This dissertation considers a stochastic dynamic system which is governed by a multidimensional diffusion process with time dependent coefficients. The control acts additively on the state of the system. The objective is to minimize the expected cumulative cost associated with the position of the system and the amount of control exerted. It is proved that Hamilton-Jacobi-Bellman’s equation of the problem has a solution, which corresponds to the optimal cost of the problem. We also investigate the smoothness of the free boundary arising from the problem.
In the second part of the dissertation, we study the backward parabolic problem for a nonlinear parabolic equation of the form u_t + Au(t) = f (t, u(t)), u(T ) = ϕ, where A is a positive self-adjoint unbounded operator and f is a Lipschitz function. The problem is ill-posed, in the sense that if the solution does exist, it will not depend continuously on the data. To regularize the problem, we use the quasi-reversibility method to establish a modified problem. We present approximated solutions that depend on a small parameter \\epsilon > 0 and give error estimates for our regularization. These
results extend some work on the nonlinear backward problem. Some numerical examples are given to justify the theoretical analysis.
","abstract_html":"<p>This dissertation considers a stochastic dynamic system which is governed by a multidimensional diffusion process with time dependent coefficients. The control acts additively on the state of the system. The objective is to minimize the expected cumulative cost associated with the position of the system and the amount of control exerted. It is proved that Hamilton-Jacobi-Bellman’s equation of the problem has a solution, which corresponds to the optimal cost of the problem. We also investigate the smoothness of the free boundary arising from the problem.</p> <p>In the second part of the dissertation, we study the backward parabolic problem for a nonlinear parabolic equation of the form u_t + Au(t) = f (t, u(t)), u(T ) = ϕ, where A is a positive self-adjoint unbounded operator and f is a Lipschitz function. The problem is ill-posed, in the sense that if the solution does exist, it will not depend continuously on the data. To regularize the problem, we use the quasi-reversibility method to establish a modified problem. We present approximated solutions that depend on a small parameter \\epsilon > 0 and give error estimates for our regularization. These</p> <p>results extend some work on the nonlinear backward problem. Some numerical examples are given to justify the theoretical analysis.</p>","abstract_has_math":false,"creators":["Nguyen, Nhat Do Minh"],"institution":null,"degree_name":"Ph.D.","degree_level":"Open Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Jose L. Menaldi"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-01-01T08:00:00Z","date_published":"2015-01-01T08:00:00Z","updated_at":"2026-07-24T06:00:17Z","subjects":["dynamic programming","free boundary","optimal stochastic control","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wayne.edu/oa_dissertations/1379","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Jose L. Menaldi"]},{"key":"dc:creator","label":"Author","values":["Nguyen, Nhat Do Minh"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2016-01-01T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["dynamic programming","free boundary","optimal stochastic control","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wayne.edu/oa_dissertations/1379"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This dissertation considers a stochastic dynamic system which is governed by a multidimensional diffusion process with time dependent coefficients. The control acts additively on the state of the system. The objective is to minimize the expected cumulative cost associated with the position of the system and the amount of control exerted. It is proved that Hamilton-Jacobi-Bellman’s equation of the problem has a solution, which corresponds to the optimal cost of the problem. We also investigate the smoothness of the free boundary arising from the problem.
In the second part of the dissertation, we study the backward parabolic problem for a nonlinear parabolic equation of the form u_t + Au(t) = f (t, u(t)), u(T ) = ϕ, where A is a positive self-adjoint unbounded operator and f is a Lipschitz function. The problem is ill-posed, in the sense that if the solution does exist, it will not depend continuously on the data. To regularize the problem, we use the quasi-reversibility method to establish a modified problem. We present approximated solutions that depend on a small parameter \\epsilon > 0 and give error estimates for our regularization. These
results extend some work on the nonlinear backward problem. Some numerical examples are given to justify the theoretical analysis.
"]},{"key":"dc:title","label":"Title","values":["On A Multi-Dimensional Singular Stochastic Control Problem: The Parabolic Case"]}]}],"canonical_facts":{"dc:contributor":["Jose L. Menaldi"],"dc:creator":["Nguyen, Nhat Do Minh"],"dc:date.available":["2016-01-01T08:00:00Z"],"dc:description.abstract":["This dissertation considers a stochastic dynamic system which is governed by a multidimensional diffusion process with time dependent coefficients. The control acts additively on the state of the system. The objective is to minimize the expected cumulative cost associated with the position of the system and the amount of control exerted. It is proved that Hamilton-Jacobi-Bellman’s equation of the problem has a solution, which corresponds to the optimal cost of the problem. We also investigate the smoothness of the free boundary arising from the problem.
In the second part of the dissertation, we study the backward parabolic problem for a nonlinear parabolic equation of the form u_t + Au(t) = f (t, u(t)), u(T ) = ϕ, where A is a positive self-adjoint unbounded operator and f is a Lipschitz function. The problem is ill-posed, in the sense that if the solution does exist, it will not depend continuously on the data. To regularize the problem, we use the quasi-reversibility method to establish a modified problem. We present approximated solutions that depend on a small parameter \\epsilon > 0 and give error estimates for our regularization. These
results extend some work on the nonlinear backward problem. Some numerical examples are given to justify the theoretical analysis.
"],"dc:identifier":["https://digitalcommons.wayne.edu/oa_dissertations/1379"],"dc:subject":["dynamic programming","free boundary","optimal stochastic control","Mathematics"],"dc:title":["On A Multi-Dimensional Singular Stochastic Control Problem: The Parabolic Case"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Open Access Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T06:00:17Z"}