In this article, we present a general methodology for control problems driven by the Brownian motion filtration including non-Markovian and non-semimartingale state processes controlled by mutually singular measures. The main result of this paper is the development of a concrete pathwise method for characterizing and computing near-optimal controls for abstract controlled Wiener functionals. The theory does not require ad hoc functional differentiability assumptions on the value process and elipticity conditions on the diffusion components. The analysis is pathwise over suitable finite dimensional spaces and it is based on the weak differential structure introduced by Leão, Ohashi and Simas [31] jointly with measurable selection arguments. The theory is applied to stochastic control problems based on path-dependent SDEs where both drift and possibly degenerated diffusion components are controlled. Optimal control of drifts for path-dependent SDEs driven by fractional Brownian motion is also discussed. We finally provide an application in the context of financial mathematics. Namely, we construct near-optimal controls in a non-Markovian portfolio optimization problem.
Some dynamical properties for a bouncer model-a classical particle of mass m falling in the presence of a constant gravitational field g and hitting elastically a periodically moving wall-in the presence of drag force that is assumed to be proportional to the particle's velocity are studied. The dynamics of the model is described in terms of a two-dimensional nonlinear mapping obtained via solution of the second Newton's law of motion. We characterize the behavior of the average velocity of the particle as function of the control parameters as well as the time. Our results show that the average velocity starts growing at first and then bends towards a regime of constant value, thus confirming that the introduction of drag force is a sufficient condition to suppress Fermi acceleration in the model.
Agradeço a Deus por te me dado a oportunidade de desenvolver essa pesquisa e por ter me capacitado a concluí-la.Agradeço a minha esposa Lidiane por todo auxílio sua compreensão sem limite durante esse período, companheirismo, apoio e por me dar suporte e a força necessária para que pudesse trilhar esseárduo caminho.Aos meus pais Alaor e Marta por todo o apoio e o esforço que fizeram para que eu pudesse chegar até aqui, me ajudando nos momentos difíceis e a minha irmã Patrícia por ter me apoiado, devo muito a eles.Agradeço ao meu orientador Prof. Dorival Leão pela paciência, amizade, dedicação e por todo conhecimento que pude adquirir com sua experiência, além da confiança depositada em mim. Agradeço também carinhosamente ao Prof. Alberto Ohashi, que também me ajudou muito para que esse trabalho pudesse ser concluído.Finalmente, ao CAPES agradeço pelo apoio financeiro cedido. In this thesis, we present a concrete methodology to calculate the -optimal controls for non-Markovian stochastic systems. A pathwise analysis and the use of the discretization structure proposed by Leão and Ohashi [36] jointly with measurable selection arguments, allows us a structure to transform an infinite dimensional problem into a finite dimensional.In this way, we guarantee a concrete description for a rather general class of stochastic problems. Seja X(t), a variável aleatória que representa o estado de um sistema estocástico no tempo t, t ∈ [0, T ], no qual Té uma constante positiva. Por exemplo, X(t)é o capital de um investidor em ativos financeiros no instante t. Neste caso, o estado do processó e quanto dinheiro o investidor tem no instante t.Assumimos queé possível controlar o estado do sistema por uma variável u(t), a qualé denominada variável controle no instante t. Para destacar a ação do controle no estado do sistema, denotaremos o sistema controlado por X u (t).No caso do capital do investidor, definimos u(t) como sendo a quantidade de ativos no tempo t. Nesse caso a quantidade de ativosé o controle e a partir deste, definimos o estado do sistema como o capital do investidor X u (t) = u(t)S(t), no qual S(t)representa o preço do ativo. Dado uma função de utilidade ξ sobre o estado do sistema X u , queremos determinar uma ação de controle que otimiza a função de utilidade ξ.Dado a base estocástica (Ω, F, P), no qual Ωé o espaço das funções contínuas
In this note, we prove an L p uniform approximation of the fractional Brownian motion with Hurst exponent 0 < H < 1 2 by means of a family of continuous-time random walks imbedded on a given Brownian motion. The approximation is constructed via a pathwise representation of the fractional Brownian motion in terms of a standard Brownian motion. For an arbitrary choice ǫ k for the size of the jumps of the family of random walks, the rate of convergence of the approximation scheme is O(ǫ
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.