Quasi-linear Parabolic Partial Differential Equations Research Articles

REDUCED-ORDER MODELS FOR THE IMPLIED VARIANCE UNDER LOCAL VOLATILITY

Implied volatility is a key value in financial mathematics. We discuss some of the pros and cons of the standard ways to compute this quantity, i.e. numerical inversion of the well-known Black–Scholes formula or asymptotic expansion approximations, and propose a new way to directly calculate the implied variance in a local volatility framework based on the solution of a quasilinear degenerate parabolic partial differential equation. Since the numerical solution of this equation may lead to large nonlinear systems of equations and thus high computation times compared to the classical approaches, we apply model order reduction techniques to achieve computational efficiency. Our method of choice for the derivation of a reduced-order model (ROM) will be proper orthogonal decomposition (POD). This strategy is additionally combined with the discrete empirical interpolation method (DEIM) to deal with the nonlinear terms. Numerical results prove the quality of our approach compared to other methods.

Probabilistic interpretation for a system of quasilinear parabolic partial differential equation combined with algebra equations

In this paper, we study a kind of system of second order quasilinear parabolic partial differential equation combined with algebra equations. Introducing a family of coupled forward–backward stochastic differential equations, and by virtue of some delicate analysis techniques, we give a probabilistic interpretation for it in the viscosity sense.

Finite element approximations of the stochastic mean curvature flow of planar curves of graphs

This paper develops and analyzes a semi-discrete and a fully discrete finite element method for a one-dimensional quasilinear parabolic stochastic partial differential equation (SPDE) which describes the stochastic mean curvature flow for planar curves of graphs. To circumvent the difficulty caused by the low spatial regularity of the SPDE solution, a regularization procedure is first proposed to approximate the SPDE, and an error estimate for the regularized problem is derived. A semi-discrete finite element method, and a space-time fully discrete method are then proposed to approximate the solution of the regularized SPDE problem. We show $$L^2$$ -convergence with rates for both, semi- and fully discrete 1methods. Computational experiments are provided to study the interplay of the geometric evolution and gradient type-noises.

Representation theorems for SPDEs via backward doubly

In this paper we establish a probabilistic representation for the spatial gradient ofthe viscosity solution to a quasilinear parabolic stochastic partial differential equations(SPDE, for short) in the spirit of the Feynman-Kac formula, without using thederivatives of the coefficients of the corresponding backward doubly stochastic differentialequations (FBDSDE, for short).

Delay-independent sliding mode control for a class of quasi-linear parabolic distributed parameter systems with time-varying delay

The sliding mode control (SMC) problem for a class of quasi-linear parabolic partial differential equation (PDE) systems with time-varying delay is considered. Firstly, the stability problem for the reduced order sliding dynamical equations is investigated and a sufficient condition for the stability of sliding motion is given. Then the SMC law, which forces the system state from any initial state to reach the sliding manifold within finite time, is designed. At last a simulation example is presented to illustrate effectiveness of the proposed method.

A new high accuracy two-level implicit off-step discretization for the system of two space dimensional quasi-linear parabolic partial differential equations

In this article, we propose a two-level compact implicit off-step discretization for the solution of the system of 2D quasi-linear parabolic partial differential equations subject to suitable initial and boundary conditions. New methods for the estimates of first order space derivatives of the solution are also derived. These methods are fourth order accurate in space and second order accurate in time, and involve only nine spatial grid points of a single compact cell. We further develop the alternating direction implicit (ADI) scheme for a general linear parabolic equation which is shown to be unconditionally stable for the heat equation in polar coordinates. The proposed methods are directly applicable to singular problems without the need of any special technique, which is the main advantage of this work. The method is effectively applied to the time dependent Navier–Stokes’ model equations in polar coordinates. Numerical examples are provided to illustrate the accuracy of the methods.

Probabilistic approach for systems of second order quasi-linear parabolic PDEs

Using the stochastic representation for second order parabolic equations, we prove the existence of local smooth solutions in Sobolev spaces for a class of second order quasi-linear parabolic partial differential equations (possibly degenerate) with smooth coefficients. As a simple application, the rate of convergence for vanishing viscosity is proved to be O(νt). Moreover, using Bismutʼs formula, we also obtain a global existence result for non-degenerate semi-linear parabolic equations. In particular, multi-dimensional Burgers equations are covered.

