| 1. |
- Frentz, Marie, 1980-, et al.
(författare)
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Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type
- 2010
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 234:1, s. 146-164
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Tidskriftsartikel (refereegranskat)abstract
- Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equations of Kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of Asian options. The purpose of this paper is to numerically solve the Cauchy problem, for a general class of second order degenerate parabolic differential operators of Kolmogorov type with variable coefficients, using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations. Furthermore, we show how to apply these results in the context of mathematical finance and option pricing. The approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation in itself. These problems are caused by the fact that the natural setting for degenerate parabolic differential operators of Kolmogorov type is that of a Lie group much more involved than the standard Euclidean Lie group of translations, the latter being relevant in the case of uniformly elliptic parabolic operators.
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| 2. |
- Frentz, Marie, 1980-, et al.
(författare)
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Non-divergence form parabolic equations associated with non-commuting vector fields : Boundary behavior of nonnegative solutions
- 2012
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Ingår i: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V. - 0391-173X .- 2036-2145. ; 11:2, s. 437-474
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Tidskriftsartikel (refereegranskat)abstract
- In a cylinder Omega(T) = Omega x (0, T) subset of R-+(n+1) we study the boundary behavior of nonnegative solutions of second order parabolic equations of the formH u = Sigma(m)(i,j=1) a(ij)(x, t)XiX (j)u - partial derivative(t)u = 0, (x, t) is an element of R-+(n+1),where X = {X-l, . . . , X-m} is a system of C-infinity vector fields inR(n) satisfying Hormander's rank condition (1.2), and Omega is a non-tangentially accessible domain with respect to the Carnot-Caratheodory distance d induced by X. Concerning the matrix-valued function A = {a(ij)}, we assume that it is real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries a(ij) are Holder continuous with respect to the parabolic distance associated with d. Our main results are: I) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem 1.1); 2) the Holder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem 1.2); 3) the doubling property for the parabolic measure associated with the operator H (Theorem 1.3). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [20, 39]. With one proviso: in those papers the authors assume that the coefficients a(ij) be only bounded and measurable, whereas we assume Holder continuity with respect to the intrinsic parabolic distance.
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| 3. |
- Frentz, Marie, 1980-, et al.
(författare)
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The obstacle problem for parabolic non-divergence form operators of Hörmander type
- 2012
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Ingår i: Journal of Differential Equations. - : Elsevier. - 0022-0396 .- 1090-2732. ; 252:9, s. 5002-2041
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we establish the existence and uniqueness of strong solutions to the obstacle problem for a class of parabolic sub-elliptic operators in non-divergence form structured on a set of smooth vector fields in Rn, X={X1,…,Xq}X={X1,…,Xq}, q⩽n, satisfying Hörmanderʼs finite rank condition. We furthermore prove that any strong solution belongs to a suitable class of Hölder continuous functions. As part of our argument, and this is of independent interest, we prove a Sobolev type embedding theorem, as well as certain a priori interior estimates, valid in the context of Sobolev spaces defined in terms of the system of vector fields.
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| 4. |
- Nyström, Kaj, 1969-, et al.
(författare)
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On Monte Carlo algorithms applied to Dirichlet problems for parabolic operators in the setting of time-dependent domains
- 2009
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Ingår i: Monte Carlo Methods and Applications. - Berlin New York : de Gruyter. - 1569-3961. ; 15, s. 11-47
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Tidskriftsartikel (refereegranskat)abstract
- Dirichlet problems for second order parabolic operators in space-time domains Ω⊂ Rn+1 are of paramount importance in analysis, partial differential equations and applied mathematics. These problems can be approached in many different ways using techniques from partial differential equations, potential theory, stochastic differential equations, stopped diffusions and Monte Carlo methods. The performance of any technique depends on the structural assumptions on the operator, the geometry and smoothness properties of the space-time domain Ω, the smoothness of the Dirichlet data and the smoothness of the coefficients of the operator under consideration. In this paper, which mainly is of numerical nature, we attempt to further understand how Monte Carlo methods based on the numerical integration of stochastic differential equations perform when applied to Dirichlet problems for uniformly elliptic second order parabolic operators and how their performance vary as the smoothness of the boundary, Dirichlet data and coefficients change from smooth to non-smooth. Our analysis is set in the genuinely parabolic setting of time-dependent domains, which in itself adds interesting features previously only modestly discussed in the literature. The methods evaluated and discussed include elaborations on the non-adaptive method proposed by Gobet [4] based on approximation by half spaces and exit probabilities and the adaptive method proposed in [3] for weak approximation of stochastic differential equations.
