arXiv research

We present a stochastic numerical method for solving fully non-linear free boundary problems of parabolic type and provide a rate of convergence under reasonable conditions on the non-linearity.

The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility which can be a function of the second derivative of the option price itself. A motivation for studying the nonlinear Black--Scholes equation with a nonlinear vola…

We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free bondary satisfies a nonlinear integral equation, and analyze it in the limit of small $ρ$ = $2r/ σ^2$, where $r$ is the interest rate and $σ$ is the volatility. We use perturbation met…

Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.

The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the…

In this paper we study a utility maximization problem with both optimal control and optimal stopping in a finite time horizon. The value function can be characterized by a variational equation that involves a free boundary problem of a fully nonlinear partial differential equation. Using the dual control method, we der…

Study curves evolving on hypersurfaces with free boundaries, preserving length.

We present a numerical approach for solving the free boundary problem for the Black-Scholes equation for pricing American style of floating strike Asian options. A fixed domain transformation of the free boundary problem into a parabolic equation defined on a fixed spatial domain is performed. As a result a nonlinear t…

We investigate qualitative and quantitative behavior of a solution of the mathematical model for pricing American style of perpetual put options. We assume the option price is a solution to the stationary generalized Black-Scholes equation in which the volatility function may depend on the second derivative of the opti…

In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, to stu…

This paper analyzes optimal consumption strategies for loss-averse investors with multiplicative habit formation.

Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial differential equations, are presented, which arise naturally in some longstanding, …

Study on stability of free boundary Willmore problem using new gradient inequality.

In this paper we investigate a nonlinear generalization of the Black-Scholes equation for pricing American style call options in which the volatility term may depend on the underlying asset price and the Gamma of the option. We propose a numerical method for pricing American style call options by means of transformatio…

We present several principal bundles of embeddings of compact manifolds (with or without boundary) whose base manifolds are nonlinear Grassmannians. We study their infinite dimensional differential manifold structure in the Fréchet category. This study is motivated by the occurrence of such objects in the geometric Lag…

Study on stock trading model with uncertain market status, proving free boundaries and optimal strategies.

In this paper we establish a connection between free boundary minimal surfaces in a ball in $\mathbb{R}^3$ and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.

Proposes neural networks for solving complex free boundary problems.

Solves a general class of free boundary Monge-Ampère equations.

We analyze and calculate the early exercise boundary for a class of stationary generalized Black-Scholes equations in which the volatility function depends on the second derivative of the option price itself. A motivation for studying the nonlinear Black Scholes equation with a nonlinear volatility arises from option p…

In this paper, we consider a free boundary problem with volume constraint. We show that positive minimizer is locally Lipschitz and the free boundary is analytic away from a singular set with Hausdorff dimension at most $n-8$.

Sharp bound on singular set dimension for specific geometric problems.

Model for corporate bond pricing with credit rating migration, solving a double free boundary problem.

Study on free boundary problems in RCD spaces, proving existence and regularity.

The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.

The study finds the optimal metrics for free boundary minimal surfaces in spherical caps.

Proves existence of mean curvature flow with surgery for free boundary surfaces.

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension $8$, using variational arguments, we also obtain solutions which are global minimizers of the correspondi…

We solve an optimal consumption problem with habit formation constraints.

The critical catenoid is uniquely determined by certain symmetries of its boundary.

The present paper is devoted to the joint motion of two immiscible incompressible liquids in porous media. The liquids have different densities and initially separated by a surface of strong discontinuity (free boundary). We discuss the results of numerical simulations for exact free boundary problems on the microscopi…

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in $\mathbb{R}^{N}$. Its free boundary is of \textquotedblleft catenoid\textquotedblright\ type. This is a higher dimensional analogy of the Hauswirth-Helein-Pacard solution in $\mathbb{R}$ (\cite{Pacard}). The exist…

In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem. We review several different techniques of const…

We study classical solutions to the one-phase free boundary problem in which the free boundary consists of smooth curves and the components of the positive phase are simply-connected. We show that if two components of the free boundary are close, then the solution locally resembles an entire solution discovered by Haus…

Study finds optimal retirement timing in uncertain wage scenarios.

We study American swaptions in the linear-rational (LR) term structure model introduced in [5]. The American swaption pricing problem boils down to an optimal stopping problem that is analytically tractable. It reduces to a free-boundary problem that we tackle by the local time-space calculus of [7]. We characterize th…

This paper is a discussion of relations between some free-boundary problems and infinite dimensional Lie groups; particularly a version of Nahm's equations for the group of Hamiltonian diffeomorphisms in two dimensions.

Study anisotropic obstacle problem for minimal surfaces using Cahn-Hoffman transform.

In this paper, we compute the Morse index for a free boundary minimal submanifold from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. As an application, we show that the Morse index…

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkow…

Anisotropic obstacle problems and Stefan problem studied with evolving surfaces.

Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to ac…

In this paper, we derive a new handy integral equation for the free-boundary of infinite time horizon, continuous time, stochastic, irreversible investment problems with uncertainty modeled as a one-dimensional, regular diffusion $X$. The new integral equation allows to explicitly find the free-boundary $b(\cdot)$ in s…

Study optimal consumption with drawdown limits over a fixed time frame.

New solutions found for a complex boundary problem.

The paper studies the properties of maps with free boundaries, focusing on the obstacle case.

We consider smooth bounded surfaces with a smooth boundary and a prescribed background metric g_0. We now consider all metrics g conformal to g_0 which have a prescribed volume M. We now minimize the first eigenvalue of the Laplace operator of g over the metrics conformal to g_0 and having the prescribed volume. We sho…

The paper studies free boundary minimal surfaces with many boundaries and their convergence to closed minimal surfaces.