Future Expectations Modeling, Random Coefficient Forward–Backward Stochastic Differential Equations, and Stochastic Viscosity Solutions

References

• [1] Agram N, Øksendal B (2014) Infinite horizon optimal control of forward-backward stochastic differential equations with delay. J. Comput. Appl. Math. 259(March):336–349.Crossref, Google Scholar
• [2] Blanchard O (1981) Output, the stock market, and interest rates. Amer. Econom. Rev. 71(1):132–143.Google Scholar
• [3] Buckdahn R, Ma J (2001) Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I. Stochastic Processes Appl. 93(2):181–204.Crossref, Google Scholar
• [4] Buckdahn R, Ma J (2001) Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part II. Stochastic Processes Appl. 93(2):205–228.Crossref, Google Scholar
• [5] Buckdahn R, Ma J (2007) Pathwise stochastic control problems and stochastic HJB equations. SIAM J. Control Optim. 45(6):2224–2256.Crossref, Google Scholar
• [6] Cadenillas A, Zapatero F (1999) Optimal central bank intervention in the foreign exchange market. J. Econom. Theory 87(1):218–242.Crossref, Google Scholar
• [7] Cheridito P, Soner HM, Touzi N, Victoir N (2007) Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Commun. Pure Appl. Math. 60(7):1081–1110.Crossref, Google Scholar
• [8] Cont R, Fournie DA (2010) Change of variable formulas for non-anticipative functionals on path space. J. Functional Anal. 259(4):1043–1072.Crossref, Google Scholar
• [9] Cont R, Fournie DA (2013) Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41(1):109–133.Crossref, Google Scholar
• [10] Crandall MG, Ishii H, Lions PL (1992) Users guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27(1):1–68.Crossref, Google Scholar
• [11] Crandall MG, Lions PL (1983) Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277(1):1–42.Crossref, Google Scholar
• [12] Cvitanić J, Ma J (1996) Hedging options for a large investor and forward-backward SDE’s. Ann. Appl. Probab. 6(2):370–398.Crossref, Google Scholar
• [13] Dornbusch R (1976) Expectations and exchange rate dynamics. J. Political Econom. 84(6):1161–1176.Crossref, Google Scholar
• [14] Duffie D, Ma J, Yong J (1995) Black’s console rate conjecture. Ann. Appl. Probab. 5(2):356–382.Crossref, Google Scholar
• [15] Dupire B (2009) Functional Itô calculus. Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS, Bloomberg L.P., New York.Google Scholar
• [16] Ekren I, Keller C, Touzi N, Zhang J (2014) On viscosity solutions of path dependent PDEs. Ann. Probab. 42(1):204–236.Crossref, Google Scholar
• [17] Ekren I, Touzi N, Zhang J (2016) Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. Ann. Probab. 44(2):1212–1253.Crossref, Google Scholar
• [18] Ekren I, Touzi N, Zhang J (2016) Vimscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. Ann. Probab. 44(4):2507–2553.Google Scholar
• [19] Englezos N, Frangos NE, Kartala XI, Yannacopoulos AN (2013) Stochastic Burgers equation and a generalization of the Cole-Hopf transformation. Stochastic Processes Appl. 123(8):3239–3272.Crossref, Google Scholar
• [20] Fleming WH, Soner HM (2006) Controlled Markov Processes and Viscosity Solutions, vol. 25 (Springer, Berlin).Google Scholar
• [21] Haadem S, Øksendal B, Proske F (2013) Maximum principle for jump diffusion processes with infinite horizon. Automatica 49(7):2267–2275.Crossref, Google Scholar
• [22] Jeanblanc-Picqué M (1993) Impulse control method and exchange rate. Math. Finance 3(2):161–177.Crossref, Google Scholar
• [23] Krugman PR (1991) Target zones and exchange rate dynamics. Quart. J. Econom. 106(3):669–682.Crossref, Google Scholar
• [24] Kunita H (1990) Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Math, vol. 24 (Cambridge University Press, Cambridge, UK).Google Scholar
• [25] Lions PL, Souganidis PE (1998) Fully nonlinear stochastic partial differential equations. Comptes Rendus Acad. Sci. I Math. 326(9):1085–1092.Google Scholar
• [26] Lions PL, Souganidis PE (1998) Fully nonlinear stochastic partial differential equations: Non-smooth equations and applications. Comptes Rendus Acad. Sci. I Math. 327(8):735–741.Google Scholar
• [27] Ma J, Protter P, Yong J (1994) Solving forward-backward stochastic differential equations explicitly - a four step scheme. Probab. Theory Related Fields 98(3):339–359.Crossref, Google Scholar
• [28] Ma J, Yong J (1997) Adapted solution of a degenerate backward SPDE, with applications. Stochastic Processes Appl. 70(1):59–84.Crossref, Google Scholar
