Ordinary Differential Equations

Walter, Wolfgang

doi:10.1007/978-1-4612-0601-9

Cited by 432 publications

(146 citation statements)

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“…When such an ω is chosen, (A.10) becomes (4.10). The latter is a first order linear ODE with continuous coefficients, so θ ⋆ j,· (ω) is continuous on [0, T ) (e.g., by Chapter 1.2 of [96]). Since our terminal inventory constraint is deterministic, we observe that…”

Section: Appendix a Section 4 Proofsmentioning

confidence: 99%

Mini-Flash Crashes, Model Risk, and Optimal Execution

Bayraktar

Munk

2017

SSRN Journal

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Abstract. Oft-cited causes of mini-flash crashes include human errors, endogenous feedback loops, the nature of modern liquidity provision, fundamental value shocks, and market fragmentation. We develop a mathematical model which captures aspects of the first three explanations. Empirical features of recent mini-flash crashes are present in our framework. For example, there are periods when no such events will occur. If they do, even just before their onset, market participants may not know with certainty that a disruption will unfold. Our mini-flash crashes can materialize in both low and high trading volume environments and may be accompanied by a partial synchronization in order submission.Instead of adopting a classically-inspired equilibrium approach, we borrow ideas from the optimal execution literature. Each of our agents begins with beliefs about how his own trades impact prices and how prices would move in his absence. They, along with other market participants, then submit orders which are executed at a common venue. Naturally, this leads us to explicitly distinguish between how prices actually evolve and our agents' opinions. In particular, every agent's beliefs will be expressly incorrect.As far as we know, this setup suggests both a new paradigm for modeling heterogeneous agent systems and a novel blueprint for understanding model misspecification risks in the context of optimal execution.

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Section: Appendix a Section 4 Proofsmentioning

confidence: 99%

Mini-Flash Crashes, Model Risk, and Optimal Execution

Bayraktar

Munk

2017

SSRN Journal

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“…It is trivial to see that Tv(1) = 0 and if we differentiate the expression for Tv(t) we obtain 12) and thus…”

Section: Some Auxiliary Resultsmentioning

confidence: 99%

Radial solutions for a nonlocal boundary value problem

Enguiça

Sánchez

2006

Boundary Value Problems

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“…By the nature of the vector field (sign of the nonlinearity), any solution ρ = ρ(r) of (3.3) as well as its derivative r n−1 ρ (r) are strictly increasing functions in a (right) neighborhood of r = 0, precisely as far as ρ(r) ≤ 0. With respect to the existence of ρ = ρ(r), we notice that the point r = 0 is a regular singularity for the equation in (3.3) (see, e.g., [14] or [13]). Precisely, this initial value problem has a unique solution, which is a holomorphic function at the point r = 0, that is,…”

Section: Resultsmentioning

confidence: 99%

Terminal value problem for singular ordinary differential equations: Theoretical analysis and numerical simulations of ground states

Palamides

Yannopoulos

2006

Boundary Value Problems

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A singular boundary value problem (BVP) for a second-order nonlinear differential equation is studied. This BVP is a model in hydrodynamics as well as in nonlinear field theory and especially in the study of the symmetric bubble-type solutions (shell-like theory). The obtained solutions (ground states) can describe the relationship between surface tension, the surface mass density, and the radius of the spherical interfaces between the fluid phases of the same substance. An interval of the parameter, in which there is a strictly increasing and positive solution defined on the half-line, with certain asymptotic behavior is derived. Some numerical results are given to illustrate and verify our results. Furthermore, a full investigation for all other types of solutions is exhibited. The approach is based on the continuum property (connectedness and compactness) of the solutions funnel (Knesser's theorem), combined with the corresponding vector field's ones.

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Ordinary Differential Equations

Cited by 432 publications

References 0 publications

Mini-Flash Crashes, Model Risk, and Optimal Execution

Mini-Flash Crashes, Model Risk, and Optimal Execution

Radial solutions for a nonlocal boundary value problem

Terminal value problem for singular ordinary differential equations: Theoretical analysis and numerical simulations of ground states

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