A stochastic control problem and related free boundaries in finance

Guan, Chonghu; Li, Xun; Xu, Zuo Quan; Yi, Fahuai

doi:10.3934/mcrf.2017021

Cited by 14 publications

(14 citation statements)

References 21 publications

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“…for (t, x) ∈ (0, T )×(K, ∞). Then V satisfies the following HJB variational equation (see Guan et al (2017)):…”

Section: Optimal Investment Stopping Problemsmentioning

confidence: 99%

“…Jian et al (2014) apply the dual transformation method to convert the nonlinear variational equation with power utility into an equivalent free boundary problem of a linear PDE and analyse qualitatively the properties of the free boundary and optimal strategies. The work is further extended in Guan et al (2017) to a problem with a call option type terminal payoff and power utility.…”

Section: Introductionmentioning

confidence: 99%

“…Finding the free boundary is much more difficult for the optimal investment stopping problem than for the American options pricing problem as the former has a nonlinear PDE in the continuation region and a non-Lipschitz continuous utility function and may have one or more free boundaries whereas the latter has a linear PDE in the continuation region and a Lipschitz continuous payoff function and a unique free boundary. The dual transformation in Jian et al (2014) and Guan et al (2017) is a step in the right direction to simplify the primal nonlinear variational equation into the dual linear variational equation, however, finding the free boundary remains a challenging and open problem.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Global Closed-Form Approximation of Free Boundary for Optimal Investment Stopping Problems

Ma¹,

Xing²,

Zheng³

2019

SIAM J. Control Optim.

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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 derive the asymptotic properties of the dual value function and the associated dual free boundary for a class of utility functions, including power and non-HARA utilities. We construct a global closed-form approximation to the dual free boundary, which greatly reduces the computational cost. Using the duality relation, we find the approximate formulas for the optimal value function, trading strategy, and exercise boundary for the optimal investment stopping problem. Numerical examples show the approximation is robust, accurate and fast.2010 MSC: 49L20, 90C46

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“…for (t, x) ∈ (0, T )×(K, ∞). Then V satisfies the following HJB variational equation (see Guan et al (2017)):…”

Section: Optimal Investment Stopping Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Global Closed-Form Approximation of Free Boundary for Optimal Investment Stopping Problems

Ma¹,

Xing²,

Zheng³

2019

SIAM J. Control Optim.

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“…Unlike these optimal stopping problems, the constrained control process is much more involved as it is directly affected by the stopping time in our problem. In control-stopping problems with no constraint, the duality approach is commonly taken to separately solve the control and stopping parts (see e.g., Karatzas and Wang (2000), Dybvig and Liu (2010), Yang and Koo (2018), Guan et al (2017), Ma et al (2019)). Here, the original problem is transformed to a pure optimal stopping dual problem without control, and the dual-state variable is unaffected by the stopping time.…”

Section: Introductionmentioning

confidence: 99%

Optimal Expansion of Business Opportunity

Chen¹,

Chiu²,

Wong³

2021

Preprint

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Any firm whose business strategy has an exposure constraint that limits its potential gain naturally considers expansion, as this can increase its exposure. We model business expansion as an enlargement of the opportunity set for business policies. However, expansion is irreversible and has an opportunity cost attached. We use the expected optimization of utility to formulate this as a novel stochastic control problem combined with an optimal stopping time, and we derive an explicit solution for exponential utility.We apply the framework to an investment and a reinsurance scenario. In the investment problem, the cost and incentives to increase the trading exposure are analyzed, while the optimal timing for an insurer to launch its reinsurance business is investigated in the reinsurance problem. Our model predicts that the additional income gained through business expansion is the key incentive for a decision to expand. Interestingly, companies may have this incentive but are likely to wait for a period of time before expanding, although situations of zero opportunity cost or specific restrictive conditions on the model parameters are exceptions to waiting. The business policy remains on the boundary of the opportunity set before expansion during the waiting period. The length of the waiting period is related to the opportunity cost, return, and risk of the expanded business.

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“…Hu, Liang and Wu [11] propose a free boundary model for pricing a corporate bond with credit rating migration. Guan et al [9] investigate an optimal stopping problem for an investor whose utility is nonsmooth and nonconcave over a finite time horizon. Mathematically, the free boundary problem can be distinguished into infinite and finite time horizon problems.…”

mentioning

confidence: 99%

Free boundary problem for an optimal investment problem with a borrowing constraint

Guan¹,

Li²,

Zhou³

et al. 2022

JIMO

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This paper considers an optimal investment problem under CRRA utility with a borrowing constraint. We formulate it into a free boundary problem consisting of a fully nonlinear equation and a linear equation. We prove the existence and uniqueness of the classical solution and present the condition for the existence of the free boundary under a linear constraint on a borrowing rate. Furthermore, we prove that the free boundary is continuous and smooth when the relative risk aversion coefficient is sufficiently small.

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A stochastic control problem and related free boundaries in finance

Cited by 14 publications

References 21 publications

Global Closed-Form Approximation of Free Boundary for Optimal Investment Stopping Problems

Global Closed-Form Approximation of Free Boundary for Optimal Investment Stopping Problems

Optimal Expansion of Business Opportunity

Free boundary problem for an optimal investment problem with a borrowing constraint

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