FBSDE Methods in Heterogeneous Agent Models

As global economies face the challenges of ageing populations, climate change and increasing inequality, the production and implementation of models capturing the full complexity of these interwoven phenomena is of paramount importance. This early stage project will apply stochastic optimal control and cutting edge, neural network (NN) based, numerical methods to produce a novel framework for analysis and computational resolution of complex, micro-founded, macroeconomic models. The proposed framework will allow for the treatment of dynamic models, incorporating multiple, interlocking, features, allowing for more accurate validation of economic theories and in the long term, policy evaluation.

Micro-founded, macroeconomic models derive, large scale, emergent economic processes, such as employment rates, wealth distributions and savings rates, from assumptions on the preferences and optimal strategies of individual economic agents against a random, stochastically modelled, environment. Compare to the derivation of emergent physical phenomena, such as the distribution of gas molecules or atoms in a crystal, from first principles. A strength of the approach is that the preferences and environments of economic agents as well as emergent economic phenomena, can often be measured empirically, allowing for accurate calibration and validation. However, as opposed to the most basic physical laws, e.g. Newtonian mechanics, the mathematical formulation of micro-founded models requires stochastic optimal control theory, resulting in highly non-linear, typically infinite dimensional, equations.

The most accurate micro-founded approaches are heterogeneous agent models, where each agent is treated completely independently. However, as this increases the mathematical and computational complexity of the model, many implementations make compromises by either pre-aggregating agents' environments, leading to homogeneous agent models, assuming time stationarity or treating only one or two dependent variables simultaneously. Given these simplifications, many important, empirically observed economic phenomena are often left out, or can only be treated in isolation; for example, time stationarity means that intergenerational transfers are ignored, homogeneity between agents removes effects from economic inequality and treating dependent variables in isolation misses potential, non-linear influences between them.

This project will leverage powerful tools of stochastic analysis, in particular the language of coupled forward-backward stochastic differential equations (FBSDE) to give a rigorous, continuous time formulations of highly complex, dynamic, heterogeneous agent models. The FBSDE formulation gives access to efficient and cutting edge, NN based numerical methods, which offer robust, computational efficiency in the presence of increasing dimensionality. The proposal builds on promising, early stage work between the PL, economists and mathematicians at BI Oslo in which a robust, rigorous mathematical formulation of a simplified heterogeneous agent model has been obtained and accompanying numerical algorithm implemented. Funding is required to expand on this early work, extending the approach to genuinely applicable economic models and to build an open sourced code-base in order to attract a community of researchers, innovation users and policy influencers.

The concrete goals of the project are to produce a new mathematical framework and computational approach, tailored to the resolution of complex, heterogeneous agent models, exhibiting the following desirable properties:


Robust to a wide range of economically relevant extensions and variants of the model.
Able to capture age, demographics and inheritance as mechanisms of wealth accumulation and transfer.
Are economically intuitive; i.e. lead naturally to expressions of practical research relevance; for example sensitivity of model outputs with respect to control variables.
Require minimal mathematical literacy to implement, extend and apply to economics research.