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Ryan Hynd

Publications and source records attributed to Ryan Hynd.

44 records · Page 3Linked to original sources
arXiv2015

On Hamilton-Jacobi-Bellman equations with convex gradient constraints

We study PDE of the form $\max\{F(D^2u,x)-f(x), H(Du)\}=0$ where $F$ is uniformly elliptic and convex in its first argument, $H$ is convex, $f$ is a given function and $u$ is the unknown. These equations are derived from dynamic programming in a wide class of stochastic singular control problems. In particular, examples of these equations arise in mathematical finance models involving transaction costs, in queuing theory, and spacecraft control problems. The main aspects of this work are to identify conditions under which solutions are uniquely defined and have Lipschitz continuous gradients. We also generalize previous results known for the case where $M\mapsto F(M,x)$ is the maximum of finitely many linear functions.

math.AP↗
arXiv2014

Infinite horizon value functions in the Wasserstein spaces

We perform a systematic study of optimization problems in the Wasserstein spaces that are analogs of infinite horizon, deterministic control problems. We derive necessary conditions on action minimizing paths and present a sufficient condition for their existence. We also verify that the corresponding generalized value functions are a type of viscosity solution of a time independent, Hamilton-Jacobi equation in the space of probability measures. Finally, we prove a special case of a conjecture involving the subdifferential of generalized value functions and their relation to action minimizing paths.

math.AP↗
arXiv2014

Option pricing in the large risk aversion, small transaction cost limit

We characterize the price of a European option on several assets for a very risk averse seller, in a market with small transaction costs as a solution of a nonlinear diffusion equation. This problem turns out to be one of asymptotic analysis of parabolic PDE, and the interesting feature is the role of a nonlinear PDE eigenvalue problem. In particular, we generalize previous work of Guy Barles and H. Mete Soner who studied this problem for a European option on a single asset.

math.AP↗
arXiv2012

Nonuniqueness of infinity ground states

In this paper, we construct a dumbbell domain for which the associated principle $\infty$-eigenvalue is not simple. This gives a negative answer to the outstanding problem posed by Juutinen-Lindquivst-Manfredi ("The $\infty$-eigenvalue problem", Arch. Ration. Mech. Anal. 148, 1999, no.2, 89-105). It remains a challenge to determine whether simplicity holds for convex domains.

math.AP↗
arXiv2011

Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control

We study the partial differential equation max{Lu - f, H(Du)}=0 where u is the unknown function, L is a second-order elliptic operator, f is a given smooth function and H is a convex function. This is a model equation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. We establish the existence of a unique viscosity solution of the Dirichlet problem that has a Holder continuous gradient. We also show that if H is uniformly convex, the gradient of this solution is Lipschitz continuous.

math.AP↗
arXiv2011

The eigenvalue problem of singular ergodic control

We consider the problem of finding a real number lambda and a function u satisfying the PDE max{lambda -Δu -f,|Du|-1}=0, for all x in R^n. Here f is a convex, superlinear function. We prove that there is a unique lambda* such that the above PDE has a viscosity solution u satisfying u(x)/|x|->1 as |x| tends to infinity. Moreover, we show that associated to lambda^* is a convex solution u^* with D^2u^* uniformly bounded and give two min-max formulae for lambda^*. lambda^* has a probabilistic interpretation as being the least, long-time averaged ("ergodic") cost for a singular control problem involving f.

math.AP↗