Effect of labour income on the optimal bankruptcy problem

In this paper we deal with the optimal bankruptcy problem for agents who can optimally allocate their consumption rate, the amount of capital invested in the risky asset, as well as their leisure time. In our framework, the agents are endowed by an initial debt, and they are required to repay their debt continuously. Declaring bankruptcy, the debt repayment is exempted at the cost of a wealth shrinkage. We implement the duality method to solve the problem analytically and conduct a sensitivity analysis to the bankruptcy cost and benefit parameters. Introducing the flexible leisure/working rate, and therefore the labour income, into the bankruptcy model, we investigate its effect on the optimal strategies. Supplementary Information The online version contains supplementary material available at 10.1007/s10479-023-05166-z.


Introduction
In this paper, we study an optimal stopping time problem, in which an agent decides her consumption-portfolio-leisure strategy as well as the optimal bankruptcy time.Her utility is described by a power utility function concerning the consumption and leisure rates.Moreover, the leisure rate should be upper bounded by a positive constant (L).Relating to the leisure rate, the agent earns the labour income with a fixed wage rate.The sum of labour and leisure rates is assumed to be constant L. As the complement, the labour rate is lower bounded by a positive constant L − L for the consideration of retaining the employment state.By determining the continuous and stopping regions of the corresponding stopping time problem, we prove that the optimal bankruptcy time is the first hitting time of the wealth process downward to a critical wealth boundary.
The idea is directly inspired by Jeanblanc et al. (2004), in which a stochastic control model is constructed to quantify the benefit of filing consumer bankruptcy in the perspective of complete debt erasure.Their research is a response to the sharp growth in bankruptcy cases between 1978 and 2003 due to the promulgation of the 1978 Bankruptcy Reform Act in American.The Act introduced two kinds of consumer bankruptcy mechanisms, which are reflected in its Chapter 7 and Chapter 13 separately: debtors following Chapter 7 to file bankruptcy are granted the debt exemption, but must undertake the liquidation of non-exempt assets.Alternatively, the mechanism in Chapter 13 adopts the reorganization procedure instead of the liquidation.Debtors are permitted to retain assets, but the debt is required to be reorganized and paid continuously from future revenues.The statistical data shows that filing bankruptcy under Chapter 7 predominates in all consumer bankruptcy cases the fixed and flexible bankruptcy costs; the debt d is the continuous-time debt repayment.As already said, the optimal bankruptcy corresponds to the first hitting time of the wealth process of a downward boundary, the bankruptcy wealth threshold.This threshold, as a function of the debt repayment d, is an increasing and convex curve.The rationale of this result is the following: a heavier debt repayment, in fact, implies that the benefit of bankruptcy becomes more attractive, therefore inducing the agent to take a higher threshold to make the bankruptcy requirement more accessible such that she can enjoy the debt exemption easily.Furthermore, the convexity of the mapping can be explained by the fact that this motivation is diminishing as the debt repayment decreases.Similar results hold true if we consider the bankruptcy threshold as a function of α, i.e., the proportion of wealth after the bankruptcy liquidation: a lower value of α indicates a higher flexible cost (a higher value of (1 − α)) and pushes the agent to set a lower wealth level to avoid suffering the bankruptcy.Our numerical results also show the non-monotonic relationship between the bankruptcy wealth threshold and F itself; this is due to the role of F , which is not only the fixed cost of bankruptcy, but, according to the model in Jeanblanc et al. (2004), in order to make the problem feasible, it also has an important role as liquidity constraint in the pre-bankruptcy period.Moreover, comparing the optimal control policies between the model with and without the bankruptcy option, we find that this additional option offers the agent a better circumstance such that the optimal consumption-portfolio-leisure policies dominate the ones without it before the bankruptcy.Whereas, after declaring bankruptcy, the agent suffers the wealth shrinkage and prefers to invest less in the risky asset for the needs of obtaining utility from consumption and leisure.In addition, we also study the impact of introducing the leisure rate as a second control variable, such that the influence of labour income can be disclosed.The numerical result indicates that the optimal consumption and portfolio policies with the flexible leisure option always prevail over the corresponding policies of the model with a full leisure rate, and therefore without labour income.
The paper is organized as follows.Section 2 formulates the corresponding optimization problem, and provides the financial market setting.Section 3 offers the value function of post-bankruptcy problem and its Legendre-Fenchel transform.In Section 4, we construct the duality between the optimal control problem with the individual's shadow price, and obtain a free boundary problem which endows us the closed-form optimal solutions.Section 5 presents the numerical tests to this model and the sensitivity analysis of the bankruptcy wealth threshold to main parameters.Finally, Section 6 concludes.Most of the proofs and computations are reported in the online appendix.

Financial Market
We first formulate the considered financial market over the infinite-time horizon.Based on the mutual fund theorem from Karatzas et al. (1997), we consider only one risky asset which dynamics follows the Geometric Brownian Motion with constant drift and diffusion coefficients.The agent faces two investment opportunities: the investment in the money market, which endows her a fixed and positive interest rate r > 0, and the risky asset, which dynamically evolves according to the stochastic differential equation (SDE) dS(t) = µS(t)dt + σS(t)dB(t), S(0) = S 0 . (2.1) Here, B(t) denotes a standard Brownian motion on the filtered probability space (Ω, F , P), {F t , 0 ≤ t < ∞} is the augmented natural filtration on this Brownian motion, and S 0 represents the initial stock price, which is assumed to be a positive constant.Since the drift and diffusion terms µ and σ are positive constants, there exists a unique solution to the SDE (2.1), S(t) = S 0 e (µ− 1 2 σ 2 )t+σB(t) .Then referring to Karatzas and Shreve (1998b), we introduce the state-price density process as H(t) ξ(t) Z(t), with ξ(t), Z(t), the discount process and an exponential martingale, respectively defined as ξ(t) e −rt , with ξ(0) = 1, Z(t) e − 1 2 θ 2 t−θB(t) , with Z(0) = 1.
Moreover, θ µ−r σ stands for the market price of risk, that is, the Sharpe-Ratio.Since the exponential martingale Z(t) is, in fact, a P-martingale, and both the number of risky assets and the dimension of the driving Brownian motions are equal to one, the financial market M defined with the above setting, M = {(Ω, F , P), B, r, µ, σ, S 0 }, is standard and complete, based on the result from (Karatzas and Shreve, 1998b, Section 1.7, Definition 7.3).Additionally, we can define an equivalent martingale measure through P(A) E Z(t)I A , ∀A ∈ F t .Then based on the Girsanov Theorem, we can get a standard Brownian motion under the P measure as B(t) B(t) + θt, ∀t ≥ 0. (2.2)

