On exact null controllability of Black-Scholes equation

summary:In this paper we discuss the exact null controllability of linear as well as nonlinear Black–Scholes equation when both the stock volatility and risk-free interest rate influence the stock price but they are not known with certainty while the control is distributed over a subdomain. The proof of the linear problem relies on a Carleman estimate and observability inequality for its own dual problem and that of the nonlinear one relies on the infinite dimensional Kakutani fixed point theorem with $L^2$ topology

Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states

In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the \-reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied in this paper: the weakly degenerate case, that is, if the reciprocal of the diffusion coefficient is summable, and the strongly degenerate case, that is, if that reciprocal isn't summable. In our main result we show that the above systems can be steered from an initial continuous state that admits a finite number of points of sign change to a target state with the same number of changes of sign in the same order. Our method uses a recent technique introduced for uniformly parabolic equations employing the shifting of the points of sign change by making use of a finite sequence of initial-value pure diffusion pro\-blems. Our interest in degenerate reaction-diffusion equations is motivated by the study of some \-energy balance models in climatology (see, e.g., the Budyko-Sellers model) and some models in population genetics (see, e.g., the Fleming-Viot model).Comment: arXiv admin note: text overlap with arXiv:1510.0420

Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations

We show Carleman estimates, observability inequalities and null controllability results for parabolic equations with non smooth coefficients degenerating at an interior point.Comment: Accepted in Memoirs of the American Mathematical Societ

Control of partial differential equations via physics-informed neural networks

This paper addresses the numerical resolution of controllability problems for partial differential equations (PDEs) by using physics-informed neural networks. Error estimates for the generalization error for both state and control are derived from classical observability inequalities and energy estimates for the considered PDE. These error bounds, that apply to any exact controllable linear system of PDEs and in any dimension, provide a rigorous justification for the use of neural networks in this field. Preliminary numerical simulation results for three different types of PDEs are carried out to illustrate the performance of the proposed methodology.This research was supported by Fundación Séneca (Agencia de Ciencia y Tecnología de la Región de Murcia (Spain)) under contract 20911/PI/18 and grant number 21503/EE/21 (mobility program Jiménez de la Espada). F. Periago acknowledges the hospitality of the Mathematics Department at University of California, Santa Barbara, where part of this work was carried out. The authors also thank professor Lu Lu for very fruitful comments on the use of DeepXDE

Black-Scholes formulae for Asian options in local volatility models

We develop approximate formulae expressed in terms of elementary functions for the density, the price and the Greeks of path dependent options of Asian style, in a general local volatility model. An algorithm for computing higher order approximations is provided. The proof is based on a heat kernel expansion method in the framework of hypoelliptic, not uniformly parabolic, partial differential equations.Asian Options, Degenerate Diffusion Processes, Transition Density Functions, Analytic Approximations, Option Pricing

Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations

We consider a parabolic problem with degeneracy in the interior of the spatial domain, and we focus on observability results through Carleman estimates for the associated adjoint problem. The novelties of the present paper are two. First, the coefficient of the leading operator only belongs to a Sobolev space. Second, the degeneracy point is allowed to lie even in the interior of the control region, so that no previous result can be adapted to this situation; however, different cases can be handled, and new controllability results are established as a consequence

Extraordinary exciton conductance induced by strong coupling

We demonstrate that exciton conductance in organic materials can be enhanced by several orders of magnitude when the molecules are strongly coupled to an electromagnetic mode. Using a 1D model system, we show how the formation of a collective polaritonic mode allows excitons to bypass the disordered array of molecules and jump directly from one end of the structure to the other. This finding could have important implications in the fields of exciton transistors, heat transport, photosynthesis, and biological systems in which exciton transport plays a key role.Comment: Main text: 5 pages, 4 figures; Supplemental: 2 pages, 1 figure. Version 2: Updated reference to related work arXiv:1409.2550. Version 3: Updated to version accepted for publication in Physical Review Letter