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In this paper, we propose the application of third-order semi-discrete central-upwind conservative schemes to option pricing partial differential equations (PDEs). Our method is a high-order extension of the recent efficient second-order “Black-Box” schemes that successfully priced several option pricing problems. We consider the Kurganov–Levy scheme and its extensions, namely the Kurganov–Noelle–Petrova and the Kolb schemes. These “Black-Box” solvers ensure non-oscillatory property and achieve desired accuracy using a third-order central weighted essentially non-oscillatory (CWENO) reconstruction. We compare the schemes using a European test case and observe that the Kolb scheme performs better. We apply the Kolb scheme to one-dimensional butterfly, barrier, American and non-linear options under the Black–Scholes model. Further, we extend the Kurganov–Levy scheme to solve two-dimensional convection-dominated Asian PDE. We also price American options under the constant elasticity of variance (CEV) model, which treats volatility as a stochastic instead of a constant as in Black–Scholes model. Numerical experiments achieve third-order, non-oscillatory and high-resolution solutions.

The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers' equation and the multi-dimensional incompressible Euler equations. The Fourier method for such problems with quadratic nonlinearities comes in two main flavors. One is the spectral Fourier method. The other is the $$2/3$$ 2 / 3 pseudo-spectral Fourier method, where one removes the highest $$1/3$$ 1 / 3 portion of the spectrum; this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors. Two main themes are discussed in this paper. First, we prove that as long as the underlying exact solution has a minimal $$C^{1+\alpha }$$ C 1 + ? spatial regularity, then both the spectral and the $$2/3$$ 2 / 3 pseudo-spectral Fourier methods are stable. Consequently, we prove their spectral convergence for smooth solutions of the inviscid Burgers equation and the incompressible Euler equations. On the other hand, we prove that after a critical time at which the underlying solution lacks sufficient smoothness, then both the spectral and the $$2/3$$ 2 / 3 pseudo-spectral Fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations. In particular, after shock formation in inviscid Burgers' equation, the total variation of bounded (pseudo-) spectral Fourier solutions must increase with the number of increasing modes and we stipulate the analogous situation occurs with the 3D incompressible Euler equations: the limiting Fourier solution is shown to enforce $$L^2$$ L 2 -energy conservation, and the contrast with energy dissipating Onsager solutions is reflected through spurious oscillations.

In this paper, we establish analyticity of the Navier–Stokes equations with small data in critical Besov spaces \({\dot{B}^{\frac{3}{p}-1}_{p,q}}\) . The main method is Gevrey estimates, the choice of which is motivated by the work of Foias and Temam (Contemp Math 208:151–180, 1997). We show that mild solutions are Gevrey regular, that is, the energy bound \({\|e^{\sqrt{t}\Lambda}v(t)\|_{E_p}>\infty}\) holds in \({E_p:=\tilde{L}^{\infty}(0,T;\dot{B}^{\frac{3}{p}-1}_{p,q})\cap \tilde{L}^{1}(0,T;\dot{B}^{\frac{3}{p}+1}_{p,q})}\) , globally in time for p < ∞. We extend these results for the intricate limiting case p = ∞ in a suitably designed E∞ space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spaces.

We prove stability estimates for the ENO reconstruction and ENO interpolation procedures. In particular, we show that the jump of the reconstructed ENO pointvalues at each cell interface has the same sign as the jump of the underlying cell averages across that interface. We also prove that the jump of the reconstructed values can be upper-bounded in terms of the jump of the underlying cell averages. Similar sign properties hold for the ENO interpolation procedure. These estimates, which are shown to hold for ENO reconstruction and interpolation of arbitrary order of accuracy and on non-uniform meshes, indicate a remarkable rigidity of the piecewise-polynomial ENO procedure.

We propose three novel methods for recovering edges in piecewise smooth functions from their possibly incomplete and noisy spectral information. The proposed methods utilize three different approaches: #1. The randomly-based sparse Inverse Fast Fourier Transform (sIFT); #2. The Total Variation-based (TV) compressed sensing; and #3. The modified zero crossing. The different approaches share a common feature: edges are identified through separation of scales. To this end, we advocate here the use of concentration kernels (Tadmor, Acta Numer. 16:305–378, 2007), to convert the global spectral data into an approximate jump function which is localized in the immediate neighborhoods of the edges. Building on these concentration kernels, we show that the sIFT method, the TV-based compressed sensing and the zero crossing yield effective edge detectors, where finitely many jump discontinuities are accurately recovered. One- and two-dimensional numerical results are presented.

We are concerned with the detection of edges--the location and amplitudes of jump discontinuities of piecewise smooth data realized in terms of its discrete grid values. We discuss the interplay between two approaches. One approach, realized in the physical space, is based on local differences and is typically limited to low-order of accuracy. An alternative approach developed in our previous work [Gelb and Tadmor, Appl. Comp. Harmonic Anal., 7, 101---135 (1999)] and realized in the dual Fourier space, is based on concentration factors; with a proper choice of concentration factors one can achieve higher-orders--in fact in [Gelb and Tadmor, SIAM J. Numer. Anal., 38, 1389---1408 (2001)] we constructed exponentially accurate edge detectors. Since the stencil of these highly-accurate detectors is global, an outside threshold parameter is required to avoid oscillations in the immediate neighborhood of discontinuities. In this paper we introduce an adaptive edge detection procedure based on a cross-breading between the local and global detectors. This is achieved by using the minmod limiter to suppress spurious oscillations near discontinuities while retaining high-order accuracy away from the jumps. The resulting method provides a family of robust, parameter-free edge-detectors for piecewise smooth data. We conclude with a series of one- and two-dimensional simulations.