The Optimization Problem
The agent optimally chooses the consumption rate, the amount of money allocated in the risky asset and the leisure rate, which are denoted as c(t), π(t) and l(t), treated as the three control variables in the optimization.The sum of the labour and leisure rate is constant and equals L. Therefore, the working rate at time t is ( L − l(t)) that enables the agent to earn a wage of w( L − l(t)), where w > 0 represents the constant wage rate.Obviously, the condition 0 ≤ l(t) ≤ L must be imposed for the positive labour income consideration.Furthermore, a realistic constraint is introduced into the model, that is, the working rate should be lower bounded by a positive constant ( L − L) for the sake of retaining the employment state.
Following Jeanblanc et al. (2004), an affine loss function is introduced for accommodating fixed and variable costs of filing bankruptcy.Let τ denote the bankruptcy time, the agent is obliged to repay continuously a positive fixed debt d until the stopping time τ , whereas this debt obligation is exempted after declaring bankruptcy, but with the fixed cost F > 0 and the variable cost (1 − α), where the proportional coefficient α takes the value in (0, 1).In more detail, the agent needs to pay a fixed toll F once for all at the time τ , and the (1 − α) proportion of the remaining wealth, which is related to the social cost, time cost and taxes cost of declaring bankruptcy.Therefore, the agent is able to keep the amount α(X(τ ) − F ) of wealth for the consumption and investment after bankruptcy.For the purpose of making sure that the agent is capable of affording the bankruptcy, the wealth level is required to cover the cost before the stopping time τ , that is, where η is a small non-negative constant to guarantee that there is still a few amounts of wealth left even after the liquidation.The bankruptcy mechanism described above entails the wealth process X(t) to satisfy the following SDE Furthermore, we assume that the agent's preference is described by a power utility function of consumption and leisure rate Setting k > 1 makes the mixed second partial derivative negative, which clarifies that consumption and leisure are substitute goods.The following lemma introduces the Legendre-Fenchel transform of the function u(c, l), which will help us to reduce the number of control variables up to a single one.Referring to (Choi et al., 2008, Section 2.2), the Legendre-Fenchel transform of the utility function is defined as and it is given in the following lemma.
Lemma 2.1.The Legendre-Fenchel transform of the utility function u(c, l) is .
Furthermore, the consumption-leisure policy reaching the supremum in (2.4) is In this framework, the primal optimization problem, which is denoted as (P ), is expressed as A(x) stands for the admissible control set and follows the definition below.
Remark 2.1.The above framework is consistent with an infinitely lived agent or an agent which death is modelled as the first jump time of an independent Poisson process.In the first case, γ is the subjective discount rate.In the second case, we have γ = γ + λ D , where γ is the subjective discount rate and λ D is the intensity of the Poisson process.In fact, if τ D is the time in which the death occurs, we have due to the independence of the Poisson process.
Definition 2.1.Given the initial wealth x ≥ F + η, A(x) is defined as the set of all admissible policies satisfying: • {c(t) : t ≥ 0} is an F t -progressively measurable and non-negative process such that • {l(t) : t ≥ 0} is an F t -progressively measurable and non-negative process such that 0 ≤ l(t) ≤ L, ∀t ≥ 0, • X(t) ≥ F + η for 0 ≤ t ≤ τ , and X(t) ≥ 0 for t > τ a.s., • represents the market value of the debt repayment reduced by the maximum amount to borrow against the future income in the pre-bankruptcy period: the agent is therefore unable to allocate the investment and consumption when the wealth level stays below it.
The subsequent proposition provides the corresponding budget constraint.
Proposition 2.1.Given any initial wealth x ≥ F + η, any strategy (τ, {c(t), π(t), l(t)} t≥0 ) ∈ A(x), the budget constraint is given by (2.5) Completing the construction of the primal optimization problem, we are going to solve it explicitly in the following sections.The gain function of the primal optimization problem (P ) can be rewritten as the expectation of two separated terms representing the pre-and post-bankruptcy part, where we define J P B (X(t); c, π, l) E ∞ t e −γ(s−t) u(c(s), l(s))ds F t in the post-bankruptcy framework, i.e., no debt repayment.We perform the backward approach, hence, begin with the postbankruptcy part by means of the dynamic programming principle.

Post-Bankruptcy Problem
We first tackle the post-bankruptcy problem, assuming without loss of generalization τ = 0, which is the optimization over an infinite time horizon through controlling the investment amount, consumption and leisure rate.Since the debt repayment is removed from the wealth process after the bankruptcy, the corresponding dynamics becomes Afterwards, based on the gain function J P B (•) defined at the end of the previous section, we can express the value function of the post-bankruptcy part as follows, The admissible control set A P B (x) is compatible with Definition 2.1, only removing the condition about the stopping time and changing the liquidity condition from "X(t) ≥ F + η for 0 ≤ t ≤ τ , and X(t) ≥ 0 for t > τ a.s." to "X(t) ≥ 0 for t ≥ 0 a.s.".Additionally, for any given initial endowment x ≥ 0 and admissible consumption-portfolio-leisure strategy {c(t), π(t), l(t)} t≥0 ∈ A P B (x), the following propositions provide us the budget and liquidity constraint to the postbankruptcy problem.
Proposition 3.1.The infinite horizon budget constraint of the post-bankruptcy problem is Proof.Similarly as in Appendix A.2, we can prove that E t 0 H(s) c(s) + wl(s) − w L ds ≤ x.Then the above budget constraint can be obtained by taking the limit as t → ∞.
Proposition 3.2.The infinite horizon liquidity constraint of the post-bankruptcy problem is Proof.The result directly comes from (Karatzas and Shreve, 1998b, Section 3.9, Theorem 9.4), more precisely, the non-negative property of X(t) and Then, we implement the methodology presented in (Karatzas and Wang, 2000;He and Pages, 1993) to establish a duality between the optimal control problem and the individual's shadow price problem through the Lagrange method.We make the following derivation of J P B (x; c, π, l), introducing a non-increasing process D P B (t) ≥ 0 and a Lagrange multiplier λ, By the Fubini's Theorem, see e.g.(Björk, 2009, Appendix A, Theorem A.48), we have and the inequality concerning J P B (x; c, π, l) can be rewritten as where D represents the set of non-negative, non-increasing and progressively measurable processes.Then, we put forward a theorem to construct the duality between the optimal consumptionportfolio-leisure problem (P P B ) and the individual's shadow price problem (S P B ).
Theorem 3.1.(Duality Theorem) Suppose D * P B (t) is the optimal solution (the arg inf) to the dual shadow price problem (S P B ), then the optimal consumption-leisure strategy to the primal problem (P P B ) satisfies c * (t) + wl * (t) = −ũ ′ (λ * e γt D * P B (t)H(t)), and we have the following relation Here λ * is the parameter λ which gives the infimum in the above equation.
It can be known from the above duality theorem that the optimal solution of the post-bankruptcy problem is transformed into finding the optimal D * P B (t).In order to solve the problem (S P B ) explicitly, we follow the approach in Davis and Norman (1990) and first provide the subsequent assumption.
Then we introduce a new function φ P B : (R + , R + ) → R as which implies that ṼPB (λ) = φ P B (0, λ).(3.4) for any t ≥ 0, with the smooth fit conditions ∂φP B ∂z (t, ẑPB ) = 0, ∂ 2 φP B ∂z 2 (t, ẑPB ) = 0.In line with (Choi et al., 2008, Appendix A), we assume that φ P B (t, z) takes the time-independent form φ P B (t, z) = e −γt v P B (z) for solving the above variational inequalities explicitly.The solution is computed in Appendix B.2 in a semi-analytical framework, i.e., as the solution of a non-linear system of equations.Once the function φ P B (•, •) is computed, and therefore ṼPB (•) is known, we can recover V P B (•) from Theorem 3.1.

