Applied Mathematics and Optimization最新文献

We investigate the convergence of numerical solution of Reflected Backward Stochastic Differential Equations (RBSDEs) using the penalization approach in a general non-Markovian framework. We prove the convergence between the continuous penalized solution and the reflected one, in full generality, at order 1/2 as a function of the penalty parameter; the convergence order becomes 1 when the increasing process of the RBSDE has a bounded density, which is a mild condition in practice. The convergence is analyzed in a.s.-sense and (mathbb {L}^p)-sense ((pge 2)). To achieve these new results, we have developed a refined analysis of the behavior of the process close to the barrier. Then we propose an implicit scheme for computing the discrete solution of the penalized equation and we derive that the global convergence order is 3/8 as a function of time discretization under mild regularity assumptions. This convergence rate is verified in the case of American put options and some numerical tests illustrate these results.

This paper is concerned with a kind of partially observed nonzero-sum differential game of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. A necessary condition in the form of maximum principle with Pontryagin s type for open-loop Nash equilibrium point of this type of partially observed game, and a verification theorem which is a sufficient condition for Nash equilibrium point are established. The theoretical results are applied to study a partially observed linear-quadratic game.

In this paper, we explore a static setting for the assessment of risk in the context of mathematical finance and actuarial science that takes into account model uncertainty in the distribution of a possibly infinite-dimensional risk factor. We study convex risk functionals that incorporate a safety margin with respect to nonparametric uncertainty by penalizing perturbations from a given baseline model using Wasserstein distance. We investigate to which extent this form of probabilistic imprecision can be approximated by restricting to a parametric family of models. The particular form of the parametrization allows to develop numerical methods based on neural networks, which give both the value of the risk functional and the worst-case perturbation of the reference measure. Moreover, we consider additional constraints on the perturbations, namely, mean and martingale constraints. We show that, in both cases, under suitable conditions on the loss function, it is still possible to estimate the risk functional by passing to a parametric family of perturbed models, which again allows for numerical approximations via neural networks.

In this paper, we study the stability in partial data inverse problems of determining the time-dependent viscosity and potential terms appearing in the Moore–Gibson–Thompson (MGT) equation in dimension (nge 2). The MGT equation, which is third order in time and of hyperbolic type, arises as a linearization of a model for nonlinear ultrasound wave propagation in viscous thermally relaxing fluids. By directly establishing some key Carleman estimates for the MGT equation and its dual, some suitable geometric optics solutions of exponential type are constructed. Then, the stability results in recovering the coefficients from partial observations on the boundary are obtained by means of the suitable geometric optics solutions together with the light ray and Fourier transforms.

Diffeological spaces firstly introduced by J. M. Souriau in the 1980 s are a natural generalization of smooth manifolds but optimization techniques are only known on manifolds so far. Generalizing these techniques to diffeological spaces is very challenging because of several reasons. One of the main reasons is that there are various definitions of tangent spaces which do not coincide. Additionally, one needs to deal with a generalization of a Riemannian space in order to define gradients which are indispensable for optimization methods. One main aim of this paper is a suitable definition of a tangent space in view to optimization methods. Based on this definition, we present a diffeological Riemannian space and a diffeological gradient, which we need for the formulation of an optimization algorithm on diffeological spaces. Moreover, in order to be able to update the iterates in an optimization algorithm on diffeological spaces, we present a diffeological retraction and the Levi-Civita connection on diffeological spaces. This paper also illustrates the novel objects by examples. Finally, we formulate the steepest descent method on diffeological spaces and apply it to an example.

In this paper, we investigate the connection between a class of doubly reflected backward stochastic differential equations, driven by a right continuous with left limits martingale M with two completely separated reflection obstacles, a stochastic Lipschitz driver f, and a generalized Dynkin game, where the game payoff is expressed in terms of a nonlinear expectation (mathcal {E}^{f,M}).

We propose three modified contact boundary conditions incorporating both the velocity and the displacement with a parameter (delta ) for the viscoelastic problem. As (delta ) approaches 0, these conditions formally reduce to the conventional Signorini, Tresca-friction, and Clarke-subdifferential type boundary conditions, respectively. Consequently, the modified conditions, as a generalization of the conventional ones, can be viewed as contact conditions in the displacement with a dynamic setting. We derive weak formulations for the viscoelastic contact model under three modified contact conditions and explore their well-posedness. Additionally, we provide bounds on the weak solutions with respect to the parameter (delta ).

In this paper, we examine a stochastic linear-quadratic control problem characterized by regime switching and Poisson jumps. All the coefficients in the problem are random processes adapted to the filtration generated by Brownian motion and Poisson random measure for each given regime. The model incorporates two distinct types of controls: the first is a conventional control that appears in the continuous diffusion component, while the second is an unconventional control, dependent on the variable z, which influences the jump size in the jump diffusion component. Both controls are constrained within general closed cones. By employing the Meyer-Itô formula in conjunction with a generalized squares completion technique, we rigorously and explicitly derive the optimal value and optimal feedback control. These depend on solutions to certain multi-dimensional fully coupled stochastic Riccati equations, which are essentially backward stochastic differential equations with jumps (BSDEJs). We establish the existence of a unique nonnegative solution to the BSDEJs. One of the major tools used in the proof is the newly established comparison theorems for multi-dimensional BSDEJs.

We consider the mixed problem for the “Kirchhoff plate equation” on an open bounded domain (Omega ) in (mathbb {R}^n, n = 1, 2, 3,, ldots ) with sufficiently smooth boundary (Gamma = partial (Omega )). Under both Dirichlet and Neumann homogeneous Boundary Conditions, the dynamical system defines a strongly continuous group of unitary operators on an appropriate function space. We then introduce a suitably devised Neumann boundary control in feedback form, as to force the new dynamic (to be well-posed and) to asymptotically decay in an optimal function space (the same space of optimal regularity and exact controllability under open-loop control, (L^2)-in time and space.) We obtain the following results: (1) uniform stabilization for (n=1); (2) polynomial/rational stability for (n = 2,3,4, ldots ); (3) and, independently, strong stabilization for any dimension n. In the present paper, we employ a frequency domain approach, based on technical PDE-estimates.

Motivated by stability issues for suspension bridges, the analysis focuses on the maximization of the torsional eigenvalues of a nonhomogeneous multi-span fish-bone plate with respect to the mass density. The incorporation of internal piers significantly impacts the spectral properties of the system. After a general spectral theorem, a characterization of the densities maximizing the first and the second torsional eigenvalue is provided, starting from the corresponding results for the nonhomogeneous Dirichlet problem. In the case where the mass of the central span is equal to its length, more explicit insight is then given, taking into account the role of the position of the piers and discussing the scenario for higher-order eigenvalues, as well.