In this thesis, we study a class of coupled forward backward stochastic differential equations (FBSDEs), called singular FBSDEs, which were first introduced in 2013, to model the evolution of emissions and the price of emissions
allowances in a carbon market such as the European Union Emissions Trading System. These FBSDEs have two key properties: the terminal condition
of the backward equation is a discontinuous function of the terminal value of
the forward equation, and the forward dynamics may not be strongly elliptic,
not even in a neighbourhood of the singularities of the terminal condition.
We first consider a model for an electricity market subject to a carbon market
with a single compliance period. We show that the carbon pricing problem
leads to a singular FBSDE. This type of model is then extended to a multiperiod emissions trading system in which cumulative emissions are compared
with a cap at multiple compliance times. We show that the multi-period
pricing problem is well-posed for various mechanisms linking the trading periods. We then introduce an infinite period model, for a carbon market with
a sequence of compliance times and no end date. We show that, under appropriate conditions, the value function for the multi-period pricing problem
converges, as the number of periods tends to infinity, to a value function for
this infinite period model, and present a setting in which this occurs. Finally, we focus on numerical investigations. For the single period model
for an electricity market with emissions trading, the processes and functions
appearing in the pricing FBSDE are chosen to model the features of the UK
energy market, using historical data. Numerical methods are used to solve
the pricing FBSDE, and the results are interpreted. In the future, these
could support policies seeking to mitigate the effects of climate change.