Sensitivity analysis in a market with memory

A general market model with memory is considered in terms of stochastic functional differential equations. We aim at representation formulae for the sensitivity analysis of the dependence of option prices on the memory. This implies a generalization of the concept of delta.Comment: Withdrawn by the authors due to an error in equation (2.6). A new work is in preparatio

Funnel control for a moving water tank

We study tracking control for a moving water tank system, which is modelled using the Saint-Venant equations. The output is given by the position of the tank and the control input is the force acting on it. For a given reference signal, the objective is to achieve that the tracking error evolves within a prespecified performance funnel. Exploiting recent results in funnel control we show that it suffices to show that the operator associated with the internal dynamics of the system is causal, locally Lipschitz continuous and maps bounded functions to bounded functions. To show these properties we consider the linearized Saint-Venant equations in an abstract framework and show that it corresponds to a regular well-posed linear system, where the inverse Laplace transform of the transfer function defines a measure with bounded total variation.Comment: 11 page

Existence and controllability of integrodifferential equations with non-instantaneous impulses in Fréchet spaces

In this paper, we investigate existence of mild solutions to a non-instantaneous integrodifferential equation via resolvent operators in the sense of Grimmer in Fréchet spaces. Utilizing the technique of measures of noncompactness in conjunction with the Darbo's fixed point theorem, we present sufficient criteria ensuring the controllability of the given problem. An illustrative example is also discussed

Resolubility of linear Cauchy problems on Fréchet spaces and a delayed Kaldor’s model

The long-run aim of this thesis is to solve delay differential equations with infinite delay of the type d dt u(t) = Au(t)+Zt −∞ u(s)k(t−s)ds+ ft,u(t) , on Fréchet spaces under an extended theory of groups of linear operators; where A is a linear operator, k(s) ⩾ 0 satisfiesR∞ 0 k(s)ds = 1 and f is a nonlinear map. In order to pursue such a goal we study a discrete delay model which explains the natural economic fluctuations considering how economic stability is affected by the role of the fiscal and monetary policies and a possible government inefficiency concerning its fiscal policy decision-making. On the other hand, we start to develop such an extended theory by considering linear Cauchy problems associated to a continuous linear operator on Fréchet spaces, for which we establish necessary and sufficient conditions for generation of a uniformly continuous group which provides the unique solution. Further consequences arises by considering pseudodifferential operators with constant coefficients defined on a particular Fréchet space of distributions, namely FL2loc, and special attention is given to the distributional solution of the heat equation on FL2loc for all time, which extends the standard solution on Hilbert spaces for positive time.El objetivo a largo plazo de esta tesis doctoral es resolver ecuaciones diferenciales de la forma d dt u(t) = Au(t)+Zt −∞ u(s)k(t−s)ds+ ft,u(t) ,en espacios de Fréchet extendiendo la teoria de grupos de operadores lineales; siendo A un operador linear, k(s)⩾0 satisfaceR∞ 0 k(s)ds = 1 y f una función no linear. Persiguiendo tal fin, estudiamos un modelo con retraso que enseña las fluctuaciones de la economía considerandocomolaestabilidadeconómicaesafectadaporlaactuacióndelgobierno,sus políticasfiscalymonetariayunaposibleineficienciadelgobiernoenloqueserefierealasu tomadedecisión.Porotrolado,damosinicioalareferidaextensióndelateoríadegrupos puesto que consideramos problemas de Cauchy lineales asociados a operadores lineales contínuos en espacios de Fréchet, para los cuales establecimos condiciones necessarias y suficientes para la generación de un grupo uniformemente contínuo en tal espacio y que proporcionelaúnicasolucióndelproblema.Consequenciasadicionalessurgencuando se considera operadores pseudodiferenciales con coeficientes constantes definidos en un particular espacio de Fréchet de distribuciones, a saber FL2loc, y una atención especial es dada a la solución distribucional de la ecuación del calor en FL2loc para todo el tiempo, la cual extiende la solución usual en espacios de Hilbert para tiempo positivo

[Book of abstracts]

USPCAPESCNPqFAPESPICMC Summer Meeting on Differential Equations (2016 São Carlos

Generalized covariation for Banach space valued processes, Ito formula and applications

This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If X and Y take respectively values in Banach spaces B1 and B2 and χ is a suitable subspace of the dual of the projective tensor product of B1 and B2 (denoted by (B1⊗̂πB2) ∗), we define the so-called χ-covariation of X and Y. If X = Y, the χ-covariation is called χ-quadratic variation. The notion of χ-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if χ is the whole space (B1⊗̂πB1) ∗ then the χ-quadratic variation coincides with the quadratic variation of a B1-valued semimartingale. We evaluate the χ-covariation of various processes for several examples of χ with a particular attention to the case B1 = B2 = C([−τ, 0]) for some τ> 0 and X and Y being window processes. If X is a real valued process, we call window process associated with X the C([−τ, 0])-valued process X: = X(·) defined by Xt(y) = Xt+y, where y ∈ [−τ, 0]. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type h = H(XT (·)), H: C([−T, 0]) − → R for not-necessarily semimartingales X with finite quadratic variation. This representation will be linked to a function u: [0, T]×C([−T, 0]) − → R solving an infinite dimensional partial differential equation

Variational principles for fluid dynamics on rough paths

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