arXiv subjects

Gugyum Ha

Publications and source records attributed to Gugyum Ha.

8 recordsLinked to original sources

arXiv2026

Finite-horizon Optimal Consumption and Investment with Costly Income Adjustment

We study optimal consumption and investment for an investor who receives labor income until a fixed retirement date, can raise the income rate irreversibly at a constant price per unit of income flow, and cannot borrow against future income. A pathwise argument shows that no income is purchased during a final period before retirement whose length depends only on the interest rate and the price. Before this period the dual problem has two free boundaries: a lower one generated by the borrowing constraint and an upper one, at which income is purchased, that escapes to infinity as the final period begins. The analysis of this problem is our main mathematical contribution. During the final period it is a single obstacle problem, which we solve, together with the regularity of its free boundary, by established methods. Before that period a partial hodograph transformation straightens both free boundaries and turns the problem into a quasilinear parabolic equation with oblique boundary conditions on a fixed interval. Comparison principles, explicit barriers and regularity theory for quasilinear equations then give existence, uniqueness and smoothness. Transforming back and gluing the two constructions yields global regularity: both free boundaries are infinitely differentiable and the value function is a classical solution at every positive wealth level. A dual budget process verifies optimality. We also derive coupled Volterra integral equations for the free boundaries, prove exponential convergence to the perpetual problem under a scalar condition, and find numerically a purchase threshold that is U-shaped in the time to retirement.

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arXiv2026

Renting, Owning, and Moving: Housing Tenure Choice with Transaction Costs

The wealth thresholds at which a household switches between renting and owning depend on its preferences, but the dates at which some owner sells or every renter buys do not. In our finite-horizon model, a household chooses consumption, a portfolio, and costly tenure changes while rents, ownership expenses, and transaction costs vary over time on announced dates. When financial markets span house-price risk, these dates are those of a deterministic minimum cost schedule for housing, discounted at the risk-adjusted housing carry and computed from a scalar reflected equation. Preferences, housing services, and labor income only locate the thresholds inside these timing windows, and house-price coefficients enter the dates only through the carry, because financial hedging offsets the exposure of the home. When owning costs at least as much as renting, the windows concern selling, and an anticipated selling surcharge closes the selling window before the surcharge peaks and keeps it closed for years afterwards. We also establish strong duality and the regularity of the thresholds and compute them by integral equations. Numerically, the windows disappear when short sales or borrowing are restricted but do not move with the preferences we vary under an uninsurable increase in the ownership cost.

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arXiv2026

Consumption and Investment with Multiple Job-Switching Opportunities

We study finite-horizon optimal switching with affine rewards in a consumption-investment model with finitely many occupational regimes. Directed shortest-path costs yield the exact natural borrowing limit. A dual transformation reduces the control problem to a one-dimensional system of variational inequalities with interconnected obstacles. Convexity and a distributional waiting inequality give local spatial $C^{1,1}$ regularity and smooth fit, including destination ties. These estimates establish strong duality and recover optimal consumption, investment, and switching strategies. A reflected backward stochastic differential equation representation yields a set-based integral formula without prescribed boundary geometry. Every connected component of the spatially regular locus is a maximal smooth graph over time. Exact examples exhibit disconnected contact sets, temporal boundaries under strict triangle costs, and a corner in an individual boundary component under additive route costs. A convergent time discretisation completes the analysis.

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arXiv2026

Consumption adjustment with mean-reverting risk premia: A singular stochastic control approach

We study a two-sided singular stochastic control problem in which changes in consumption carry utility costs and the risk premium follows the mean-reverting Kim-Omberg model. Optimal consumption is reflected between two moving barriers given by thresholds that depend on the current risk premium. These thresholds are the stopping boundaries of a Dynkin game for a two-dimensional diffusion driven by a single Brownian motion. A sharp occupation estimate shows that they are Lipschitz in logarithmic scale, uniformly in the cost of reducing consumption, and that smooth fit holds along the diffusion direction at every contact point. Outside a bounded interval of risk premia and at most two further points, the thresholds are infinitely differentiable. As the reduction cost grows, optimal consumption converges to the ratcheting policy.

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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.

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arXiv2026

The Obstacle Problem Arising from the American Chooser Option

We study the finite-horizon obstacle problem associated with the American chooser option. Its obstacle is the maximum of two independently determined American call and put values and has a spatial kink where these values cross. Using truncation, penalization, and comparison, we construct a local strong Sobolev solution and prove uniqueness within an exponential-growth class. We show that contact occurs only on the smooth exercise-payoff branches and that both contact regions have nonempty sections at every time. We then establish strict monotonicity, terminal limits, and local Lipschitz continuity of the two free boundaries. Localized one-phase equations, quantitative nondegeneracy, and parabolic boundary Harnack estimates yield $C^\infty$ regularity on the open time interval.

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arXiv2026

Finite-Horizon Portfolio Choice, Labor Supply, and Early Retirement under Borrowing Constraints

We study a finite-horizon optimal consumption and portfolio problem with labor supply flexibility and an irreversible early retirement option under a borrowing constraint. The agent chooses consumption, risky investment, and leisure before retirement, while after retirement labor income disappears and leisure is fixed at its maximal level. Preferences are described by a Cobb--Douglas utility, and wealth must remain nonnegative. {Using a dual martingale method, we transform the primal problem into a zero-sum stopper--singular-controller game. The associated dual value is characterized by a min--max parabolic variational inequality with obstacle and gradient constraints. We show that the maximal strong solution of the resulting variational inequality is the unique admissible strong solution whose gradient-constrained free boundary, namely the binding boundary, is monotone increasing in calendar time. A verification argument then identifies this strong solution with the value of the stopper--singular-controller game, and duality recovers the optimal retirement, consumption, leisure, and portfolio policies.} The numerical analysis recovers the value function and optimal policies, and illustrates how labor supply flexibility affects consumption, portfolio choice, and retirement timing under borrowing constraints.

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arXiv2026

Finite-Horizon Optimal Consumption and Investment with Time-Varying Job-Switching Costs

In this paper, we study the finite-horizon problem of an economic agent's optimal consumption, investment, and job-switching decisions. The key new feature of our model is that the job-switching cost is time-varying. This extension leads to a novel mathematical characterization: the agent's dual problem reduces to a parabolic double obstacle problem with time-dependent upper and lower obstacles. By employing rigorous PDE theory, we establish not only the existence and uniqueness of the solution to this double obstacle problem, but also the smoothness of the two free boundaries that emerge from it. Building on these results, we characterize the agent's optimal consumption, portfolio, and job-switching strategies.

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