Figure 1.
Proving that $V\leq W$ .
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October
2018, 11(5): 793-824.
doi: 10.3934/dcdss.2018050
The vanishing viscosity limit for a system of H-J equations related to a debt management problem
Department of Mathematics, Penn State University, University Park, PA 16802, USA |
The paper studies a system of Hamilton-Jacobi equations, arising from a model of optimal debt management in infinite time horizon, with exponential discount and a bankruptcy risk. For a stochastic model with positive diffusion, the existence of an equilibrium solution is obtained by a topological argument. Of particular interest is the limit of these viscous solutions, as the diffusion parameter approaches zero. Under suitable assumptions, this (possibly discontinuous) limit can be interpreted as an equilibrium solution to a non-cooperative differential game with deterministic dynamics.
Mathematics Subject Classification:
Primary: 34B15, 35Q91, 49N70, 91A23.
Citation:
Alberto Bressan, Yilun Jiang. The vanishing viscosity limit for a system of H-J equations related to a debt management problem. Discrete & Continuous Dynamical Systems - S,
2018, 11
(5)
: 793-824.
doi: 10.3934/dcdss.2018050
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References:
| [1] |
H. Amann,
Invariant sets and existence theorems for semilinear parabolic equation and elliptic system, J. Math. Anal. Appl., 65 (1978), 432-467.
doi: 10.1016/0022-247X(78)90192-0. |
| [2] |
M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, 1997.
doi: 10.1007/978-0-8176-4755-1. |
| [3] |
G. Barles,
Solutions de Viscosité des Équations de Hamilton-Jacobi, Springer-Verlag, Paris, 1994. |
| [4] |
G. Barles, A. Briani and E. Chasseigne,
A Bellman approach for two-domains optimal control problems in $R^N$, ESAIM Control Optim. Calc. Var., 19 (2013), 710-739.
doi: 10.1051/cocv/2012030. |
| [5] |
G. Barles and E. Chasseigne,
(Almost) everything you always wanted to know about deterministic control problems in stratified domains, Netw. Heterog. Media, 10 (2015), 809-836.
doi: 10.3934/nhm.2015.10.809. |
| [6] |
R. Barnard and P. Wolenski,
Flow invariance on stratified domains, Set-Valued Var. Anal., 21 (2013), 377-403.
doi: 10.1007/s11228-013-0230-y. |
| [7] |
A. Bressan and Y. Hong, Optimal control problems on stratified domains, Netw. Heter. Media, 2 (2007), 313–331 (with Y. Hong). Errata Corrige in Netw. Heter. Media, 8 (2013), 625.
doi: 10.3934/nhm.2007.2.313. |
| [8] |
A. Bressan and Y. Jiang,
Optimal open-loop strategies in a debt management problem, Analysis & Appl., 16 (2018), 133-157.
doi: 10.1142/S0219530517500038. |
| [9] |
A. Bressan, A. Marigonda, K. Nguyen and M. Palladino,
A stochastic model of optimal debt management and bankruptcy, SIAM J. Financial Math., 8 (2017), 841-873.
doi: 10.1137/16M1095019. |
| [10] |
A. Bressan and K. Ngyuen,
A game theoretical model of debt and bankruptcy, ESAIM: Control, Optim. Calc. Variat., 22 (2016), 953-982.
doi: 10.1051/cocv/2016030. |
| [11] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics, Springfield Mo. 2007. |
| [12] |
M. G. Crandall and P. L. Lions,
Viscosity solutions of Hamilton-Jacobi equations, Trans. American Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
| [13] |
C. De Zan and P. Soravia,
Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients, Interfaces Free Bound, 12 (2010), 347-368.
doi: 10.4171/IFB/238. |
| [14] |
M. Garavello and P. Soravia,
Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost, Nonlinear Differential Equations Appl, 11 (2004), 271-298.
doi: 10.1007/s00030-004-1058-9. |
| [15] |
M. Garavello and P. Soravia,
Representation formulas for solutions of the HJI equations with discontinuous coefficients and existence of value in differential games, J. Optim. Theory Appl., 130 (2006), 209-229.
doi: 10.1007/s10957-006-9099-3. |
| [16] |
G. Nuño and C. Thomas, Monetary policy and Sovereign Debt Vulnerability, Working document n. 1517, Banco de España Publications, 2015.Google Scholar |
| [17] |
Z. Rao, A. Camilli and H. Zidani,
Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations, J. Differential Equations, 257 (2014), 3978-4014.
doi: 10.1016/j.jde.2014.07.015. |
| [18] |
S. E. Shreve, Stochastic Calculus for Finance. Ⅱ. Continuous-time Models, Springer-Verlag, New York, 2004. |
show all references
References:
| [1] |
H. Amann,
Invariant sets and existence theorems for semilinear parabolic equation and elliptic system, J. Math. Anal. Appl., 65 (1978), 432-467.
