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End date: 2024 × Start date: 2021 × End date: 2023 × Start date: 2020 × DDC: 330 × Type: article × WGL Institute: WIAS ×

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    On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints
    (Berlin ; Heidelberg : Springer, 2021) Berthold, Holger; Heitsch, Holger; Henrion, René; Schwientek, Jan
    We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.
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    Short-dated smile under rough volatility: asymptotics and numerics
    (London : Taylor & Francis, 2021) Friz, Peter K.; Gassiat, Paul; Pigato, Paolo
    In Friz et al. [Precise asymptotics for robust stochastic volatility models. Ann. Appl. Probab, 2021, 31(2), 896–940], we introduce a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small-noise formulae for option prices, using the framework [Bayer et al., A regularity structure for rough volatility. Math. Finance, 2020, 30(3), 782–832]. We investigate here the fine structure of this expansion in large deviations and moderate deviations regimes, together with consequences for implied volatility. We discuss computational aspects relevant for the practical application of these formulas. We specialize such expansions to prototypical rough volatility examples and discuss numerical evidence.
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    Optimal Control Problems with Sparsity for Tumor Growth Models Involving Variational Inequalities
    (Dordrecht [u.a.] : Springer Science + Business Media, 2022) Colli, Pierluigi; Signori, Andrea; Sprekels, Jürgen
    This paper treats a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly coupled and, furthermore, the cost functional contains a nondifferentiable term like the L1-norm in order to include sparsity effects which is of utmost relevance, especially time sparsity, in the context of cancer therapies as applying a control to the system reflects in exposing the patient to an intensive medical treatment. To cope with the difficulties originating from the variational inequality in the state system, we employ the so-called deep quench approximation in which the convex part of the double obstacle potential is approximated by logarithmic functions. For such functions, first-order necessary conditions of optimality can be established by invoking recent results. We use these results to derive corresponding optimality conditions also for the double obstacle case, by deducing a variational inequality in terms of the associated adjoint state variables. The resulting variational inequality can be exploited to also obtain sparsity results for the optimal controls.
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    Semitractability of optimal stopping problems via a weighted stochastic mesh algorithm
    (Oxford [u.a.] : Wiley-Blackwell, 2020) Belomestny, Denis; Kaledin, Maxim; Schoenmakers, John
    In this paper, we propose a Weighted Stochastic Mesh (WSM) algorithm for approximating the value of discrete- and continuous-time optimal stopping problems. In this context, we consider tractability of such problems via a useful notion of semitractability and the introduction of a tractability index for a particular numerical solution algorithm. It is shown that in the discrete-time case the WSM algorithm leads to semitractability of the corresponding optimal stopping problem in the sense that its complexity is bounded in order by (Formula presented.) with (Formula presented.) being the dimension of the underlying Markov chain. Furthermore, we study the WSM approach in the context of continuous-time optimal stopping problems and derive the corresponding complexity bounds. Although we cannot prove semitractability in this case, our bounds turn out to be the tightest ones among the complexity bounds known in the literature. We illustrate our theoretical findings by a numerical example. © 2020 Wiley Periodicals LLC