In macroeconomics, much theoretical progress has been made in understanding when distributions matter for aggregates. For example, newer heterogeneous agent models deliver strikingly different implications for monetary and fiscal policies than what the traditional representative agent models do, and also allow us to study the distributional implications of different policies across households. This class of models can incorporate the potentially rich interactions between inequality and the macroeconomy: on the one hand, inequality shapes macroeconomic aggregates; on the other hand, macroeconomic shocks and policies affect inequality. 

However, economic models which describe the evolution of a distribution of agents are challenging to analyze. For this research program, mathematical methods based on stochastic analysis and partial differential equations, such as mean field games and stochastic control theory, have proven fruitful. Conversely, the formulation of economic models with distributions of agents provides motivation for new mathematical research questions.

To develop this field at the intersection of economics and mathematics, mathematicians and economists can benefit substantially from collaborating and exchange ideas.

• Bernt Øksendal - University of Oslo

• Kurt Mitman - Stockholm University

• Avi Mayorcas - Cambridge University 

• Pontus Rendahl - Copenhagen Business School

• Kristina Rognlien Dahl - BI Norwegian Business School

• Galo Nuño - Bank of Spain

• Francesco Lippi - LUISS University

• Arne Løkka - London School of Economics

• SeHyoun Ahn - Norges Bank

• Danial Ali Akbari - University of Oslo

• Sigmund Ellingsrud - BI Norwegian Business School

For the program as a pdf, click here.

Stochastics and partial differential equations in macroeconomics

December 1st and 2nd, 2022, Room A2-020

BI Norwegian Business School, Nydalsveien 37, Oslo

Thursday December 1st

08:30 - 08:55

Coffee

Morning session 08:55-12:00

08:55-09:00

Welcome

09:00 - 09:45

Avi Mayorcas (Technische Universität Berlin)

Stochastic Analysis and Optimal Control Theory

With the aim of (hopefully) starting a conversation between mathematicians and economists, in this talk I will recap some core concepts from continuous time stochastic analysis and optimal control theory. I will aim to discuss topics that mathematicians are typically interested in and hopeful space for fruitful collaboration with economists and social scientists.

10:00 - 10:45

Pontus Rendahl (Copenhagen Business School)

Continuous vs. discrete time: Some computational insights

Solving a workhorse incomplete markets model in continuous time is many times faster compared to its discrete time counterpart. This paper dissects the computational discrepancies and identifies the key bottlenecks. The implicit finite difference method – a commonly used tool in continuous time – accounts for a large share of the difference. This method is shown to be a special case of Howard’s improvement algorithm, efficiently implemented by relying on sparse matrix operations. By representing the policy function with a transition matrix it is possible to formulate a similar procedure in discrete time, which effectively eliminates the differences in run-times entirely.

11:00 - 11:45

Kurt Mitman (IIES, Stockholm University)

Micro heterogeneity and macroeconomic policy

How does the economy respond to a change in monetary policy? How does an increase in government spending affect output, consumption an employment? How should the government design transfer programmes to stabilize the economy in recessions? The Global Financial Crisis and COVID pandemic have renewed interest in these questions and revealed the shortcomings of DSGE-dominated macro.

Lunch 12:00-13:15

Afternoon session I 13:15-15:00

13:15 - 14:00

Avi Mayorcas (Technische Universität Berlin)

Mean Field Games in Macroeconomics A Case Study: Time dependent mean field game models for household wealth

The overriding theme of this talk is to (hopefully) spark a discussion regarding the benefits and drawbacks of rigorous continuum approximations, averaging and scaling limits in mathematics as they apply to problems of economic and social science modelling. The framework for the talk will centre around a presentation of mathematical, stochastic, optimal control leading to a description of the mean field game paradigm - one theoretical approximation to strategic games with a large number of players. Towards the end of the talk I will discuss some ongoing work with S. Ellingsrud and F. Harang applying these ideas to models of wealth dynamics. 

