Herbert Egger

Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates

This paper investigates the stable identification of local volatility surfaces σ(S, t) in the Black–Scholes/Dupire equation from market prices of European Vanilla options. Based on the properties of the parameter-to-solution mapping, which assigns option prices to given volatilities, we show stability and convergence of approximations gained by Tikhonov regularization. In the case of a known term-structure of the volatility surface, in particular, if the volatility is assumed to be constant in time, we prove convergence rates under simple smoothness and decay conditions on the true volatility. The convergence rate analysis sheds light onto the importance of an appropriate a priori guess for the unknown volatility and the nature of the ill-posedness of the inverse problem, caused by smoothing properties and the nonlinearity of the direct problem. Finally, the theoretical results are illustrated by numerical experiments.

Preconditioning Landweber iteration in Hilbert scales

In this paper we investigate convergence of Landweber iteration in Hilbert scales for linear and nonlinear inverse problems. As opposed to the usual application of Hilbert scales in the framework of regularization methods, we focus here on the case s≤0, which (for Tikhonov regularization) corresponds to regularization in a weaker norm. In this case, the Hilbert scale operator L−2s appearing in the iteration acts as a preconditioner, which significantly reduces the number of iterations needed to match an appropriate stopping criterion. Additionally, we carry out our analysis under significantly relaxed conditions, i.e., we only require instead of which is the usual condition for regularization in Hilbert scales. The assumptions needed for our analysis are verified for several examples and numerical results are presented illustrating the theoretical ones.

Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation

Consider the semilinear parabolic equation −ut (x, t) + uxx + q(u) = f( x, t), with the initial condition u(x, 0) = u0(x), Dirichlet boundary conditions u(0 ,t )= ϕ0(t), u(1 ,t )= ϕ1(t) and a sufficiently regular source term q(·), which is assumed to be known ap riorion the range of u0(x). We investigate the inverse problem of determining the function q(·) outside this range from measurements of the Neumann boundary data ux(0 ,t )= ψ0(t), ux(1 ,t )= ψ1(t). Via the method of Carleman estimates, we derive global uniqueness of a solution (u, q) to this inverse problem and H¨ older stability of the functions u and q with respect to errors in the Neumann data ψ0 ,ψ 1, the initial condition u0 and the ap rioriknowledge of the function q (on the range of u0). The results are illustrated by numerical tests. The results of this paper can be extended to more general nonlinear parabolic equations.

Electrical Forces Determine Glomerular Permeability

There is ongoing controversy about the mechanisms that determine the characteristics of the glomerular filter. Here, we tested whether flow across the glomerular filter generates extracellular electrical potential differences, which could be an important determinant of glomerular filtration. In micropuncture experiments in Necturus maculosus, we measured a potential difference across the glomerular filtration barrier that was proportional to filtration pressure (-0.045 mV/10 cm H₂O). The filtration-dependent potential was generated without temporal delay and was negative within Bowmans space. Perfusion with the cationic polymer protamine abolished the potential difference. We propose a mathematical model that considers the relative contributions of diffusion, convection, and electrophoretic effects on the total flux of albumin across the filter. According to this model, potential differences of -0.02 to -0.05 mV can induce electrophoretic effects that significantly influence the glomerular sieving coefficient of albumin. This model of glomerular filtration has the potential to provide a mechanistic theory, based on experimental data, about the filtration characteristics of the glomerular filtration barrier. It provides a unique approach to the microanatomy of the glomerulus, renal autoregulation, and the pathogenesis of proteinuria.

