infection induces programmed cell death, caspase activation, and release of high-mobility group box 1 protein in A549 alveolar epithelial cells: inhibition by methyl prednisolone"</p><p>Respiratory Research 2008;9(1):39-39.</p><p>Published online 1 May 2008</p><p>PMCID:PMC2390540.</p><p></p