arXiv subjects

Takwon Kim

Publications and source records attributed to Takwon Kim.

6 recordsLinked to original sources

arXiv2026

Sharp asymptotics for a free boundary touching a degenerate boundary: the perpetual Heston put

We study the exercise boundary of the perpetual American put in the Heston model near zero variance. There the pricing operator degenerates, and the exercise boundary, a free boundary in the stock--variance half-plane, touches the boundary of the half-plane. No Feller condition is imposed. The boundary is analytic at positive variance and extends as a regular $C^1$ curve tangent to the zero-variance axis. Its approach to the axis follows one of three laws, according to whether the Feller ratio $2κθ/σ^2$ is larger than, equal to, or smaller than one: a logarithmic correction to linear order, an exponential correction in the square root of the logarithm, or a fractional power. In the first two cases the leading coefficients are explicit and the tangent is not Hölder continuous at the axis. In the third case the exponent is selected by a possibly nonsymmetric stable operator on the axis, and the amplitude is finite and positive.

math.AP↗

arXiv2026

Interior smoothness and zero-variance geometry for finite-maturity Heston American puts

We study finite-maturity American puts in the Heston model without the Feller condition. At positive variance, the exercise boundary is smooth and strictly decreasing in remaining maturity and variance. At zero variance, the endpoint is locally Lipschitz in remaining maturity, the boundary moves with bounded speed in maturity uniformly down to zero variance, and it extends continuously to the axis. Each boundary slice at positive maturity has a regular $C^1$ closure tangent to the zero-variance axis, its stock threshold has variance derivative tending to minus infinity, and both partial derivatives of the inverse boundary graph vanish along the slice. The continuation premium at the axis contact stock is of order variance, and as a consequence smooth fit fails in the variance direction at the axis. At expiry the whole boundary, including its axis edge, converges to the strike. We give quantitative boundary gap bounds, an exact limiting weighted boundary Hessian, and vertical identities retaining the maturity term. Exact boundary gap equivalents remain open.

q-fin.MF↗

arXiv2026

Consumption and Investment When Welfare Floors Change over Time

We study consumption and investment over a finite horizon when continuation utility must stay above a welfare floor that changes over time, and the floor at the horizon acts as a terminal wealth floor. A cumulative multiplier reduces the problem to a parabolic obstacle problem. When relative risk aversion exceeds one, the intervention boundary exists at every interior date. When it is below one, the boundary exists exactly at the dates where a discounted transformation of the floor is locally increasing and sets a strict running record, so it can disappear and later reenter. We prove local regularity of the boundary, construct the dual value from the obstacle solution, and verify the optimal cumulative multiplier, whose moments are controlled through its representation as a running supremum. Strong duality identifies the exact lower endpoint of feasible wealth. With stochastic income in a complete market, the floor becomes the value of an outside option, which links the boundary to limited commitment and endogenous borrowing capacity.

math.OC↗

arXiv2026

Finite-Horizon Reversible Investment under Multi-Factor Dynamics

We study a finite-horizon reversible investment problem in which a risk-neutral firm adjusts capacity at a proportional purchase cost and a lower salvage value under multi-factor geometric Brownian motion. Via the singular control--optimal switching correspondence, the marginal value of capacity solves a family of parabolic double-obstacle problems. We prove existence, uniqueness and local Sobolev regularity of the strong solution, characterize investment, waiting and disinvestment regions by continuous, strictly separated free boundaries, and verify optimality of the reflected capacity process. Numerically, joint demand improvements shift both boundaries super-additively, 1.5--2.7 times as strongly at the disinvestment boundary, depending on factor correlation.

q-fin.MF↗

arXiv2026

Liouville theorems for nonlocal Lane--Emden inequalities with degenerate kernels

We establish a Liouville-type theorem for the inequality $\mathcal{L}_K u \ge |u|^q$ in $\mathbb{R}^n$, where $\mathcal{L}_K$ is a translation-invariant, degenerate elliptic integro-differential operator whose kernel is even and satisfies $0\le K(z)\le Λ|z|^{-n-2s}$. No lower ellipticity bound is imposed on the kernel, and no sign condition is imposed on the solution. We prove that every such solution is trivial when $1 \le q \le \frac{n}{n-2s}$ if $n > 2s$, and for every $q \ge 1$ if $n \le 2s$. The proof combines a test function method with a dyadic decomposition of the nonlocal tail and uses neither the maximum principle nor a fundamental solution.

math.AP↗

arXiv2023

Generalized Schauder Theory and its Application to Degenerate/Singular Parabolic Equations

In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form $$u_t = a^{i'j'}u_{i'j'} + 2 x_n^{γ/2} a^{i'n} u_{i'n} + x_n^γ a^{nn} u_{nn} + b^{i'} u_{i'} + x_n^{γ/2} b^n u_{n} + c u + f \quad (γ\leq1).$$ When the equation above is singular, it can be derived from Monge--Ampère equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution $u$, which is called $s$-polynomial. To prove $C_s^{2+α}$-regularity and higher regularity of the solution $u$, we establish generalized Schauder theory which approximates coefficients of the operator with $s$-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap method to obtain higher regularity.

math.AP↗