Analytic approach to solve a degenerate parabolic PDE for the Heston model

We present an analytic approach to solve a degenerate parabolic problem associated with the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second‐order degenerate elliptic operator of the parabolic PDE. We use this approach to prove, under appropriate assumptions on some involved unknown parameters, the existence and uniqueness of weak solutions to the parabolic problem on unbounded subdomains of the half‐plane. Copyright © 2017 John Wiley & Sons, Ltd.     

美式利率期权定价的抛物型变分不等式

应用PDE方法对美式利率期权定价问题进行理论分析.在CIR利率模型下美式利率期权定价问题可归结为一个退化的一维抛物型变分不等式.通过引入惩罚函数证明了该变分不等式的解的存在唯一性,然后研究了自由边界的一些性质,如单调性,光滑性和自由边界在终止期的位置.     

A multicomponent flow model in deformable porous media

We propose a model for multicomponent flow of immiscible fluids in a deformable porous medium accounting for capillary hysteresis. Oil, water, and air in the soil pores offer a typical example of a real situation occurring in practice. We state the problem within the formalism of continuum mechanics as a slow diffusion process in Lagrange coordinates. The balance laws for volumes, masses, and momentum lead to a degenerate parabolic PDE system. In the special case of a rigid solid matrix material and three fluid components, we prove under further technical assumptions that the system is mathematically well posed in a small neighborhood of an equilibrium.     

Nonlinear degenerate parabolic equations with Neumann boundary condition

We study Neumann problem for a class of nonlinear degenerate parabolic PDE. A typical nonlinearity we have in mind is, for instance, β(u)=−1/u(u>0). We establish a necessary and sufficient condition on given data for existence of solution.     

A multiresolution space‐time adaptive scheme for the bidomain model in electrocardiology

The bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time‐dependent ODE modeling the evolution of the gating variable. In the simpler subcase of the monodomain model, the elliptic PDE reduces to an algebraic equation. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge‐Kutta‐Fehlberg‐type adaptive time integration. A series of numerical examples demonstrates that these methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, the optimal choice of the threshold for discarding nonsignificant information in the multiresolution representation of the solution is addressed, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed‐up, memory compression, and errors in different norms. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Congested traffic dynamics, weak flows and very degenerate elliptic equations

Starting from a model of traffic congestion, we introduce a minimal-flow-like variational problem whose solution is characterized by a very degenerate elliptic PDE. We precisely investigate the relations between these two problems, which can be done by considering some weak notion of flow for a related ODE. We also prove regularity results for the degenerate elliptic PDE, which enables us in some cases to apply the DiPerna–Lions theory.     

Discrete and continuous maximum principles for parabolic and elliptic operators

The major qualitative properties of linear parabolic and elliptic operators/PDEs are the different maximum principles (MPs). Another important property is the stabilization property (SP), which connects these two types of operators/PDEs. This means that under some assumptions the solution of the parabolic PDE tends to an equilibrium state when t→∞, which is the solution of the corresponding elliptic PDE. To solve PDEs we need to use some numerical methods, and it is a natural requirement that these qualitative properties are preserved on the discrete level. In this work we investigate this question when a two-level discrete mesh operator is used as the discrete model of the parabolic operator (which is a one-step numerical procedure for solving the parabolic PDE) and a matrix as a discrete elliptic operator (which is a linear algebraic system of equations for solving the elliptic PDE). We clarify the relation between the discrete parabolic maximum principle (DPMP), the discrete elliptic maximum principle (DEMP) and the discrete stabilization property (DSP). The main result is that the DPMP implies the DSP and the DEMP.     

