Programme and Reading


Please find the details of the course and the suggested reading.

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The course will run from 13:30 to 17:30 each day (GMT+1)


Monday 21st June

  • Introduction to the course
  • Dynamic programming in continuous time I


Tuesday 22nd June

  • Dynamic programming in continuous time II
  • Dynamic programming in continuous time III


Wednesday 23rd June

  • Introduction to machine learning and deep neural networks in macroeconomics I
  • Introduction to machine learning and deep neural networks in macroeconomics II


Thursday 24th June

  • Solving high-dimensional dynamic programming problems using deep nueral networks
  • Introduction to heterogeneous-agent models in continuous time


Friday 25th June

  • Solving heterogeneous agent models with aggregate shocks using neural networks
  • Optimal policies with heterogeneous agents



1. Dynamic programming in continuous time.

Achdou, Y., J.-M. Lasry, P.-L. Lions and B. Moll (2017). “Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach,” mimeo.

Barles, G. and P. E. Souganidis (1991). “Convergence of Approximation Schemes for Fully Nonlinear Second Order Equations,” Journal of Asymptotic Analysis 4, 271-283.

Björk, T. (2009). Arbitrage Theory in Continuous Time, Oxford University Press. Chapters 4-7, 19.

Brunnermeier, M. K, T. Eisenbach, and Y. Sannikov (2013). “Macroeconomics With Financial Frictions: A Survey,” Advances In Economics And Econometrics. Cambridge University Press.

Fleming, W. H. and H. M. Soner (2006). Controlled Markov Processes and Viscosity Solutions, Springer. Chapter 9.

Merton, R. C. (1969). “Lifetime Portfolio Selection under Uncertainty: the Continuous-Time Case,” Review of Economics and Statistics 51 (3), 247–257.

Nuño, G. and C. Thomas (2019). “Monetary Policy and Sovereign Debt Sustainability,” Bank of Spain Working Paper.

Øksendal, B. (2007). Stochastic Differential Equations: An Introduction with Applications. Springer. Chapters 3-5, 11.

Parra-Alvarez. J.-C. (2018). “A comparison of numerical methods for the solution of continuous-time DSGE models,” Macroeconomic Dynamics, 22(6), 1555-1583.

Van Handel, R. (2007). Lectures notes on Stochastic Calculus, Filtering, and Stochastic Control, Chapters 5, 8.


2. Deep learning and reinforcement learning.

Barto A. and R. S. Sutton (2018). Reinforcement Learning, an Introduction. MIT Press.

Duarte, V. (2018). “Machine Learning for Continuous-Time Finance,” mimeo.

Fernández-Villaverde, J,. G. Nuño, G. Sorg-Langhans, and M. Vogler (2019), “Solving High-Dimensional Dynamic Programming Problems using Deep Learning,” mimeo.

Goodfellow, I., Bengio, Y., and Courville, A. (2016). Deep Learning. MIT Press.

Sirignano, J. and K. Spiliopoulos (2018). “DGM: A deep learning algorithm for solving partial differential equations,” Journal of Computational Physics.


3. Heterogeneous agent models in continuous time.

Achdou, Y., J.-M. Lasry, P.-L. Lions and B. Moll (2017). “Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach,” mimeo.

Ahn, S., G. Kaplan, B. Moll, T. Winberry and C. Wolf (2018). “When Inequality Matters for Macro and Macro Matters for Inequality,” NBER Macroeconomics Annual.

Aiyagari, R., (1994), “Uninsured Idiosyncratic Risk and Aggregate Saving,” Quarterly Journal of Economics, 109 (3), 659-84.

Algan, Y., Allais, O., Den Haan, W. J., and Rendahl, P. (2014). Solving and simulating models with heterogeneous agents and aggregate uncertainty. In Schmedders, K. and Judd, K. L., editors, Handbook of Computational Economics 3, 277–324.

Bewley, T. (1986). “Stationary Monetary Equilibrium with a Continuum of Independently Fluctuating Consumers.” In Contributions to Mathematical Economics in Honor of Gerard Debreu, ed. Werner Hildenbrand and Andreu Mas-Collel. Amsterdam: NorthHolland

Boppart, Krusell and Mitman (2018). “Exploiting MIT Shocks in Heterogeneous-Agent Economies: The Impulse Response as a Numerical Derivative,” Journal of Economic Dynamics and Control 89, 68-92.

Fernández-Villaverde, J., S. Hurtado and G. Nuño (2019). “Financial Frictions and the Wealth Distribution,” NBER Working Paper No. 26302.

Heathcote J., K. Storesletten and G. L. Violante (2009). “Quantitative Macroeconomics with Heterogeneous Households,” Annual Review of Economics 1, 319-54.

Huggett, M. (1993). “The Risk-free rate in Heterogeneous-agent Incomplete-insurance Economies,” Journal of economic Dynamics and Control 17 (5-6), 953-969.

Kaplan, G., B. Moll and G. Violante (2018). “Monetary Policy According to HANK,” American Economic Review 108(3), 697–743.

Krusell, P. and A. Smith (1998). “Income and Wealth Heterogeneity in the Macroeconomy,” Journal of Political Economy 106(5), 867–896.


4. Optimal policies with heterogeneous agents.

Bigio. S. and Y. Sannikov (2019). “A Model of Credit, Money, Interest, and Prices,” mimeo.

Bigio, S., G. Nuno and J. Passadore (2019). “A Framework for Debt-Maturity Management,” NBER Working Paper No. 25808.

Dávila, J., J. H. Hong, P. Krusell and J. V. Ríos-Rull (2012). “Constrained Efficiency in the Neoclassical Growth Model With Uninsurable Idiosyncratic Shocks," Econometrica, 80(6), pp. 2431-2467.

Lucas, R. and B. Moll (2014). “Knowledge Growth and the Allocation of Time,” Journal of Political Economy, 122 (1), pp. 1-51.

Luenberger D. (1969). Optimization by Vector Space Methods, Ed. Wiley-Interscience, NJ.

Nuño, G. and B. Moll (2018). “Social Optima in economies with Heterogeneous Agents,” Review of Economic Dynamics, 28, pp. 150-180.

Nuño, G. and C. Thomas (2019). “Optimal Monetary Policy with Heterogeneous Agents,” Bank of Spain Working Paper.