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious ${\cal O}(1)$ Gibbs oscillations in the neighborhood of edges and an overall deterioration to the unacceptable first-order convergence rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this is achieved by mollification. Here we utilize a modified version of the two-parameter family of spectral mollifiers introduced by Gottlieb & Tadmor [GoTa85]. The ubiquitous one-parameter, finite-order mollifiers are based on dilation. In contrast, our mollifiers achieve their high resolution by an intricate process of high-order cancelation. To this end, we first implement a localization step using edge detection procedure, [GeTa00a, GeTa00b]. The accurate recovery of piecewise smooth data is then carried out in the direction of smoothness away from the edges, and adaptivity is responsible for the high resolution. The resulting adaptive mollifier greatly accelerates the convergence rate, recovering piecewise analytic data within exponential accuracy while removing spurious oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a robust, general-purpose ``black box'' procedure for accurate post processing of piecewise smooth data.

We study the $L^1$ -stability and error estimates of general approximate solutions for the Cauchy problem associated with multidimensional Hamilton-Jacobi (H-J) equations. For strictly convex Hamiltonians, we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors. We then demonstrate this general theory for two types of approximations: approximate solutions constructed by the vanishing viscosity method, and by Godunov-type finite difference methods. If we let $\epsilon$ denote the `small scale' of such approximations (– the viscosity amplitude $\epsilon$ , the spatial grad-size $\Delta x$ , etc.), then our $L^1$ -error estimates are of ${\cal O}(\epsilon)$ , and are sharper than the classical $L^\infty$ -results of order one half, ${\cal O}(\sqrt{\epsilon})$ . The main building blocks of our theory are the notions of the semi-concave stability condition and $L^1$ -measure of the truncation error. The whole theory could be viewed as a multidimensional extension of the $Lip^\prime$ -stability theory for one-dimensional nonlinear conservation laws developed by Tadmor et. al. [34,24,25]. In addition, we construct new Godunov-type schemes for H-J equations which consist of an exact evolution operator and a global projection operator. Here, we restrict our attention to linear projection operators (first-order schemes). We note, however, that our convergence theory applies equally well to nonlinear projections used in the context of modern high-resolution conservation laws. We prove semi-concave stability and obtain $L^1$ -bounds on their associated truncation errors; $L^1$ -convergence of order one then follows. Second-order (central) Godunov-type schemes are also constructed. Numerical experiments are performed; errors and orders are calculated to confirm our $L^1$ -theory.

A third-order accurate Godunov-type scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: 1. A non-oscillatory piecewise-quadratic reconstruction of pointvalues from their given cell averages; and 2. A central differencing based on staggered evolution of the reconstructed cell averages. This results in a third-order central scheme, an extension along the lines of the second-order central scheme of Nessyahu and Tadmor \cite{NT}. The scalar scheme is non-oscillatory (and hence – convergent), in the sense that it does not increase the number of initial extrema (– as does the exact entropy solution operator). Extension to systems is carried out by componentwise application of the scalar framework. In particular, we have the advantage that, unlike upwind schemes, no (approximate) Riemann solvers, field-by-field characteristic decompositions, etc., are required. Numerical experiments confirm the high-resolution content of the proposed scheme. Thus, a considerable amount of simplicity and robustness is gained while retaining the expected third-order resolution.

We consider the 2 x 2 hyperbolic system of isentropic gas dynamics, in both Eulerian or Lagrangian variables (also called the p-system). We show that they can be reformulated as a kinetic equation, using an additional kinetic variable. Such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws. A new phenomenon occurs here, namely that the advection velocity is now a combination of the macroscopic and kinetic velocities. Various applications are given: we recover the invariant regions, deduce new L°° estimates using moments lemma and prove L°° — w* stability for 7 > 3.

It has been a long open question whether the pseudospectral Fourier method without smoothing is stable for hyperbolic equations with variable coefficients that change signs. In this work we answer this question with a detailed stability analysis of prototype cases of the Fourier method. We show that due to weighted \(L^2\)-stability, the \(N\)-degree Fourier solution is algebraically stable in the sense that its \(L^2\) amplification does not exceed \({O}(N)\). Yet, the Fourier method is weakly \(L^2\) -unstable in the sense that it does experience such \({O}(N)\) amplification. The exact mechanism of this weak instability is due the aliasing phenomenon, which is responsible for an \({O}(N)\) amplification of the Fourier modes at the boundaries of the computed spectrum. Two practical conclusions emerge from our discussion. First, the Fourier method is required to have sufficiently many modes in order to resolve the underlying phenomenon. Otherwise, the lack of resolution will excite the weak instability which will propagate from the slowly decaying high modes to the lower ones. Second -- independent of whether smoothing was used or not, the small scale information contained in the highest modes of the Fourier solution will be destroyed by their \({O}(N)\) amplification. Happily, with enough resolution nothing worse can happen.

Rosenau has recently proposed a regularized version of the Chapman-Enskog expansion of hydrodynamics. This regularized expansion resembles the usual Navier-Stokes viscosity terms at law wave-numbers, but unlike the latter, it has the advantage of being a bounded macroscopic approximation to the linearized collision operator. The behavior of Rosenau regularization of the Chapman-Enskog expansion (RCE) is studied in the context of scalar conservation laws. It is shown that thie RCE model retains the essential properties of the usual viscosity approximation, e.g., existence of traveling waves, monotonicity, upper-Lipschitz continuity..., and at the same time, it sharpens the standard viscous shock layers. It is proved that the regularized RCE approximation converges to the underlying inviscid entropy solution as its mean-free-path epsilon approaches 0, and the convergence rate is estimated.