Primal Optimization Problem
Once we solved the post-bankruptcy problem, we first of all have to deal with the jump in the wealth at bankruptcy time.We introduce a subset of the primal optimization problem's admissible control set, A 1 (x) ⊂ A(x), inside which any policy maximises the gain function of the post-bankruptcy problem.That is to say, for any (τ, {c(t), π(t), l(t)}) ∈ A 1 (x), the following holds, Here X x,c,π,l (τ ) is the wealth at time τ given an initial wealth x and assuming the policies c, π, l for consumption, allocation in the risky asset and leisure rate, respectively.Then, from the dynamic programming principle, the whole optimization problem is converted into denoting the value function at the moment of bankruptcy as Therefore, it can be observed that the relationship between U (•) and the post-bankruptcy value function Simple computations give us the Legendre-Fenchel transform of U (x), that is, Ũ (z) After obtaining the optimal solution for the post-bankruptcy problem, we now reduce the primal optimization problem by fixing the stopping time.Defining a set of admissible controls corresponding to a fixed stopping time τ ∈ T as and a utility maximization problem as J(x; c, π, l, τ ), (P τ ) the problem (P ) can be transformed into an optimal stopping time problem, Then, we put forward the liquidity constraint for the optimal bankruptcy problem.
Proposition 4.1.The liquidity constraint of the considered problem is Following the same technique as in the post-bankruptcy problem, considering the budget and liquidity constraints (2.5) and (4.2), we introduce a real number λ > 0, the Lagrange multiplier, and a non-increasing continuous process D(t) > 0. We obtain As in (He and Pages, 1993, Section 4), the individual's shadow price problem inspired by the above inequality is defined as below, Ṽτ (x) inf Hereafter, we provide a theorem to construct the duality between the individual's shadow price problem (S τ ) and the optimal consumption-portfolio-leisure problem (P τ ).
Theorem 4.1.(Duality Theorem) Suppose D * (t) is the optimal solution (the arg inf) to Problem coincide with the optimal solution to Problem (P τ ), and we have the following relationship Here λ * is the parameter λ which gives the infimum in the above equation.
Furthermore, the duality theorem makes the value function of Problem (P ) conforms to the following derivation,

Let us introduce a new function Ṽ (λ) sup
τ ∈T Ṽτ (λ).As in (Karatzas and Wang, 2000, Section 8, Theorem 8.5), the value function under the condition that Ṽ (λ) exists and is differentiable for any λ > 0. Hence, the objective optimization problem contains two steps: The first step involves an optimal stopping time problem, and the second refers to obtain the optimum λ * achieving the infimum part.We begin with the first optimization problem related to the individual's shadow price.Before this, following the method of Davis and Norman (1990), an assumption is imposed on the process D(t).
Assumption 4.1.The non-increasing process D(t) is absolutely continuous with respect to t.Hence, there is a non-negative process ψ(t) such that dD(t) = −ψ(t)D(t)dt.
Introducing a new process Z(t) λD(t)e γt H(t), the value function of the dual problem (S τ ) is rewritten as We consider a new generalized optimization problem it can be observed that Ṽ (λ) = φ(0, λ), which indicates the solution of Ṽ (λ) is resorted to solve the above generalized problem.We start with the infimum part through defining the dynamic programming principle gives us the subsequent Bellman equation The following characterizations hold for the optimum ψ * (the arg min): We first focus on the optimal stopping time problem (4.4) and present a lemma to determine its continuous and stopping regions.But before this, one relationship should be noticed.With the optimum ψ * (t), the function φ inf (t, z) can be rewritten as Fixing the time with t = τ , φ inf (τ, z) can be treated as a single variable function of z, that is, From this equation, and Equation (4.1) and ( 4.3), we obtain Ũ considering the convex property of Ũ (z), we directly obtain the relationship there exists z such that the continuous region of the optimal stopping problem (4.4) with the state variable Z(t) is Ω 1 = {0 < z < z}, and the stopping region is Ω 2 = {z ≥ z}.z is the boundary that separates Ω 1 and Ω 2 .
Proof.See Appendix C.3 After obtaining the continuous region Ω 1 = {0 < z < z}, we can treat the stopping time of bankruptcy as the first hitting time of process Z(t) to the boundary z from the inner region of Ω 1 .Figure 4.1 describes the relationship between the optimal stopping time and the continuous and stopping regions.The optimal bankruptcy time is the moment when process Z(t) first touches the boundary, which is represented with the red dotted line, from within the continuous region.Hence, the stopping time satisfies τ = inf{t ≥ 0 : Z(t) ≥ z} and will be proved to be finite with probability one under a sufficient constraint with the following lemma.Besides, it should be clear that z, corresponding to the bankruptcy threshold, is an upper barrier for the process Z(t), and therefore a lower barrier of the wealth process, as we will show in Remark 4.1.
Lemma 4.2.Under the assumption we have P (τ z < τ 0 ) = 1, with two stopping times, τ z = inf Subsequently, combining Condition (4.3) and the optimal stopping time problem (4.4), we get the free boundary problem which characterizes the function φ(•, •), considering two different cases: (1) Variational Inequalities assuming z < ẑ: Find the free boundaries z > 0 (Bankruptcy), ẑ > 0 ((F + η)-wealth level), and a function φ(•, for any t ≥ 0, with the smooth fit conditions Furthermore, in the period up to the stopping time τ , which corresponds to the interval 0 < z < z, we need to consider that whether the constraint 0 ≤ l(t) ≤ L is trigged or not, which is related to the boundary ỹ introduced in Lemma 2.1.Hence, the problem of the pre-bankruptcy part is divided into two cases, 0 < ỹ ≤ z and 0 < z < ỹ.Then combining with the two cases of the post-bankruptcy part, we have the following framework of partition for the primal optimization problem (P ).
As in the post-bankruptcy problem, we assume that φ(t, z) adopts a time-independent form, φ(t, z) = e −γt v(z), then the solution is obtained, see Appendix C.5.We would like to stress that only one of the seven cases admits a solution.
After acquiring the closed form of v(z), (Karatzas and Wang, 2000, Section 8, Theorem 8.5) indicates that keeps true for a unique λ * > 0 under the differentiable property of v(•).Then the wealth threshold of bankruptcy, namely x, can be calculated from the relationship x = −v ′ (z).Therefore, given any initial wealth x ≥ F + η, we get the optimal Lagrange multiplier λ * through solving the equation, x = −v ′ (λ * ).Furthermore, since the optimum λ * is the initial value of process (3.1), Z * (t), the optimal wealth process follows X * (t) = −v ′ (Z * (t)).The optimal bankruptcy time satisfies τ * = inf t≥0 {X * (t) ≤ x}.Moreover, recalling Lemma 2.1, the optimal consumption and leisure strategies are and the optimal portfolio strategy is π ), which can be obtained from (He and Pages, 1993, Section 5, Theorem 3).
Remark 4.1.In Figure 4.1, we plot the relationship between the optimal bankruptcy time and the continuous and stopping regions with respect to Z * (t), showing that the optimal bankruptcy time is the first time the process Z * (t) touch the upper barrier z.The same plot can be done with respect to X * (t) = −v ′ (Z * (t)): the convex property of v(•), see (Karatzas and Shreve, 1998b, Section 3.4, Lemma 4.3), indicates that X * (t) is a decreasing function of Z * (t), therefore, in this case the optimal bankruptcy time is the first time the process X * (t) touch a lower barrier x = −v ′ (z).

