scholarly journals A non-local free boundary problem arising in a theory of financial bubbles

2014 ◽  
Vol 372 (2028) ◽  
pp. 20130404 ◽  
Author(s):  
Henri Berestycki ◽  
Regis Monneau ◽  
José A. Scheinkman
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Existence And Uniqueness ◽  
Obstacle Problem ◽  
Speculative Bubbles ◽  
Financial Bubbles ◽  
Uniqueness Of The Solution ◽  
Non Local ◽  
The Cost

We consider an evolution non-local free boundary problem that arises in the modelling of speculative bubbles. The solution of the model is the speculative component in the price of an asset. In the framework of viscosity solutions, we show the existence and uniqueness of the solution. We also show that the solution is convex in space, and establish several monotonicity properties of the solution and of the free boundary with respect to parameters of the problem. To study the free boundary, we use, in particular, the fact that the odd part of the solution solves a more standard obstacle problem. We show that the free boundary is and describe the asymptotics of the free boundary as c , the cost of transacting the asset, goes to zero.

scholarly journals Numerical Treatment to a Non-local Parabolic Free Boundary Problem Arising in Financial Bubbles

2018 ◽  
Vol 45 (1) ◽  
pp. 59-73
Author(s):  
Avetik Arakelyan ◽  
Rafayel Barkhudaryan ◽  
Henrik Shahgholian ◽  
Mohammad Salehi
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Numerical Treatment ◽  
Financial Bubbles ◽  
Non Local

A PARABOLIC–HYPERBOLIC FREE BOUNDARY PROBLEM MODELLING TUMOR TREATMENT WITH VIRUS

2007 ◽  
Vol 17 (01) ◽  
pp. 63-80 ◽  
Author(s):  
YOUSHAN TAO ◽  
HUI ZHANG
Keyword(s):  
Immune Response ◽  
Free Boundary ◽  
Global Existence ◽  
Boundary Problem ◽  
Tumor Cells ◽  
Free Boundary Problem ◽  
Sensitivity Coefficient ◽  
Semilinear Parabolic Equation ◽  
Replication Competent Virus ◽  
Uniqueness Of The Solution

We consider a procedure for cancer therapy which consists of injecting replication-competent viruses into the tumor. The viruses infect tumor cells, replicate inside them, and eventually cause their death. As infected cells die, the viruses inside them are released and then proceed to infect adjacent tumor cells. However, a major factor influencing the efficacy of virus agents is the immune response that may limit the replication and spread of the replication-competent virus. The competition between tumor cells, a replication-competent virus and an immune response is modelled as a free boundary problem for a nonlinear system of partial differential equations, where the free boundary is the surface of the tumor. In this model, the immune response equation is a semilinear parabolic equation, including a chemotaxis term which is used to describe the movement of the immune response induced by gradients of the infected cell density. Under the assumption that the chemotactic sensitivity coefficient is small compared with the diffusion coefficient of the immune response, we prove the global existence and uniqueness of the solution of this free boundary problem. For large chemotactic coefficient, the global existence is still open.

Iterative scheme for an elliptic non-local free boundary problem

2015 ◽  
Vol 95 (12) ◽  
pp. 2794-2806 ◽  
Author(s):  
R. Barkhudaryan ◽  
M. Juráš ◽  
M. Salehi
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Iterative Scheme ◽  
Non Local

A free boundary problem coming from the perpetual American call options with utility

2012 ◽  
Vol 24 (2) ◽  
pp. 231-271 ◽  
Author(s):  
SONG LIPING ◽  
YU WANGHUI
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Financial Market ◽  
Free Boundary Problem ◽  
Utility Functions ◽  
Call Options ◽  
Line Method ◽  
Uniqueness Of The Solution ◽  
A Free Boundary Problem

A free boundary problem, which comes from the model of the perpetual American call options with utility functions in financial market, is investigated. It is a degenerative parabolic free boundary problem and is studied by the line method. The existence, regularity and uniqueness of the solution as well as some properties of the free boundary are established.

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