14:15 - 15:00

SeHyoun Ahn (Norges Bank)

Sparse Grids with Differential Operators for Economics

We give an intuitive presentation of the sparse grid finite-difference operator without the imprecise abstractions from the applied math literature (Griebel, 1998). Given the intuitive presentation, we show that the standard approach fails to guarantee convergence for the finite-difference method over the sparse grids. We provide an alternate formulation with a convergence proof for the solution over the sparse grid to the correct viscosity solution generalizing the standard Barles and Souganidis (1991) framework for regular grid. We also address the (potential) non-convergence of the standard semi-implicit iteration method of Achdou et al. (2017) in solving the discretized system over sparse grid with an adjustment from the globalization of Newton’s methods. Finally, we apply the new sparse grid with differential operators to example problems from economics, and show that we get a potential compression of 50%∼99% in grid points for equal accuracy.

15:00-15:30

Coffee break

Afternoon session II 15:30-16:30 (short presentations)

Danial Ali Akbari (University of Oslo)

A Heterogeneous Analytic Climate Economy

We develop an analytic climate economy with heterogeneous agents in continuous time. The model links integrated assessment components, parametric assumptions, and calibration approaches directly to their policy impacts, in particular the social cost of carbon. The model incorporates two novel aspects. First, by solving the model in continuous time we can better estimate the impact of short term dynamics on the social cost of carbon, compared to discrete-time models where increments are typically 5 or 10 years. As a result we are also able to evaluate whether the external validity of such estimates with larger time steps are salient in the short run. Second, by incorporating idiosyncratic risk, we can see how within globe (region) inequality impacts policy estimates, in particular the social cost of carbon. We find that even an efficient social planner would price carbon higher when idiosyncratic risk applies to the wealth of the population.

Sigmund Ellingsrud (BI Norwegian Business School)

The finite horizon/OLG Huggett model in continuous time

We formulate the model and argue why it is interesting. With a calibrated model, solved using numerical methods, we present two numerical results.

Friday December 2nd

08:30 – 09:00

Coffee

Morning session 09:00-12:00

09:00 -09:45

Anton Yurchenko-Tytarenko (University of Oslo)

Volterra sandwiched volatility model: Markovian approximation and hedging

We propose a new market model with stochastic volatility driven by a general Hölder continuous Gaussian Volterra process, i.e. the resulting price is not a Markov process. On the one hand, it is consistent with the empirically observed phenomenon of market memory, but, on the other hand, brings a vast amount of issues of a technical nature, especially in optimization problems. In the talk, we describe a way to obtain a Markovian approximation to the model as well as exploit it for the numerical computation of the optimal hedge. Two numerical methods are considered: Nested Monte Carlo and Least Squares Monte Carlo. The results are illustrated by simulations.

10:00- 10:45

Galo Nuño (Bank of Spain)

Optimal policies with heterogeneous agents

The computation of optimal policies in models with heterogeneous agents is at the frontier of research in macroeconomics, with exciting new applications in the design of monetary, fiscal and macropudential policies. In this talk, I will revise the main technical challenges in computing optimal policies, as well as some proposed solutions and diverse applications.

11:00 - 11:45

Bernt Øksendal (University of Oslo)

Optimal stopping of conditional McKean-Vlasov jump diffusions and applications to economics

We study the problem of optimal stopping of conditional McKean-Vlasov  (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal. To achieve this, we combine the state equation for the conditional McKean-Vlasov equation with the associated stochastic Fokker-Planck equation for the conditional law of the solution of the state. This gives us a Markovian system which can be handled by using a version of the Dynkin formula.

We illustrate our result by solving explicitly two optimal stopping problems for conditional McKean-Vlasov jump diffusions. More specifically,  (i) we first find the optimal time to sell in a market with common noise and jumps, and next,  (ii) we find the stopping time to quit a project whose state is modelled by a jump diffusion, when the performance functional involves the conditional mean of the state.

The talk is based on recent joint work with Nacira Agram, KTH, Stockholm, Sweden

Lunch 12:00-13:30

The workshop is free of charge, and open for anyone interested. 

To attend you will need to register HERE 

The workshop will be held at BI Norwegian Business School, located in Nydalen, Oslo. 

• Karin Kinnerud - BI Norwegian Business School - webpage  

• Karl Harmenberg - BI Norwegian Business School - webpage 

• Fabian Harang - BI Norwegian Business School - webpage