Numerical Analysis
We now implement the sensitivity analysis with respect to the input parameters.As baseline parameters, we consider the ones listed in Table 5.1.These inputs satisfy conditions (2.3), (4.6) and (4.7).The parameters r, µ, σ, γ and α are directly taken from Jeanblanc et al. (2004).Whereas the fixed bankruptcy toll and the debt repayment amount in their study are F = 400$ and d = 125$, we set F = 0.96 and d = 0.3 such that the ratios of F and d in our and their research keep consistent.The same consideration is also applied for the setting of the initial wealth x.As discussed in Section 4, seven cases should be considered simultaneously and only one must be verified: in fact with these parameters, only Case 2, "0 < z < αỹ ≤ ẑ < αẑ P B ", admits a solution.With the above input parameters, we derive the output parameters: B 2 , z, ẑ, ỹ, the set of wealth thresholds {x, x, x}, the optimal Lagrange multiplier λ * and the value function V (x).The results are listed in Table 5.2 and 5.3.The f val, i.e., the value of the function at its zero, in Tables 5.2 represents the maximum error generating from using the f solve function of MATLAB to solve the non-linear equations (C.6), (C.7) and (C.8): as expected, the f val value is close to zero, i.e., the algorithm correctly solve the system of equations.
We first of all want to stress that in this case the optimal bankruptcy time is 0, since the initial wealth, 6.6, is lower than the optimal bankruptcy barrier x = 10.0651,i.e., the starting wealth is inside the stopping region.
Sensitivity of optimal solutions with respect to the risk aversion coefficient k: In this part, we use the Monte Carlo method to simulate the single path of optimal wealth process and consumption-portfolio-leisure strategy by taking different values of k for discovering the sensitivity of optimal solutions to the risk aversion coefficient.Parameters are the ones in Table 5.1, with the exception of the initial wealth which is set to 25 instead of 6.6 for observing the bankruptcy mechanism.The agent with a higher value of k prefers to have a higher wealth threshold for declaring bankruptcy (x = 7.4777 for k = 2, x = 10.0651 for k = 3 and x = 12.4239 for k = 4).This is reasonable, as shown in Figure 5.1: the more risk-averse agent tends to smooth the consumption and leisure, and invest less in the risky asset such that maintaining a relatively higher wealth level.Moreover, from the optimal trajectories of leisure, it can be observed that the relatively low fixed and flexible bankruptcy costs, F = 0.96 and 1 − α = 0.1, and the high bankruptcy wealth thresholds enable the agent to enjoy the maximum leisure rate L = 0.8 even after suffering the wealth shrinkage caused by declaring bankruptcy.Therefore, the leisure processes corresponding to different k values are fully identical.Finally, in Figure 5.1 we zoom close to the  Sensitivity of the optimal bankruptcy threshold with respect to the market risk premium jointly with the risk aversion coefficient: The parameter θ = µ−r σ , which is the Sharpe-Ratio, measures the market risk premium.For the purpose of discovering the relationship between x and θ, we keep r, σ constant and change the value of µ from 0.035 to 0.2, with an interval of 0.0075, which leads the value of θ to change from 0.1 to 1.2. Figure 5.2 shows that, with a lower risk premium, that is, a smaller value of θ, the agent prefers to set a higher wealth threshold to more easily get rid of the debt thanks to the bankruptcy.Contrarily, a better market performance entails the agent a stronger ability to bear the debt repayment; hence, she sets a lower wealth threshold to avoid suffering the bankruptcy costs.Furthermore, there is a positive relationship between the optimal wealth threshold of bankruptcy and the risk aversion level k, which is already clarified through Figure 5.1.Sensitivity of the optimal bankruptcy threshold with respect to the debt repayment, the fixed and flexible bankruptcy cost: The wealth process suffers a shrinkage through an affine function α(X(τ )−F ) declaring bankruptcy, and the debt repayment is exempted.Hence, F and (1 − α) can be treated as the fixed and flexible cost of bankruptcy, and d is the benefit of bankruptcy.In order to discover the influence of the bankruptcy option, we provide Figures 5.3-5.4 to illustrate the sensitivity of optimal wealth threshold of bankruptcy with respect to the coefficients α, d and F .First considering the sensitivity of bankruptcy wealth threshold to the flexible cost coefficient, we can observe that x is an increasing function of α.The rationale is the following: since α represents the proportion of wealth held after bankruptcy, a lower value of α indicates a higher cost such that the agent prefers to set a lower wealth threshold to avoid suffering the wealth shrinkage from bankruptcy.As for the relationship between x and d, it can be observed the same increasing and convex curve.When the debt repayment is higher, which implies that the benefit of bankruptcy is more attractive, the agent tends to Figure 5.4: Risk Aversion Coefficient, Fixed Bankruptcy Cost and Bankruptcy Wealth Threshold take a higher threshold such that the wealth process satisfies the bankruptcy requirement x more easily to enjoy the debt exemption.However, this incentive becomes weaker as the debt repayment decreases, which leads to the convexity of the considering mapping.Finally, in Figure 5.4 we provide a three-dimensional image to explain the sensitivity of bankruptcy wealth threshold to the parameter F jointly with k.Since the value of F adjusted according to Jeanblanc et al. ( 2004) is relatively low, the liquidity constraint triggered by F + η is easy to be covered by the labour income.Thus, the role of F is more related to the fixed cost of bankruptcy such that a higher value of F will make the agent set a lower wealth threshold to avoid suffering the bankruptcy, which results in a monotonic decreasing relationship between x and F .However, F is not only the cost of bankruptcy, but also can be regarded as the liquidity constraint to limit the agent's investment behaviour.In order to reflect the phenomenon that the role of F is the trade-off between the liquidity constraint boundary of the pre-bankruptcy period and the fixed cost of bankruptcy, we conduct the same three-dimensional image with different input parameters, particularly, we follow the baseline of inputs in Table 5.1 and only change the values of r = 0.02, µ = 0.07, σ = 0.15, γ = 0.1 and α = 0.7 to satisfy Condition (4.6) also for larger values of F .From Figure 5.5, we find that the relationship between x and F is not monotonous.When the risk aversion level is low, a positive relationship between F and the bankruptcy threshold of wealth is observed.This is because a larger F value, which is treated as the collateral recalling that X(t) ≥ F + η before bankruptcy, will reduce the agent's available capital and limit her investment behaviour.Therefore, the agent will set a higher bankruptcy wealth threshold to get rid of the limitation of the liquidity constraint.Whereas for the agent with deep risk aversion, liquidity constraints are less restrictive, and the parameter F plays more as the role of the bankruptcy cost.