• [29] Ma J, Yong J (1998) Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics, vol. 1702 (Springer, New York).Google Scholar
• [30] Maslowski B, Veverka P (2014) Suffficient stochastic maximum principle for discounted control problem. Appl. Math. Opt. 70(2):225–252.Crossref, Google Scholar
• [31] Miller M, Weller P (1995) Stochastic saddlepoint systems: Stabilization policy and the stock market. J. Econom. Dyn. Control 19(1–2):279–302.Crossref, Google Scholar
• [32] Mundaca G, Øksendal B (1998). Optimal stochastic intervention control with application to the exchange rate. J. Math. Econom. 29(2):225–243.Crossref, Google Scholar
• [33] Neely CJ, Weller P, Corbae D (1995) Endogenous realignments and the sustainability of a target zone. CEPR Discussion Paper No. 1253, Centre for Economic Policy Research, London.Google Scholar
• [34] Pardoux E (1999) BSDEs’ weak convergence and homogenizations of semilinear PDEs. Clarke FH, Stern RJ, eds. Nonlinear Analysis Differential Equations and Control, NATO Science Series, vol. 528 (Kluwer Academic, Dordrecht, Netherlands), 503–549.Crossref, Google Scholar
• [35] Pardoux E (1998) Backward stochastic differential equations and viscosity solutions of semilinear parabolic and elliptic PDEs of second order. Decreusefond L, Øksendal B, Gjerde J, Üstünel AS, eds. Stochastic Analysis and Related Topics VI, Progress in Probability, vol. 42 (Birkhäuser, Boston), 79–127.Crossref, Google Scholar
• [36] Pardoux E, Pradeilles F, Rao Z (1997) Probabilistic interpretation of a system of semilinear parabolic PDEs. Ann. Inst. H. Poincare B Probab. Statist. 33(4):467–490.Crossref, Google Scholar
• [37] Pardoux E, Tang S (1999) Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Related Fields 114(2):123–150.Crossref, Google Scholar
• [38] Peng S (1991) Probabilistic interpretation for systems of semilinear parabolic PDEs. Stochastics Stochastic Rep. 37(1–2):61–74.Crossref, Google Scholar
• [39] Peng S (1992) A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics Stochastic Rep. 38(2):119–134.Crossref, Google Scholar
• [40] Peng S (2011) Note on viscosity solution of path-dependent PDE and G-martingales. Working paper, Shandong University, Shandong, China.Google Scholar
• [41] Peng S, Shi Y (2000) Infinite horizon forward-backward stochastic differential equations. Stochastic Processes Appl. 85(1):75–92.Crossref, Google Scholar
• [42] Peng S, Wu Z (1999) Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim. 37(3):825–843.Crossref, Google Scholar
• [43] Shi JT, Wu Z (2006) The maximum principle for fully coupled forward-backward stochastic control system. Acta Automatica Sinica 32(2):161–169.Google Scholar
• [44] Soner HM, Touzi N, Zhang J (2012) Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153(1–2):149–190.Crossref, Google Scholar
• [45] Tang S, Zang F (2013) Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete Continuous Dynamical Systems, A 35(11):5521–5553.Google Scholar
• [46] Wu Z (1999) The comparison theorem of FBSDE. Statist. Probab. Lett. 44(1):1–6.Crossref, Google Scholar
• [47] Wu Z (1998) Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. J. Systems Sci. Complexity 11(3):249–259.Google Scholar
• [48] Wu Z, Xu M (2009) Comparison theorems for forward backward SDEs. Statist. Probab. Lett. 79(4):426–435.Crossref, Google Scholar
• [49] Yannacopoulos AN (2005) A novel approach to exchange rate control using controlled backward stochastic differential equations. Ekonomia 8(1):74–91.Google Scholar
• [50] Yannacopoulos AN (2008) Rational expectation models: An approach using forward-backward stochastic differential equations. J. Math. Econom. 44(3–4):251–276.Crossref, Google Scholar
• [51] Yin J (2008) On solutions of a class of infinite horizon FBSDEs. Statist. Probab. Lett. 78(15):2412–2419.Crossref, Google Scholar
• [52] Yong J (2006) Linear forward-backward stochastic differential equations with random coefficients. Probab. Theory Related Fields 135(1):53–83.Crossref, Google Scholar
• [53] Yong J (2010) Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim. 48(6):4119–4156.Crossref, Google Scholar
• [54] Zhang L, Shi Y (2010) Comparison theorems of infinite horizon forward-backward stochastic differential equations. Working paper, Beijing University of Posts and Telecommunications, Beijing.Google Scholar