doi: 10.1016/0022-247X(78)90192-0. |
| [2] |
M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, 1997.
doi: 10.1007/978-0-8176-4755-1. |
| [3] |
G. Barles,
Solutions de Viscosité des Équations de Hamilton-Jacobi, Springer-Verlag, Paris, 1994. |
| [4] |
G. Barles, A. Briani and E. Chasseigne,
A Bellman approach for two-domains optimal control problems in $R^N$, ESAIM Control Optim. Calc. Var., 19 (2013), 710-739.
doi: 10.1051/cocv/2012030. |
| [5] |
G. Barles and E. Chasseigne,
(Almost) everything you always wanted to know about deterministic control problems in stratified domains, Netw. Heterog. Media, 10 (2015), 809-836.
doi: 10.3934/nhm.2015.10.809. |
| [6] |
R. Barnard and P. Wolenski,
Flow invariance on stratified domains, Set-Valued Var. Anal., 21 (2013), 377-403.
doi: 10.1007/s11228-013-0230-y. |
| [7] |
A. Bressan and Y. Hong, Optimal control problems on stratified domains, Netw. Heter. Media, 2 (2007), 313–331 (with Y. Hong). Errata Corrige in Netw. Heter. Media, 8 (2013), 625.
doi: 10.3934/nhm.2007.2.313. |
| [8] |
A. Bressan and Y. Jiang,
Optimal open-loop strategies in a debt management problem, Analysis & Appl., 16 (2018), 133-157.
doi: 10.1142/S0219530517500038. |
| [9] |
A. Bressan, A. Marigonda, K. Nguyen and M. Palladino,
A stochastic model of optimal debt management and bankruptcy, SIAM J. Financial Math., 8 (2017), 841-873.
doi: 10.1137/16M1095019. |
| [10] |
A. Bressan and K. Ngyuen,
A game theoretical model of debt and bankruptcy, ESAIM: Control, Optim. Calc. Variat., 22 (2016), 953-982.
doi: 10.1051/cocv/2016030. |
| [11] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics, Springfield Mo. 2007. |
| [12] |
M. G. Crandall and P. L. Lions,
Viscosity solutions of Hamilton-Jacobi equations, Trans. American Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
| [13] |
C. De Zan and P. Soravia,
Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients, Interfaces Free Bound, 12 (2010), 347-368.
doi: 10.4171/IFB/238. |
| [14] |
M. Garavello and P. Soravia,
Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost, Nonlinear Differential Equations Appl, 11 (2004), 271-298.
doi: 10.1007/s00030-004-1058-9. |
| [15] |
M. Garavello and P. Soravia,
Representation formulas for solutions of the HJI equations with discontinuous coefficients and existence of value in differential games, J. Optim. Theory Appl., 130 (2006), 209-229.
doi: 10.1007/s10957-006-9099-3. |
| [16] |
G. Nuño and C. Thomas, Monetary policy and Sovereign Debt Vulnerability, Working document n. 1517, Banco de España Publications, 2015.Google Scholar |
| [17] |
Z. Rao, A. Camilli and H. Zidani,
Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations, J. Differential Equations, 257 (2014), 3978-4014.
doi: 10.1016/j.jde.2014.07.015. |
| [18] |
S. E. Shreve, Stochastic Calculus for Finance. Ⅱ. Continuous-time Models, Springer-Verlag, New York, 2004. |
Figure 2.
A limiting dynamics. Here the coincidence set, where $V = W$ and the dynamics is stationary, is $\Omega = \{ 0, b_1\}$ . Restricted to the two open intervals $I_1 = \,]0, b_1[\,$ and $I_2 = \,]b_1,M[\,$ the dynamics is piecewise smooth. Namely, there exists intermediate points $\bar x_i\in I_i$ such that $\dot x <0$ for $x<\bar x_i$ and $\dot x > 0$ for $x> \bar x_i$ .
Figure 3.
Left: the case where $x<\hat x$ . For $(r+\rho(x))V>H^{max}(x, p(x))$ the equation (4.22) has no solution. At a point where $(r+\rho(x))V<H^{max}(x, p(x))$ , it determines two distinct values $F^-,F^+$ for $\xi$ . Right: the case where $x>\hat x$ . For any $(r+\rho(x))V>\rho(x)B$ , the equation (4.22) determines a unique solution $\xi = F^-$ .
Figure 4.
Proving the convergence $V_n'\to V'$ .
Figure 5.
The functions $\zeta, \zeta_n$ in (4.31)-(4.33) and the smooth approximation $\zeta^\varepsilon $ considered at (4.54). Setting $\zeta^\varepsilon _n = \min\{\zeta^\varepsilon ,\zeta_n\}$ , one has $\|\zeta^\varepsilon -\zeta^\varepsilon _n\|_{{\bf{L}}^1}\to 0$ as $n\to\infty$ .
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