Influence of the bankruptcy option:
To study the influence of introducing the bankruptcy option, we also solve a pure optimal control problem without optimal stopping, see Appendix D. In Figure 5.6, we take the inputs baseline in Table 5.1 and show the optimal controls (as a function of the initial wealth) for the case with and without bankruptcy option models to reveal its influence.In the second case, we consider both the case with a liquidity constraint X(t) > F + η and without liquidity constraint.From Figure 5.6, five phenomena should be noticed by comparing the two curves with liquidity constraint.First, due to the additional bankruptcy option, the value function is always greater than the value function without such an option at any given initial wealth level.Second, before the occurring of optimal stopping time, the additional option offers the agent a better circumstance, and the optimal consumption-portfolio-leisure policies always dominates the corresponding policies without bankruptcy option model.Third, in order to meet the needs of obtaining utility from consumption and leisure, the agent with the bankruptcy option and a low initial wealth immediately file bankruptcy, facing a shrinkage of wealth, which causes a downward jump in consumption and allocation in the risky asset.Fourth, when a liquidity constraint is considered, the optimal leisure rate decreases for low values of the initial wealth x, since the agent needs to spend more time working to get a larger wage, therefore not exploiting the full leisure, to face debt and liquidity constraint.Finally, the amount of money allocated in the risky asset is 0 as the wealth level drops to the liquidity constraint boundary (F + η = 0.9601).This is to avoid that the wealth process violates the liquidity constraint due to the risky asset's fluctuation.However, the optimal consumption always keeps positive even as the wealth approaches the liquidity boundary since the agent continues to obtain the labour income.Additionally, we can observe that the optimal solutions without the liquidity constraint always dominates the solutions with this extra constraint.

Influence of the labour income:
In order to investigate the influence of introducing the leisure rate as an additional control variable and thus the labour income, we first of all compare the results with the optimal bankruptcy model in Jeanblanc et al. (2004), therefore with full leisure and no labour income, to conduct the numerical analysis to discover the sensitivity with respect to the presence of leisure rate and labour income.Figure 5.7 shows that there exists a downward jumps of optimal control strategies for both full and selectable leisure models due to the shrinkage of wealth at the moment of declaring bankruptcy.Moreover, it can be observed that the optimal consumption and portfolio policies of the optimization problem introducing the leisure as a control variable always keep dominating the corresponding policies of the model with full leisure rate since the agent can earn the additional income from labour.Meanwhile, comparing the wealth levels corresponding to the downward jumps, we can observe that the agent with the full leisure rate tends to have a higher bankruptcy wealth threshold such that more wealth can be taken into the post-bankruptcy period to support the further consumption (x = 10.0651 for the "with labour income case", x = 14.6091 for the "without labour income -full leisure-case").Secondly, in Figure 5.8 we discover the sensitivity of bankruptcy wealth threshold with respect to different values of L within the optimization model considering leisure selection.Figure 5.8 shows that the bankruptcy wealth threshold is not a monotonic function of L; the relationship between these two variables works in a complex way and can be analysed briefly into two separated pieces: one piece is with a relatively low value of L, and another is with higher L value.Since L represents the upper boundary of the leisure variable l(t), ( L − L) represents the minimum mandatory working rate.In the first part, the low enough value of L obliges the agent to allocate most of the time on work, thereby gaining enough labour income to afford the debt repayment.Hence, she prefers to set a smaller wealth threshold for bankruptcy to avoid encountering the wealth shrinkage.However, for the second part, the restriction on the optimal leisure choice caused by L becomes weaker as its value increases, the agent gains the utility from taking more leisure, which leads to a decreasing labour income, a decreasing consumption since leisure and consumption are substitute, and a corresponding lower bankruptcy wealth threshold.Finally, we conduct the sensitivity analysis to the wage rate and present the result in Figure 5.9, in which an inverse relationship occurs between x and w.Because the economic of the agent becomes worse with a lower wage rate w, she tends to set a higher critical wealth level such that more wealth is retained for supporting the post-bankruptcy life.

