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- Alghamdi, Azza
(författare)
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Approximation of pluricomplex Green functions : A probabilistic approach
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This PhD thesis focuses on probabilistic methods of approximation of pluricomplex Green functions and is based on four papers.The thesis begins with a general introduction to the use of pluricomplex Green functions in multidimensional complex analysis and a review of their main properties. This is followed by short description of the main results obtained in the enclosed papers.In Paper I, we study properties of the metric space of pluriregular sets, that is zero sets of continuous pluricomplex Green functions. The best understood non-trivial examples of such sets are composite Julia sets, obtained by iteration of finite families of polynomial mappings in several complex variables. We prove that the so-called chaos game is applicable in the case of such sets. We also visualize some composite Julia sets using escape time functions and Monte Carlo simulation.In Paper II, we extend results in Paper I to the case of infinite compact families of proper polynomials mappings. With composition as the semigroup operation, we generate families of infinite iterated function systems with compact attractors. We show that such attractors can be approximated probabilistically in a manner of the classic chaos game.In Paper III, we study numerical approximation and visualisation of pluricomplex Green functions based on the Monte-Carlo integration. Unlike alternative methods that rely on locating a sequence of carefully chosen finite sets of points satisfying some optimal conditions for approximation purposes, our approach is simpler and more direct by relying on generation of pseudorandom points. We examine numerically the errors of approximation for some simple geometric shapes for which the pluricomplex Green functions are known. If the pluricomplex Green functions are not known, the errors in Monte Carlo integration can be expressed with the aid of statistics in terms of confidence intervals.Finally, in Paper IV, we study how perturbations of an orthonomalization procedure influence the resulting approximate Bergman functions. To this end we consider the concept of near orthonormality of a finite set of vectors in an inner product space, understood as closeness of the Gram matrix of those vectors to the identity matrix. We provide estimates for the errors resulting from using nearly orthogonal bases instead of orthogonal ones. The motivation for this work comes from Paper III: when Gram matrices are calculated via Monte Carlo integration, the outcomes of standard orthogonalisation algorithms are nearly orthonormal bases.
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| 2. |
- 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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| 6. |
- 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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| 7. |
- Auscher, Pascal, et al.
(författare)
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L2 well-posedness of boundary value problems and the Kato square root problem for parabolic systems with measurable coefficients
- 2020
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Ingår i: Journal of the European Mathematical Society (Print). - : European Mathematical Society Publishing House. - 1435-9855 .- 1435-9863. ; 22:9, s. 2943-3058
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a first order strategy to study boundary value problems of parabolic systems with second order elliptic part in the upper half-space. This involves a parabolic Dirac operator at the boundary. We allow for measurable time dependence and some transversal dependence in the coefficients. We obtain layer potential representations for solutions in some classes and prove new well-posedness and perturbation results. As a byproduct, we prove for the first time a Kato estimate for the square root of parabolic operators with time dependent coefficients. This considerably extends prior results obtained by one of us under time and transversal independence. A major difficulty compared to a similar treatment of elliptic equations is the presence of non-local fractional derivatives in time.
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| 8. |
- Auscher, Pascal, et al.
(författare)
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The Dirichlet problem for second order parabolic operators in divergence form
- 2018
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Ingår i: Journal de l'École polytechnique — Mathématiques. - : Cellule MathDoc/CEDRAM. - 2270-518X.
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Tidskriftsartikel (refereegranskat)abstract
- We study parabolic operators $\cH = \partial_t-\div_{\lambda,x} A(x,t)\nabla_{\lambda,x}$ in the parabolic upper half space $\mathbb R^{n+2}_+=\{(\lambda,x,t):\ \lambda>0\}$. We assume that the coefficients are real, bounded, measurable, uniformly elliptic, but not necessarily symmetric. We prove that the associated parabolic measure is absolutely continuous with respect to the surface measure on $\mathbb R^{n+1}$ in the sense defined by $A_\infty(\mathrm{d} x\d t)$. Our argument also gives a simplified proof of the corresponding result for elliptic measure.
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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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