| 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. |
- Aly, Sidi Mohamed, et al.
(författare)
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Market making and portfolio liquidation under uncertainty
- 2014
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Ingår i: International Journal of Theoretical and Applied Finance. - 1793-6322 .- 0219-0249. ; 17:5
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Tidskriftsartikel (refereegranskat)abstract
- Market making and optimal portfolio liquidation in the context of electronic limit order books are of considerably practical importance for high frequency (HF) market makers as well as more traditional brokerage firms supplying optimal execution services for clients. In general, the two problems are based on probabilistic models defined on certain reference probability spaces. However, due to uncertainty in model parameters or in periods of extreme market turmoil, ambiguity concerning the correct underlying probability measure may appear and an assessment of model risk, as well as the uncertainty on the choice of the model itself, becomes important, as for a market maker or a trader attempting to liquidate large positions, the uncertainty may result in unexpected consequences due to severe mispricing. This paper focuses on the market making and the optimal liquidation problems using limit orders, accounting for model risk or uncertainty. Both are formulated as stochastic optimal control problems, with the controls being the spreads, relative to a reference price, at which orders are placed. The models consider uncertainty in both the drift and volatility of the underlying reference price, for the study of the effect of the uncertainty on the behavior of the market maker, accounting also for inventory restriction, as well as on the optimal liquidation using limit orders.
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| 7. |
- Avelin, Benny, 1984-, et al.
(författare)
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Boundary estimates for solutions to operators of p-Laplace type with lower order terms
- 2011
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Ingår i: Journal of Differential Equations. - : Elsevier BV. - 0022-0396 .- 1090-2732. ; 250:1, s. 264-291
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study the boundary behavior of solutions to equations of the form∇⋅A(x,∇u)+B(x,∇u)=0, in a domain Ω⊂Rn, assuming that Ω is a δ-Reifenberg flat domain for δ sufficiently small. The function A is assumed to be of p-Laplace character. Concerning B, we assume that |∇ηB(x,η)|⩽c|η|p−2, |B(x,η)|⩽c|η|p−1, for some constant c, and that B(x,η)=|η|p−1B(x,η/|η|), whenever x∈Rn, η∈Rn∖{0}. In particular, we generalize the results proved in J. Lewis et al. (2008) [12] concerning the equation ∇⋅A(x,∇u)=0, to equations including lower order terms.
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| 8. |
- Lundström, Niklas L.P. 1980-, et al.
(författare)
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On a two-phase free boundary condition for p-harmonic measures
- 2009
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Ingår i: Manuscripta mathematica. - : Springer. - 0025-2611 .- 1432-1785. ; 129:2, s. 231-249
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Tidskriftsartikel (refereegranskat)abstract
- Let Ωi⊂Rn,i∈{1,2} , be two (δ, r 0)-Reifenberg flat domains, for some 0<δ<δ^ and r 0 > 0, assume Ω1∩Ω2=∅ and that, for some w∈Rn and some 0 < r, w∈∂Ω1∩∂Ω2,∂Ω1∩B(w,2r)=∂Ω2∩B(w,2r) . Let p, 1 < p < ∞, be given and let u i , i∈{1,2} , denote a non-negative p-harmonic function in Ω i , assume that u i , i∈{1,2}, is continuous in Ω¯i∩B(w,2r) and that u i = 0 on ∂Ωi∩B(w,2r) . Extend u i to B(w, 2r) by defining ui≡0 on B(w,2r)∖Ωi. Then there exists a unique finite positive Borel measure μ i , i∈{1,2} , on R n , with support in ∂Ωi∩B(w,2r) , such that if ϕ∈C∞0(B(w,2r)) , then∫Rn|∇ui|p−2⟨∇ui,∇ϕ⟩dx=−∫Rnϕdμi.Let Δ(w,2r)=∂Ω1∩B(w,2r)=∂Ω2∩B(w,2r) . The main result proved in this paper is the following. Assume that μ 2 is absolutely continuous with respect to μ 1 on Δ(w, 2r), d μ 2 = kd μ 1 for μ 1-almost every point in Δ(w, 2r) and that logk∈VMO(Δ(w,r),μ1) . Then there exists δ~=δ~(p,n)>0 , δ~<δ^ , such that if δ≤δ~ , then Δ(w, r/2) is Reifenberg flat with vanishing constant. Moreover, the special case p = 2, i.e., the linear case and the corresponding problem for harmonic measures, has previously been studied in Kenig and Toro (J Reine Angew Math 596:1–44, 2006).
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| 9. |
- Nyström, Kaj, 1969-, et al.
(författare)
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Remarks on the Skorohod Problem and Reflected Levy Driven SDEs in Time-dependent Domains
- 2015
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Ingår i: Stochastics. - : Informa UK Limited. - 1744-2508 .- 1744-2516. ; 87:5, s. 747-765
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Tidskriftsartikel (refereegranskat)abstract
- We consider the Skorohod problem for cadlag functions, and the subsequent construction of solutions to normally reflected stochastic differential equations driven by Levy processes, in the setting of non-smooth and time-dependent domains.
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| 10. |
- Önskog, Thomas, 1979-, et al.
(författare)
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Weak approximation of obliquely reflected diffusions in time-dependent domains
- 2010
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Ingår i: Journal of Computational Mathematics. - : Global Science Press. - 0254-9409 .- 1991-7139. ; 28:5, s. 579-605
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Tidskriftsartikel (refereegranskat)abstract
- In an earlier paper, we proved the existence of solutions to the Skorohod problem with oblique reflection in time-dependent domains and, subsequently, applied this result to the problem of constructing solutions, in time-dependent domains, to stochastic differential equations with oblique reflection. In this paper we use these results to construct weak approximations of solutions to stochastic differential equations with oblique reflection, in time-dependent domains in R^d, by means of a projected Euler scheme. We prove that the constructed method has, as is the case for normal reflection and time-independent domains, an order of convergence equal to 1/2 and we evaluate the method empirically by means of two numerical examples. Furthermore, using a well-known extension of the Feynman-Kac formula, to stochastic differential equations with reflection, our method gives, in addition, a Monte Carlo method for solving second order parabolic partial differential equations with Robin boundary conditions in time-dependent domains.
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