Predictive control of uncertain nonlinear parabolic PDE systems using a Galerkin/neural-network-based model

In this paper, a model predictive control (MPC) scheme for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities, arising in the context of transport-reaction processes, is proposed. A spatial operator of a parabolic PDE system is characterized by a spectrum that can be partitioned into a finite slow and an infinite fast complement. In this view, first, Galerkin method is used to derive a set of finite dimensional slow ordinary differential equation (ODE) system that captures the dominant dynamics of the initial PDE system. Then, a Multilayer Neural Network (MNN) is employed to parameterize the unknown nonlinearities in the resulting finite dimensional ODE model. Finally, a Galerkin/neural-network-based ODE model is used to predict future states in the MPC algorithm. The proposed controller is applied to stabilize an unstable steady-state of the temperature profile of a catalytic rod subject to input and state constraints.     

Exact solutions via invariant approach for Black‐Scholes model with time‐dependent parameters

We analyze the Black‐Scholes model with time‐dependent parameters, and it is governed by a parabolic partial differential equation (PDE). First, we compute the Lie symmetries of the Black‐Scholes model with time‐dependent parameters. It admits 6 plus infinite many Lie symmetries, and thus, it can be reduced to the classical heat equation. We use the invariant criteria for a scalar linear (1+1) parabolic PDE and obtain 2 sets of equivalence transformations. With the aid of these equivalence transformations, the Black‐Scholes model with time‐dependent parameters transforms to the classical heat equation. Moreover, the functional forms of the time‐dependent parameters in the PDE are determined via this method. Then we use the equivalence transformations and known solutions of the heat equation to establish a number of exact solutions for the Black‐Scholes model with time‐dependent parameters.     

The Quenching Set of a MEMS Capacitor in Two-Dimensional Geometries

The formation of finite time singularities in a nonlinear parabolic fourth order partial differential equation (PDE) is investigated for a variety of two-dimensional geometries. The PDE is a variant of a canonical model for Micro–Electro Mechanical systems (MEMS). The singularities are observed to form at specific points in the domain and correspond to solutions whose values remain finite but whose derivatives diverge as the finite time singularity is approached. This phenomenon is known as quenching. An asymptotic analysis reveals that the quenching set can be predicted by simple geometric considerations suggesting that the phenomenon described is generic to higher order parabolic equations which exhibit finite time singularity.     

Homogenization of periodic semilinear parabolic degenerate PDEs

In this paper a second order semilinear parabolic PDE with rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some part of Rd${\mathbf{R}}^d$.     

On the Doubly Singular Equation γ(u)t = Δpu

Abstract

We prove that local weak solutions of a nonlinear parabolic equation with a doubly singular character are locally continuous. One singularity occurs in the time derivative and is due to the presence of a maximal monotone graph; the other comes up in the principal part of the PDE, where the p-Laplace operator is considered. The paper extends to the singular case 1 < p < 2, the results obtained previously by the second author for the degenerate case p > 2; it completes a regularity theory for a type of PDEs that model phase transitions for a material obeying a nonlinear law of diffusion.     

A Harnack Inequality for Degenerate Parabolic Equations

In this paper we apply the Moser iteration method to degenerate parabolic divergence structure equations. Under some conditions we get a Harnack inequality for weak solutions and from it derive Hölder estimates for weak solutions of uniformly degenerate parabolic equations and the continuity of weak solutions of non-uniformly degenerate parabolic equations.     

A Power Penalty Approach to Numerical Solutions of Two-Asset American Options

This paper aims to develop a power penalty method for a linear parabolic variational inequality (Ⅵ) in two spatial dimensions governing the two-asset Ameri-can option valuation. This method yields a two-dimensional nonlinear parabolic PDE containing a power penalty term with penalty constant λ>1 and a power parameter k>0. We show that the nonlinear PDE is uniquely solvable and the solution of the PDE converges to that of the VI at the rate of order (λ<-k/2>). A fitted finite volume method is designed to solve the nonlinear PDE, and some numerical experiments are performed to illustrate the usefulness of this method.     

REAL HYPERSURFACES EVOLVING BY LEVI CURVATURE: SMOOTH REGULARITY OF SOLUTIONS TO THE PARABOLIC LEVI EQUATION

We prove, with a real analysis technique, the smooth regularity of classical solutions to a nonlinear degenerate parabolic PDE with initial data C ^2,α. This equation arises in the study of the geometric properties of the motion by the trace of the Levi form of a real hypersurface in C ² with Levi curvature different from zero at every point and which is locally the graph of a C ^2,α function.     