Conclusion
In this work the optimal consumption-portfolio-leisure and bankruptcy problem concerning a power utility function has been solved semi-analytically.By the Legendre-Fenchel transform, we have established the duality between the optimization problem with the individual's shadow price problem, which results in a system of variational inequalities then enables us to obtain the closedform solutions.The optimal wealth and control strategies are represented as functions of wealth's dual process, Z(t).Then we have proved that the optimal policy for the agent is to file bankruptcy at the first hitting time of the optimal wealth process to a critical wealth level, which is the boundary separating the continuous and stopping regions of the corresponding stopping time model.We have also conducted the sensitivity analysis of this wealth threshold to critical parameters.The bankruptcy wealth threshold is the increasing function of both d, which can be treated as the benefit of declaring bankruptcy, and α, with 1−α representing the flexible cost of bankruptcy.Whereas, the non-monotonic relationship between the bankruptcy wealth threshold and F is because F performs a trade-off between the liquidity constraint boundary in the pre-bankruptcy period and the fixed cost of bankruptcy.Regarding the effect of labour income, we show that the bankruptcy wealth threshold is a concave function of the upper bound of the leisure rate, L, that is, it first increases and then decreases: a high value for L permits the agent to have large utility from leisure, while a low value of L "forces" the agent to get a large wage, even if low utility from leisure.Moreover, the bankruptcy wealth threshold is strictly decreasing for the wage rate since a worse economic situation requires more wealth to support the post-bankruptcy period.

A Appendix of Section 2
A.1 Proof of Lemma 2.1 In order to get rid of the constraint 0 ≤ l ≤ L, we begin with the maximization of {u(c, l) − (c + wl)y} with l ∈ R. From the first-order derivative conditions with respect to c and l, we obtain the following equations (A.1) The above system entails the optimal consumption and leisure policies as Then the remaining constraint of the Legendre-Fenchel transform of Equation (2.4) is l ≤ L. Since the optimal leisure plan l also satisfies the constraint l ≤ L under the condition y ≥ ỹ.Conversely, this constraint comes into force to make the optimal leisure to be L for the interval y < ỹ.Thereafter, we can summarize as follows, The first equation in (A.1) implies the relationship between the optimal consumption and leisure, Finally, we can deduce the Legendre-Fenchel transform ũ(y) directly by substituting ĉ and l into Equation (2.4).

A.2 Proof of Proposition 2.1
Referring to (Karatzas and Shreve, 1998b, Section 3.3, Remark 3.3), we first apply the Itô's formula to ξ(t)X(t), ∀t ∈ [0, τ ], in which B(t) is the Brownian motion under the P measure mentioned in Equation (2.2).Taking the integral on both sides of the above equation from 0 to τ , we obtain The left-hand side can be rewritten as from the definition of admissible control set ensures that the left-hand side is bounded below by the constant d−w L r , so the Itô integral on the right-hand side is proved to be a P-supermartingale by means of Fatou's Lemma.Then, taking the expectation on both sides under the P measure, we have which endows us with the desired budget constraint through converting the measure to P,

