Numerical Methods for Pricing Multi-Asset Options Yuwei Chen Master of Science Graduate Department of Computer Science University of Toronto 2017 We consider the pricing of two-asset European and American options by numerical Partial Differential Equation (PDE) methods, and compare the results with certain analytical formulae. Two cases of options are tested: exchange option and spread option. ...

We develop a class of lters based upon the numerical solution of high-order elliptic problems in l R d that allow for independent determination of order and cut-oo wave number and that default to classical Fourier-based lters in homogeneous domains. However, because they are based on the solution of a PDE, the present lters are not restricted to applications in tensor-product-based geometries a...

A self-organizing joint system classical oscillator–random environment is considered within the framework of a complex probabilistic process that satisfies Langevin-type stochastic differential equation. Various types randomness generated by are considered. In limit statistical equilibrium (SEq), second-order partial equations (PDE) derived describe distribution environmental fields. The mathem...

An effective nonlinear second-order PDE image denoising and restoration model is described in this paper. The proposed anisotropic diffusion-based filtering scheme is based on some novel versions of the edge-stopping function and the conductance parameter. A consistent numerical approximation scheme is constructed for this continuous model. Our PDE-based smoothing technique provides an efficien...

This paper introduces a novel lowest-order discontinuous Petrov Galerkin (dPG) finite element method (FEM) for the Poisson model problem. The ultra-weak formulation allows for piecewise constant and affine ansatz functions and for piecewise affine and lowest-order Raviart-Thomas test functions. This lowest-order discretization for the Poisson model problem allows for a direct proof of the discr...

Artificial viscosity can be combined with a higher-order discontinuous Galerkin (DG) discretization to resolve a shock layer within a single cell. However, when a piecewiseconstant artificial viscosity model is employed with an otherwise higher-order approximation, element-to-element variations in the artificial viscosity arising at the shock induce oscillations in state gradients and pollute t...

— We study the microlocal analyticity of solutions u of the nonlinear equation ut = f(x, t, u, ux) where f(x, t, ζ0, ζ) is complex-valued, real analytic in all its arguments and holomorphic in (ζ0, ζ). We show that if the function u is a C2 solution, σ ∈ CharLu and 1 i σ([Lu, Lu]) < 0 or if u is a C3 solution, σ ∈ CharLu, σ([Lu, Lu]) = 0, and σ([Lu, [Lu, Lu]]) 6= 0, then σ / ∈WFau. Here WFau de...

(1.3) 2(1 + α)∆w + 2β div ( αD(|Dw|2)− (α∆w + β|Dw|2)Dw) = div(T (Dw, ·))− (E′(w − w)− E′(w − w)) where E′(w − w) = 1 volM ∫ E′(w − w). Chang, Gursky and Yang [CGY] proved that any F -minimizing solution u ∈ W 2,2(M) to (1.3) is actually smooth. It was asked in [CGY] whether any weak solution u ∈ W 2,2(M) is smooth. Indeed, Uhlenbeck and Viaclovsky [UV] confirmed this recently and proved the sm...

In this paper a model order reduction method for a nonlinear elliptic-parabolic system is developed. Systems of this type arise from mathematical models for lithium ion batteries. A non-intrusive reduced order approach based on proper orthogonal decomposition (POD) is presented. In addition to this the interpolation method introduced by Barrault et al. [3] is applied in order to achieve efficie...

Inverse problems are of utmost importance in many fields of science and engineering. In the variational approach inverse problems are formulated as PDE-constrained optimization problems, where the optimal estimate of the uncertain parameters is the minimizer of a certain cost functional subject to the constraints posed by the model equations. The numerical solution of such optimization problems...