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| 5. |
- Önskog, Thomas, 1979-, et al.
(författare)
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Pricing and hedging of financial derivatives using a posteriori error estimates and adaptive methods for stochastic differential equations
- 2010
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 235, s. 563-592
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Tidskriftsartikel (refereegranskat)abstract
- The efficient and accurate calculation of sensitivities of the price of financial derivatives with respect to perturbations of the parameters in the underlying model, the so-called `Greeks', remains a great practical challenge in the derivative industry. This is true regardless of whether methods for partial differential equations or stochastic differential equations (Monte Carlo techniques) are being used. The computation of the `Greeks' is essential to risk management and to the hedging of financial derivatives and typically requires substantially more computing time as compared to simply pricing the derivatives. Any numerical algorithm (Monte Carlo algorithm) for stochastic differential equations produces a time-discretization error and a statistical error in the process of pricing financial derivatives and calculating the associated `Greeks'. In this article we show how a posteriori error estimates and adaptive methods for stochastic differential equations can be used to control both these errors in the context of pricing and hedging of financial derivatives. In particular, we derive expansions, with leading order terms which are computable in a posteriori form, of the time-discretization errors for the price and the associated `Greeks'. These expansions allow the user to simultaneously first control the time-discretization errors in an adaptive fashion, when calculating the price, sensitivities and hedging parameters with respect to a large number of parameters, and then subsequently to ensure that the total errors are, with prescribed probability, within tolerance.
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| 6. |
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| 7. |
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| 8. |
- Ataei, Alireza, et al.
(författare)
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The Kato square root problem for weighted parabolic operators
- 2023
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Ingår i: Analysis & PDE. - 2157-5045 .- 1948-206X.
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Tidskriftsartikel (refereegranskat)abstract
- We give a simplified and direct proof of the Kato square root estimate for parabolic operators with elliptic part in divergence form and coefficients possibly depending on space and time in a merely measurable way. The argument relies on the nowadays classical reduction to a quadratic estimate and a Carleson-type inequality. The precise organization of the estimates is different from earlier works. In particular, we succeed in separating space and time variables almost completely despite the non-autonomous character of the operator. Hence, we can allow for degenerate ellipticity dictated by a spatial A2-weight, which has not been treated before in this context.
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| 9. |
- Avelin, Benny, et al.
(författare)
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A Galerkin type method for kinetic Fokker-Planck equations based on Hermite expansions
- 2023
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Ingår i: Kinetic and Related Models. - 1937-5093 .- 1937-5077.
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain (0,T)×D×Rd, where D is either Td or Rd. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from $\href{arXiv:1902.04037v2}{Alb+21}$ and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term.
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| 10. |
- Avelin, Benny, et al.
(författare)
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Boundary behavior of solutions to the parabolic p-Laplace equation
- 2019
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Ingår i: Analysis & PDE. - : Mathematical Sciences Publishers. - 2157-5045 .- 1948-206X. ; 12:1, s. 1-42
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Tidskriftsartikel (refereegranskat)abstract
- We establish boundary estimates for non-negative solutions to the $p$-parabolic equation in the degenerate range $p>2$. Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA-domains together with sharp boundary decay estimates. If the underlying domain is $C^{1,1}$-regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and Hölder continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waiting time phenomena present which plays a fundamental role throughout the paper. In particular, conditions on these waiting times rule out well-known examples of explicit solutions violating the boundary Harnack principle.
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