Game-Theoretical Model of Debt Management Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen September 3, 2014 Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
Control System Consider the control system of the borrower x = I ( x ) x − u ( t ), ˙ x (0) = ¯ x , where ◮ x : total debt measured as a fraction of the yearly income of the borrower. ◮ I ( x ): interest rate payed on the debt at a given time. ◮ u ( t ) ∈ [0 , 1]: is the control variable for the borrower which represents the payment rate as a fraction of the income. Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
Optimal control problem The total cost � ∞ 0 e − rt L ( u ( t )) dt , J ( u , ¯ x ) = where ◮ r is the discounted rate. ◮ L ( u ) is the cost to the borrower for implementing the control u. ′′ > 0 , L ′ > 0 , L (0) = 0 , u → 1 − L ( u ) = + ∞ lim L Our goal is to minimize the total cost, i.e., min u ( . ) J ( u , ¯ x ) Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
Hamliton Jacobi Equation The value function V (¯ x ) = min u ( . ) J ( u , ¯ x ) By using the principle of dynamic programming, V is a solution of the Hamilton-Jacobi equation rV ( x ) = H ( x , V ′ ( x )) , with H ( x , ξ ) = { L ( ω ) − ξω + ξ I ( x ) x } . The optimal control u ∗ ( x ) = arg min ω ∈ [0 , 1] { L ( ω ) − V ′ ( x ) ω } Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
Difference between an explicit and an implicit ODE Figure : Difference between an explicit and an implicit ODE Figure 2 Right plot show a standard ODE and the left plot shows the ODE in our case. Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
Differential Inclusion rV ( x ) = H ( x , V ′ ( x )) Differential inclusion � � F − ( x , V ( x )) , F + ( x , V ( x )) V ′ ( x ) ∈ . ◮ If V ′ = F − ( x , V ), the optimal control u ∗ ( x ) < I ( x ) x , so x > 0 and the debt increases. ˙ ◮ If V ′ = F + ( x , V ), the optimal control u ∗ ( x ) > I ( x ) x , so x < 0 and the debt decreases. ˙ Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
Initial Data Figure : Choosing initial data We should the point x 0 such that: V ′ ( x 0 ) = W ′ ( x 0 ) V ( x 0 ) = W ( x 0 ) and Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
Example 1 If ¯ x > 5, then ˙ x < 0, lim t →∞ x ( t ) = 5 . If 0 < ¯ x < 5 then ˙ x > 0 lim t →∞ x ( t ) = 5 Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
The Bankruptcy Cost model Let ρ : [0 , X B ) → [0 , ∞ ) be instantaneous risk of bankruptcy which depends on the total debt such that ρ ( x ) is not decreasing and ρ (0) = 0 , x → x B − ρ ( x ) = lim + ∞ . x B = max amount of debt the borrow will be able to pay back. The total cost function is now � t � ∞ J ( u , ¯ x ) = exp {− rt − ρ ( x ( s )) ds } ( ρ ( x ( t )) B + L ( u ( t ))) d t 0 0 where: B =bankruptcy cost to the borrower. Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
With Similar arguments and by using the principle of dynamic programming, The equations are modified to ( r + ρ ( x )) V ( x ) = H ( x , V ′ ( x )) , with H ( x , ξ ) = { L ( ω ) − ξω + ξ I ( x ) x + ρ ( x ) B } . The optimal control u ∗ ( x ) = arg min ω ∈ [0 , 1] { L ( ω ) − V ′ ( x ) ω } Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
Differential Inclusion The implicit ODE leads to a differential inclusion: � � F − ( x , V ( x )) , F + ( x , V ( x )) V ′ ( x ) ∈ . ◮ If we choose V ′ = F − ( x , V ) ≤ ξ # ( x ), the optimal control u ∗ ( x ) < I ( x ) x , so ˙ x > 0 and the debt increases. ◮ If we choose V ′ = F + ( x , V ) ≥ ξ # ( x ), the optimal control u ∗ ( x ) > I ( x ) x , so ˙ x < 0 and the debt decrease. Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
Example 2 Solution=Lower envelope of blue and green curves. If ¯ x < 9 . 5, then ˙ x < 0, lim t →∞ x ( t ) = 0. If ¯ x > 9 . 5, then ˙ x > 0, lim t →∞ x ( t ) = ∞ , and bankruptcy occurs. Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
The Game Theoretical Model A pool of risk neutral lenders who charge an interest α ( x ) depending on the bankruptcy risk ρ ( x ). H ( x , α ( x ) , V ′ ( x )) ODE: ( r + ρ ( x )) V ( x ) = Hamiltonian: H ( x , ξ ) = { L ( ω ) − ξω + ξα ( x ) x + ρ ( x ) B } [ α ( x ) + λ ][ α ( x ) − r − ρ ( x )] α ′ ( x ) = α ( x ) x − u ( x ) u ∗ ( x ) ω ∈ [0 , 1] { L ( ω ) − V ′ ( x ) ω } Optimal control: = arg min Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
Example 3 If ¯ x < 7 . 5, then ˙ x < 0, lim t →∞ x ( t ) = 0. If ¯ x > 7 . 5, then bankruptcy occurs. Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Game-Theoretical Model of Debt Management Postdoc: Khai Nguyen
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