Safe-Parking Framework for Fault-Tolerant Control of Transport−Reaction Processes

This work considers the problem of handling actuator faults in transport−reaction processes described by quasi-linear parabolic partial differential equations (PDEs) subject to input constraints. To this end, first, by exploiting the separation between the fast and slow eigenmodes of the parabolic spatial differential operator in combination with Galerkin’s method, a finite set of ordinary differential equations (ODEs) that captures the dominant dynamics of the PDE system are constructed. This finite ODE system is used to develop a Lyapunov-based model predictive controller which provides an explicit characterization of the set of initial conditions from where closed-loop stability of the parabolic PDE system is guaranteed. This control design is then subsequently used to develop a safe-parking framework for handling faults. In particular, faults which preclude the possibility of maintaining operation at the nominal equilibrium distribution, using the existing robust or reconfiguration-based fault-tolerant control approaches are considered. The key idea is to temporarily maintain the process at an appropriate “safe-park” distribution using the available depleted control action. This “safe-park” distribution is chosen to prevent onset of hazardous situations as well as to ensure smooth resumption of nominal operation upon fault repair. Utilizing the stability region characterization provided by the developed predictive controller, safe-park distributions from the safe-park candidates (equilibrium distributions subject to the remaining functioning and fail-safe values of the failed actuators) are chosen to preserve closed-loop stability upon fault repair. The proposed framework is illustrated on a diffusion−reaction process.

Alternative Forms of the Harnack Inequality for Non-Negative Solutions to Certain Degenerate and Singular Parabolic Equations

Non-negative solutions to quasi-linear, degenerate or singular parabolic partial differential equations, of p -Laplacian type for p > 2_N_/( N +1), satisfy Harnack-type estimates in some intrinsic geometry ([2, 3]). Some equivalent alternative forms of these Harnack estimates are established, where the supremum and the infimum of the solutions play symmetric roles, within a properly redefined intrinsic geometry. Such equivalent forms hold for the non-degenerate case p = 2 following the classical work of Moser ([5, 6]), and are shown to hold in the intrinsic geometry of these degenerate and/or parabolic p.d.e.’s. Some new forms of such an estimate are also established for 1 < p < 2.

Construction of solutions to quasilinear partial differential equations

We propose a method for constructing solutions to a class of quasilinear parabolic partial differential equations (PDEs) basing on a new property of these equations. The method applies to quasilinear hyperbolic and elliptic equations as well. The results of this article broaden the class of exact solutions to the quasilinear equations, in particular, to the nonlinear heat equations, the equations of chemical kinetics and mathematical biology.

Existence of solutions for a model of multiphase flow in porous media applied to gas migration in underground nuclear waste repository

We prove existence of solutions for a new model of two compressible and partially miscible phase flow in porous media, applied to gas migration in underground nuclear waste repository. This model, modelling fully and partially water saturated situations, consists of a coupled system of quasilinear parabolic partial differential equations. We seek a new set of variables in order to obtain a system which belongs to the class of equations considered by Alt and Luckhaus such that it would be possible to use their existence theorem.

A model of multiphase flow and transport in porous media applied to gas migration in underground nuclear waste repository

We prove existence of solutions for a new model of two compressible and partially miscible phase flow in porous media, applied to gas migration in an underground nuclear waste repository. This model, modeling fully and partially water saturated situations, consist of a coupled system of quasilinear parabolic partial differential equations. We seek a new set of variables in order to obtain a system which belongs to the class of equations considered by Alt and Luckhaus such that it would be possible to use their existence theorem. A simulation of a numerical test case is performed in order to numerically demonstrate the ability of this model to take in account the appearance of one phase. To cite this article: F. Smaï, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Well-Posedness and Convergence to the Steady State for a Model of Morphogen Transport

Well-posedness and large time convergence to the unique steady state are shown for a model which describes the spreading of morphogens by a nonlinear transport mechanism (transcytosis) and couples a quasilinear parabolic partial differential equation with an ordinary differential equation. A simpler model which assumes linear transport is also investigated for comparison. The analysis of both models requires the construction of specific Liapunov functionals. The study is supplemented by numerical simulations of the sensitivity of the models to the variation of their parameters.

On nonlinear systems consisting of different types of differential equations

We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation where both equations contain functional dependence on the unknown functions. Then we consider a system which consists of a quasilinear parabolic partial differential equation, a first order ordinary differential equation and an elliptic partial differential equation. These systems were motivated by models describing diffusion and transport in porous media with variable porosity.