B Appendix of Section 3 B.1 Proof of Theorem 3.1
Before proving Theorem 3.1, we insert a lemma which helps us to prove the theorem.
Lemma B.1.For any given initial wealth x ≥ 0, and any given and progressively measurable consumption and leisure processes, c(t), l(t), satisfying sup with T standing for the set of F t -stopping times, there exists a portfolio process π(t) making X x,c,π,l (t) ≥ 0, ∀t ≥ 0, holds almost surely.
Then we move to the proof of Duality Theorem 3.1.Following (He and Pages, 1993, Section 4, Theorem 1), the proof mainly contains two aspects: the first part is to show the admissibility of c * (t) and l * (t), and the second part is to claim that they are the optimal consumption-leisure strategy to Problem (P P B ).
(1) We begin with verifying that any consumption-leisure strategy satisfying c * (t) + wl * (t) = −ũ ′ (λ * e γt D * P B (t)H(t)) is admissible.Taking any stopping time τ from T , which is the set of F t -stopping times, we can define a process , where ǫ a positive constant.It is evident that D ǫ (t) is a non-negative, non-increasing, and progressively measurable process, that is, D ǫ (t) ∈ D. Let us define a function is the optimal solution of Problem (S P B ), and x ≥ 0, we get The above inequalities give us lim sup ǫ↓0 The decreasing property of ũ( Since τ can be any F -stopping time in the set T , there exists a portfolio strategy π * (t) that makes the corresponding wealth process satisfying X x,c * ,π * ,l * (t) ≥ 0, ∀t ≥ 0 based on the result from Lemma B.1.
(2) In this part, we claim that c * (t) and l * (t) are the optimal consumption and leisure to Problem (P P B ) under the liquidity constraint.Taking an arbitrary consumption strategy {c(t), π(t), l(t)} ∈ A P B (x), the proof of Lemma B.1 guarantees that there exists a process ζ(t) Since X x,c,π,l (t) ≥ 0 a.s., we obtain the following inequality with any process D(t) ∈ D, where T is any time satisfying T ≥ t.Because D(t) is bounded variation, we can integrate by parts and get Taking the expectation under the P measure on both sides and replacing Equation (B.1), we obtain then, by Lebesgue's Monotone Convergence Theorem, we have The above inequality keeps true for any admissible control strategy c(t), π(t), l(t) and any nonnegative, non-increasing process D(t); furthermore, we will show that the inequality changes into equality with the given c * (t), l * (t) and D Applying the Fatou's lemma, we obtain separately We denote the optimal consumption solution of the above problem is c * (t).From the Lagrange method, we know that c * (t) + w l * (t) = −ũ ′ ( λe γt D * P B (t)H(t)), where λ > 0 is the Lagrange multiplier.The constraint of problem (P ′ P B ) takes equality when λ = λ * .Then, the condition c * (t) + wl * (t) = −ũ ′ (λ * e γt D * P B (t)H(t)) implies that c * (t) and l * (t) are the optimal control policies of Problem (P ′ P B ).Moreover, since the maximum utility of primal problem (P P B ) is upper bounded by the maximum utility of (P ′ P B ), we can conclude that c * (t) and l * (t) are also the optimal consumption and leisure solutions of Problem (P P B ).
Case 1. 0 < ỹ ≤ ẑPB : Condition (V 3) in (3.4) results in a differential equation, which has the solution For avoiding the explosion of term z n1 when z goes to 0, we set B 11,P B = 0. Then four parameters are left to be determined, which are B 21,P B , B 12,P B , B 22,P B and ẑPB .To accomplish this task, we use the smooth conditions at z = ỹ and z = ẑPB to construct a four-equation system: Case 2. 0 < ẑPB < ỹ: The same argument with the previous case, we first handle Condition (V 3) in (3.4).Recalling Lemma 2.1, the corresponding interval 0 < z < ẑPB restricts the function ũ(z) only takes the form ũ(z) = A 1 z δ(1−k) δ(1−k)−1 − wLz, which is identical with the piece of 0 < z < ỹ in Case 1.Hence, the differential equation from (V 3) has the same solution, only changing the parameters' notations from B 11,P B to B 1,P B , and B 21,P B to B 2,P B , that is, We set the coefficient B 1,P B = 0 for the reason that the term z n1 goes to ∞ as z approaches 0, which violates the boundedness assumption of v P B (z). Subsequently, using the smooth condition at z = ẑPB , a two-equation system is established to determine the exact values of B 2,P B and ẑPB : Let us first introduce a process M τ (t) H(t)X(t)+ t 0 c(s)+d+wl(s)−w L H(s)ds, ∀t ∈ [0, τ ].Following (Karatzas and Shreve, 1998b, Section 3.9, Theorem 9.4), we firstly claim that M τ (t) is a P-martingale.Let c(t) and l(t) be the consumption and leisure processes such that For any fixed stopping time τ ∈ T , we define ζ(τ (Karatzas and Shreve, 1998b, Section 3.3, Theorem 3.5) implies that there exists a portfolio process C.2 Proof of Theorem 4.1 The proof here is consistent with Appendix B.1, but some modification is needed due to the stop-time embedding.We first introduce a lemma, which will be used in the proof of the duality theorem.
Lemma C.1.For any given initial wealth x ≥ F +η, any F t -stopping time τ with P(τ < ∞) = 1, any F τ -measurable random variable Q with P(Q ≥ F +η) = 1 under the P measure, and any given progressively measurable consumption and leisure processes c(t) ≥ 0, 0 ≤ l(t) ≤ L, satisfying sup where S stands for the set of F t -stopping times before the fixed stopping time τ , and Proof.Following the similar argument with (He and Pages, 1993, Appendix, Lemma 1), we first define a new process From the properties of processes c(t), l(t) and H(t), it can be observed that E[K(t)] < ∞, which implies {K(τ )} τ ∈S is uniformly integrable.Therefore, there exists a Snell envelope of K(t) denoted as K(t).It is a super-martingale under the P measure and satisfies By the Doob-Meyer Decomposition Theorem from (Karatzas and Shreve, 1998a, Section 1.4, Theorem 4.10), the super-martingale K(t) can be decomposed into where M (t) is a uniformly integrable martingale under the P measure with the initial value M (0) = 0, Ā(t) is a strictly increasing process with the initial value Ā(0) = 0.According to the Martingale Representation Theorem from (Björk, 2009, Section 11.1, Theorem 11.2), M (t) can be expressed as with an F-adapted process ρ(t) satisfying It can be verified that , and the martingale property of M (t).Further, because of K(0) = sup by constructing a contradiction.Let us assume that τ * attains the supremum within the expression of K(0), i.e., K(0) = K(τ * ), and introduce a stopping time as τ inf{0 ≤ t ≤ τ : K(t) > K(τ * )}.Since τ ∈ S, we have K(τ ) ≤ K(τ * ), which is contradictory to the definition of τ .Then, based on the fact that X we can conclude that X(t) ≥ 0, a.s.. Additionally, the process X(t) is re-expressed in terms of the martingale M (t) as As for the wealth process by implementing the Itô's formula to H(t)X x,c,π,l (t) and adopting the portfolio strategy as it is rewritten as d(H(t)X x,c.π,l (t)) = ρ(t)dB(t) − (c(t) + wl(t) − w L + d)H(t)dt.Taking the integral from t to τ , and then the conditional expectation w.r.t.F t on both sides of the above equation, we obtain With the aid of the above lemma, we can complete the statement and proof of Duality Theorem 4.1.Referring to (He and Pages, 1993, Section 4, Theorem 1), the proof procedure is divided into two aspects: the first part is focused on the admissibility of c * (t) and l * (t), and the second part is revolved around claiming that c * (t) and l * (t) are the optimal consumption-leisure strategy to the primal optimization problem.
(2) Then we turn to claim that c * (t) and l * (t) are the optimal consumption and leisure to the problem (P τ ).Taking an arbitrary control strategy {c(t), π(t), l(t)} ∈ A τ (x), the proof of Lemma C.1 guarantees that there exists a process ζ(t) satisfying Taking the expectation under the P measure on both sides and replacing Equation (C.5), we obtain The above inequality keeps true for any admissible control strategy {c(t), π(t), l(t)} and any nonnegative, non-increasing process D(t).Furthermore, we will show that the inequality changes into equality with the given c * (t), l * (t) and D * (t).We first define a new process where ǫ is a small enough constant.Following the same argument in the first part proof, we have L( Dǫ (t)) ≥ L(D * (t)), and lim sup Applying the Fatou's lemma, we obtain separately subject to We denote the optimal solutions of the above problem as c * (t) and l * (t).The Lagrange method endows us where λ > 0 is the Lagrange multiplier.The constraint of Problem (P ′ τ ) takes equality when λ = λ * .Then, the condition c * (t) + wl * (t) = −ũ ′ (λ * e γt D * (t)H(t)) implies that c * (t) and l * (t) are the optimal control policies of the problem (P ′ τ ).Moreover, since the maximum utility of primal problem (P τ ) is upper bounded by the maximum utility of (P ′ τ ), we can conclude that c * (t) and l * (t) are also the optimal consumption and leisure solution to the problem (P τ ).

