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# Finite horizon portfolio selection problems with stochastic borrowing constraints

• * Corresponding author: Junkee Jeon

The first author is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Grant No. NRF-2017R1C1B1001811)

• In this paper we investigate the optimal consumption and investment problem with stochastic borrowing constraints for a finitely lived agent. To be specific, she faces a credit limit which is a constant fraction of the present value of her stochastic labor income at each time. By using the martingale approach and transformation into an infinite series of optimal stopping problems which has the same characteristic as finding the optimal exercise time of an American option. We recover the value function by establishing a duality relationship and obtain the integral equation representation solution for the optimal consumption and portfolio strategies. Moreover, we provide some numerical illustrations for optimal consumption and investment policies.

Mathematics Subject Classification: Primary: 91G10, 91G80; Secondary: 90C15.

 Citation: • • Figure 1.  Free boundary $z^{\star}(t)$. Parameter values are given by $\mu = 0.05, \sigma = 0.2, r = 0.01, \beta = 0.05, \gamma = 2, \mu_{I} = 0.012, \sigma_{I} = 0.1, \nu = 0.3 \;\;\mbox{and}\; T = 10$

Figure 2.  Simulated paths of wealth to income ratio $X^{*}/I$, portfolio to income ratio $\pi^{*}/I$, consumption to income ratio $c^{*}/I$, the process $y^{*}D^{*}$, and the process $D^{*}$. Parameter values are given by $\mu = 0.05, \sigma = 0.2, r = 0.01, \beta = 0.05, \gamma = 2, \mu_{I} = 0.012, \sigma_{I} = 0.1, \nu = 0.3 \;\;\mbox{and}\; T = 30$

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