Intrinsic Harnack inequalities for quasi-linear singular parabolic partial differential equations

Intrinsic Harnack estimates for non-negative solutions of singular, quasi-linear, parabolic equations, are established, including the prototype p -Laplacean equation (1.4) below. For p in the super-critical range \frac{2N}{N+1}<p<2 , the Harnack inequality is shown to hold in a parabolic form, both forward and backward in time, and in a elliptic form at fixed time. These estimates fail for the heat equation ( p\to2 ). It is shown by counterexamples, that they fail for p in the sub-critical range 1<p\le \frac{2N}{N+1} . Thus the indicated super-critical range is optimal for a Harnack estimate to hold. The novel proofs are based on measure theoretical arguments, as opposed to comparison principles and are sufficiently flexible to hold for a large class of singular parabolic equation including the porous medium equation and its quasi-linear versions.

Numerical pricing of options using high-order compact finite difference schemes

We consider high-order compact (HOC) schemes for quasilinear parabolic partial differential equations to discretise the Black–Scholes PDE for the numerical pricing of European and American options. We show that for the heat equation with smooth initial conditions, the HOC schemes attain clear fourth-order convergence but fail if non-smooth payoff conditions are used. To restore the fourth-order convergence, we use a grid stretching that concentrates grid nodes at the strike price for European options. For an American option, an efficient procedure is also described to compute the option price, Greeks and the optimal exercise curve. Comparisons with a fourth-order non-compact scheme are also done. However, fourth-order convergence is not experienced with this strategy. To improve the convergence rate for American options, we discuss the use of a front-fixing transformation with the HOC scheme. We also show that the HOC scheme with grid stretching along the asset price dimension gives accurate numerical solutions for European options under stochastic volatility.

ROBUST FAULT DETECTION AND HANDLING IN CONTROL OF UNCERTAIN TRANSPORT-REACTION PROCESSES

This paper presents an integrated robust fault detection and fault-tolerant control architecture for transport-reaction processes modeled by quasi-linear parabolic partial differential equations with uncertain variables, control constraints and control actuator faults. Using an appropriate reduced-order model that captures the dominant process dynamics, the proposed architecture comprises a family of robustly stabilizing bounded feedback controllers with explicitly characterized stability and uncertainty attenuation properties, a performance- based fault detection scheme and a high-level supervisor that reconfigures the control actuators upon fault detection in a way that maintains robust closed-loop stability. The key idea is to shape the closed-loop performance via robust control in a way that facilitates the design of robust fault detection rules that are less sensitive to the adverse effects of uncertainty. The results are demonstrated using a non-isothermal tubular reactor example with recycle.

Discretization of forward–backward stochastic differential equations and related quasi-linear parabolic equations

Efficient numerical algorithms for a class of forward-backward stochastic differential equations (FBSDEs) and related quasi-linear parabolic partial differential equations are proposed. The quasi-linear parabolic equation is solved by new layer methods which are constructed by means of a probabilistic approach. The proposed algorithms for solving FBSDEs are based on the four-step scheme of Ma, Protter and Yong. Convergence theorems are proved. Results of some numerical experiments are presented.

Determination of a time-dependent parameter in a one-dimensional quasi-linear parabolic equation with temperature overspecification

In this work, the Adomian decomposition method is used to find the solution to a one-dimensional quasi-linear parabolic partial differential equation with a time-dependent unknown function. A wide class of physical phenomena is modelled by non-classical parabolic initial-boundary value problems. Thus the theoretical behaviour and numerical approximation of these problems have been active areas of research. The decomposition procedure first proposed by the American mathematician G. Adomian (1923–1996) is useful for obtaining both exact solutions to, and numerical approximations of, various kinds of linear and nonlinear problem. The Adomian decomposition method, which accurately computes the series solution, is of great interest in science and engineering. It provides a solution as a convergent series with components that can be elegantly computed. Sufficient conditions for convergence and stability of the approximate solution are given and the results of numerical experiments are presented.

Homeomorphism of solutions to backward SDEs and applications

In this paper we study the homeomorphic properties of the solutions to one dimensional backward stochastic differential equations under suitable assumptions, where the terminal values depend on a real parameter. Then, we apply them to the solutions for a class of second order quasilinear parabolic partial differential equations.