Defining an operator
, the continuous region of the corresponding optimal stopping time problem is expressed as z, the continuous region can be rewritten as Ω 1 = {z : h(z) > 0}.Because Ũ (z) takes the piecewise form of the last equality comes from Condition (B.2) on the interval 0 < z < αẑ P B .From Condition (4.5), it can be observed that the function h(z) only takes the form on the considering interval 0 < z < ẑ.We first extend the domain of h(z) to the whole positive real line, and discover the existence and uniqueness of its zero z, then discuss the magnitude between z and ẑ.The decreasing property of ũ(z) leads to ũ(z) − ũ z α > 0; hence, a necessary condition to ensure that there is at least one zero point is put forward as rF − d + w L − w L α < 0. Afterwards, for the sake of determining the curvature of h(z), we take the second-order derivative and obtain therefore we focus on the sign of function ũ′′ (z) − 1 α 2 ũ′′ z α .Since ũ(z) is strictly decreasing and convex on (0, ∞) and adopts the piecewise form, we discover the sign of ũ′′ (z) − 1 α 2 ũ′′ z α on three different intervals: z < αỹ, αỹ ≤ z < ỹ and z ≥ ỹ.
The first term inside the square bracket has the following inequality relationship , as for the second term, we have .
Then we can determine the sign of ũ′′ (z) − 1 Hence, we can summarize that h ′′ (z) < 0, which means the function h(z) is strictly concave for z > 0. Additionally, combining with the subsequent facts we can conclude that there exists a unique z such that {z > 0 : h(z) > 0} = {0 < z < z}.Moreover, solutions of the primal optimization problem are discussed in two different cases, z < ẑ and z ≥ ẑ.
C.4 Proof of Lemma 4.2 Referring to (Jeanblanc et al., 2004, Appendix, A.2), we make use of the scale function to calculate the probability of the stopping time.Related to the diffusion process of Z(t), the scale function is where f (x) = (γ − r)x, g(x) = −θx are the drift and diffusion coefficients of dZ(t) and a is an arbitrary constant located in (0, z).Then we obtain b a e −2 y a γ−r θ 2 x dx dy.It follows that in order to prove that the above probability is one instead of depending on the initial value z, it suffices to show that s(0) = −∞.Since < 0, which is equal to γ > r + θ 2 2 .Moreover, following the same argument, it can be easily proven that P (τ 0 < ∞) = 1 with the condition (4.7), which also gives us P (τ z < ∞) = 1.
C.5 Solutions of Variational Inequalities (4.8) and (4.9) Case 1. 0 < z < ẑ < αỹ ≤ αẑ P B : The differential equation generated from Condition (V 3) in the system (4.8)takes the solution as Since n 1 < 0, the term z n1 will suffer the explosion as z goes to 0. Therefore, we set the coefficient B 1 = 0 to meet the boundedness assumption of v(z).Forward, the smooth fit condition v(z) = Ũ (z) leads to the subsequent two-equation system to resolve the parameters B 2 and z.
Case 2. 0 < z < αỹ ≤ ẑ ≤ αẑ P B : The Condition (V 3) from (4.8) leads to the following differential equation since the condition z < ỹ remains unchanged, the above equation keeps the same compared with the previous sections, hence, takes the identical solution as Furthermore, because the condition z < αỹ also coincides with Case 1, we have the same smooth fit condition v(z) = Ũ (z), which results in the identical values of parameters B 2 and z.The difference from the previous case occurs at the boundary ẑ.Because of αỹ ≤ ẑ ≤ αẑ P B , the function Ũ (z) adopts a different form at the point ẑ; furthermore, (V 1) and (V 6) in the system (4.8)produces a different equation, that is, to acquire the value of ẑ.
Case 3. 0 < αỹ < z < ẑ ≤ αẑ P B & z < ỹ: Under the same condition z < ỹ with the previous cases, the differential equation generating from (V 3) of (4.8) follows the identical solution, Then the smooth fit condition v(z) = Ũ (z) enables us to determine the values of parameters B 2 and z with the following two-equation system: Besides, from Condition (V 1) and (V 6) of (4.8), we have the smooth fit condition Ũ ′ (ẑ) = −(F +η).
Combining with the prerequisite αỹ ≤ ẑ ≤ αẑ P B , we have the subsequent equation, which gives us the exact value of parameter ẑ.
Case 4. 0 < z < ẑ ≤ αẑ P B < αỹ < ỹ: Same with the previous cases, Condition (V 3) of (4.8) forces the function v(z) to take the solution as Then, considering the smooth fit condition v(z) = Ũ (z), a two-equation system is established to resolve the parameters B 2 and z: Combining the condition Ũ ′ (ẑ) = −(F + η) with the prerequisite ẑ ≤ ẑPB , we get the following equation which enables us to obtain the exact value of ẑ.
Since z > ỹ, Lemma 2.1 shows that the function ũ(z) takes two different forms on the corresponding interval with ỹ as the separating threshold.Hence, the solution of the above differential equation inherits this piecewise property and has the following resolution, To meet the boundedness assumption of v(z), we set B 11 = 0 to avoid the explosion of term z n1 as z goes to 0. Then, the same argument with the previous cases, we need to construct a four-equation system, which resorts to the smooth conditions at the point ỹ and z, to determine the boundary z, and the coefficient of function v(z), i.e., B 21 , B 12 , B 22 : Meanwhile, considering the prerequisite αỹ < ẑ < αẑ P B and the smooth fit condition Ũ ′ (ẑ) = −(F+η), we have which gives us the value of ẑ.
Case 6. 0 < ỹ ≤ ẑ & z ≥ ẑ: The Condition (V 3) of (4.9) endows us a partial differential equation with the solution Due to the same considerations as before, we set B 11 = 0 to meet the boundedness of v(z).Then, with the smooth fit conditions at ẑ and ỹ, we can construct a four-equation system to determine the values of parameters B 21 , B 12 , B 22 and ẑ: • C 0 condition at z = ỹ Case 7. 0 < ẑ < ỹ & z ≥ ẑ: The prerequisite ẑ < ỹ makes the function ũ(z) of the form δ(1−k)−1 − wLz.Then Condition (V 3) in (4.9) has the solution We set B 1 = 0 for avoiding the explosion of the term B 1 z n1 when z goes to 0. Then a two-equation system is created to solve the value of parameters B 2 and ẑ explicitly:

D No Optimal Bankruptcy Problem
To study the influence of introducing the bankruptcy option, we also solve a pure optimal control problem without optimal stopping, which is defined subsequently as The subscript nob indicates that the considering variables and functions are concerned with the no bankruptcy option model.Moreover, the admissible control set A nob (x) is almost consistent with the definition of A P B (x) except the liquidity condition.A nob (x) adopts "X nob (t) ≥ F + η, a.s., ∀t ≥ 0" instead of "X(t) ≥ 0, a.s., ∀t ≥ 0".Then we provide the budget and liquidity constraints as: Following the same argument with the post-bankruptcy part, we can solve the optimal control problem with two different cases: • Case 1. 0 < ỹ ≤ ẑnob : the optimal consumption-portfolio-leisure strategy is and the optimal wealth process is • Case 2. 0 < ẑnob < ỹ: the optimal consumption-portfolio-leisure strategy defined on the interval 0 < Z * nob (t) ≤ ẑnob is , and the optimal wealth process is

Figure 5 . 1 :
Figure 5.1: Trajectories of Optimal Wealth and Control Strategies w.r.t.Risk Aversion Coefficient

Figure 5
Figure 5.2: Market Risk Premium, Risk Aversion Coefficient and Bankruptcy Threshold

Figure 5
Figure 5.5: Risk Aversion Coefficient, Fixed Bankruptcy Cost and Bankruptcy Wealth Threshold

Figure
Figure 5.6: Influence of Bankruptcy Option

FigureFigure 5
Figure 5.7: Influence of Labour Income
Assumption 3.1.The non-increasing process D P B (t) is absolutely continuous with respect to t.Hence, there exists a process ψ P B (t) ≥ 0 such that dD P B (t) = −ψ P B (t)D P B (t)dt.