On decoupling of volatility smile and term structure in inverse option pricing

Correct pricing of options and other financial derivatives is of great importance to financial markets and one of the key subjects of mathematical finance. Usually, parameters specifying the underlying stochastic model are not directly observable, but have to be determined indirectly from observable quantities. The identification of local volatility surfaces from market data of European vanilla options is one very important example of this type. As with many other parameter identification problems, the reconstruction of local volatility surfaces is ill-posed, and reasonable results can only be achieved via regularization methods. Moreover, due to the sparsity of data, the local volatility is not uniquely determined, but depends strongly on the kind of regularization norm used and a good a priori guess for the parameter. By assuming a multiplicative structure for the local volatility, which is motivated by the specific data situation, the inverse problem can be decomposed into two separate sub-problems. This removes part of the non-uniqueness and allows us to establish convergence and convergence rates under weak assumptions. Additionally, a numerical solution of the two sub-problems is much cheaper than that of the overall identification problem. The theoretical results are illustrated by numerical tests.

Nonlinear regularization methods for ill-posed problems with piecewise constant or strongly varying solutions

In this paper we consider nonlinear ill-posed problems with piecewise constant or strongly varying solutions. A class of nonlinear regularization methods is proposed, in which smooth approximations to the Heavyside function are used to reparameterize functions in the solution space by an auxiliary function of levelset type. The analysis of the resulting regularization methods is carried out in two steps: first, we interpret the algorithms as nonlinear regularization methods for recovering the auxiliary function. This allows us to apply standard results from regularization theory, and we prove convergence of regularized approximations for the auxiliary function; additionally, we obtain the convergence of the regularized solutions, which are obtained from the auxiliary function by the nonlinear transformation. Second, we analyze the proposed methods as approximations to the levelset regularization method analyzed in [Fr¨ uhauf F, Scherzer O and Leit˜ ao A 2005 Analysis of regularization methods for the solution of ill-posed problems involving discontinuous operators SIAM J. Numer. Anal. 43 767–86], which follows as a limit case when the smooth functions used for the nonlinear transformations converge to the Heavyside function. For illustration, we consider the application of the proposed algorithms to elliptic Cauchy problems, which are known to be severely ill-posed, and typically allow only for limited reconstructions. Our numerical examples demonstrate that the proposed methods provide accurate reconstructions of piecewise constant solutions also for these severely ill-posed benchmark problems. (Some figures in this article are in colour only in the electronic version)

A MIXED VARIATIONAL FRAMEWORK FOR THE RADIATIVE TRANSFER EQUATION

We present a rigorous variational framework for the analysis and discretization of the radiative transfer equation. Existence and uniqueness of weak solutions are established under rather general assumptions on the coefficients. Moreover, weak solutions are shown to be regular and hence also strong solutions of the radiative transfer equation. The relation of the proposed variational method to other approaches, including least-squares and even-parity formulations, is discussed. Moreover, the approximation by Galerkin methods is investigated, and simple conditions are given, under which stable quasi-optimal discretizations can be obtained. For illustration, the approximation by a finite element PN approximation is discussed in some detail.

Efficient solution of a three-dimensional inverse heat conduction problem in pool boiling

In this paper, we consider a three-dimensional transient inverse heat conduction problem arising in pool boiling experiments, i.e., the reconstruction of the surface heat flux from pointwise temperature observations inside a heater. We show that the inverse problem is ill-posed and utilize Tikhonov regularization and conjugate gradient methods together with a discrepancy stopping rule for a stable solution. We investigate the proper choice of regularization terms, which not only affects stability of the reconstructions but also greatly influences the quality of reconstructions in the case of limited observations. For the numerical solution of the governing partial differential equation, a space-time finite element method is used. This allows us to compute exact gradients for the discretized Tikhonov functional, and enables the use of conjugate gradient methods for the solution of the regularized inverse problem. We discuss further aspects of an efficient implementation, including a multilevel optimization strategy, together with an implementable stopping criterion. Finally, the proposed algorithms are applied to the reconstruction of local boiling heat fluxes from experimental data.

Analysis and Regularization of Problems in Diffuse Optical Tomography

In this paper we consider the regularization of the inverse problem of diffuse optical tomography by standard regularization methods with quadratic penalty terms. We therefore investigate in detail the properties of the associated forward operators and derive continuity and differentiability results, which are based on derivation of

An Lp theory for stationary radiative transfer

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