Global existence for a haptotaxis model of cancer invasion with tissue remodeling

This paper considers a 3×3 haptotaxis model of cancer invasion with tissue remodeling. The model consists of a parabolic haptotaxis partial differential equation (PDE) describing the evolution of tumor cell density, an ordinary differential equation modeling the proteolysis and the remodeling of the extracellular matrix (ECM), and a parabolic PDE governing the evolution of the matrix degrading enzyme (MDE) concentration. In addition to random diffusion, tumor cells are biased towards higher ECM density, which is referred to as haptotaxis. Under a restrictive assumption on the coefficients, the global existence and uniqueness of classical solutions to the model in two dimensions, along with the boundedness of solutions in two and three dimensions, is proved by establishing some delicate a priori estimates.     

Ibragimov-type invariants for a system of two linear parabolic equations

We obtain new semi-invariants for a system of two linear parabolic type partial differential equations (PDEs) in two independent variables under equivalence transformations of the dependent variables only. This is achieved for a class of systems of two linear parabolic type PDEs that correspond to a scalar complex linear (1 + 1) parabolic equation. The complex transformations of the dependent variables which map the complex scalar linear parabolic PDE to itself provide us with real transformations that map the corresponding system of linear parabolic type PDEs to itself with different coefficients in general. The semi-invariants deduced for this class of systems of two linear parabolic type equations correspond to the complex Ibragimov invariants of the complex scalar linear parabolic equation. We also look at particular cases of the system of parabolic type equations when they are uncoupled or coupled in a special manner. Moreover, we address the inverse problem of when systems of linear parabolic type equations arise from analytic continuation of a scalar linear parabolic PDE. Examples are given to illustrate the method implemented.     

Entropy solutions to a strongly degenerate anisotropic convection-diffusion equation with application to utility theory

We study the deterministic counterpart of a backward-forward stochastic differential utility, which has recently been characterized as the solution to the Cauchy problem related to a PDE of degenerate parabolic type with a conservative first order term. We first establish a local existence result for strong solutions and a continuation principle, and we produce a counterexample showing that, in general, strong solutions fail to be globally smooth. Afterward, we deal with discontinuous entropy solutions, and obtain the global well posedness of the Cauchy problem in this class. Eventually, we select a sufficient condition of geometric type which guarantees the continuity of entropy solutions for special initial data. As a byproduct, we establish the existence of an utility process which is a solution to a backward-forward stochastic differential equation, for a given class of final utilities, which is relevant for financial applications.     

An optimal control problem in image processing

As a starting point, we present a control problem in mammographic image processing which leads to non-standard penalty terms and involves a degenerate parabolic PDE which has to be controlled in the coefficients. We then discuss the classical conditional gradient method from constrained optimization and propose a generalization for non-convex functionals which covers the conditional gradient method as well as the recently proposed iterative shrinkage method of Daubechies, Defrise and De Mol for the solution of linear inverse problems with sparsity promoting penalty terms. We prove that this new algorithm converges. This also gives a deeper understanding of the iterative shrinkage method. Further, we show an application to the above-mentioned control problem in image processing. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Feedback theory extended for proving generation of contraction semigroups

Recently, the following novel method for proving the existence of solutions for certain linear time-invariant PDEs was introduced: The operator associated with a given PDE is represented by a (larger) operator with an internal loop. If the larger operator (without the internal loop) generates a contraction semigroup, the internal loop is accretive, and some non-restrictive technical assumptions are fulfilled, then the original operator generates a contraction semigroup as well. Beginning with the undamped wave equation, this general idea can be applied to show that the heat equation and wave equations with damping are well-posed. In the present paper we show how this approach can benefit from feedback techniques and recent developments in well-posed systems theory, at the same time generalizing the previously known results. Among others, we show how well-posedness of degenerate parabolic equations can be proved.