Suchergebnisse

This note demonstrates that a linear approximation algorithm based on Chow's Lagrangean one‐dimensional backward stochastic differential equations (BSDEs) converges under reasonable assumptions.

Some background on ordinary differential equations -- Pragmatic introduction to stochastic differential equations -- Itô calculus and stochastic differential equations -- Probability distributions and statistics of SDEs -- Statistics of linear stochastic differential equations -- Useful theorems and formulas for SDEs -- Numerical simulation of SDEs -- Approximation of non-linear SDEs -- Filtering and smoothing theory -- Parameter estimation in SDE models -- Stochastic differential equations in machine learning

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. ; Open Access

We prove an L2-regularity result for the solutions of Forward Backward Doubly Stochastic Differentiel Equations (F-BDSDEs in short) under globally Lipschitz continuous assumptions on the coefficients. Therefore, we extend the well known regularity results established by Zhang (2004) for Forward Backward Stochastic Differential Equations (F-BSDEs in short) to the doubly stochastic framework. To this end, we prove (by Malliavin calculus) a representation result for the martingale component of the solution of the F-BDSDE under the assumption that the coefficients are continuous in time and continuously differentiable in space with bounded partial derivatives. As an (important) application of our L2-regularity result, we derive the rate of convergence in time for the (Euler time discretization based) numerical scheme for F-BDSDEs proposed by Bachouch et al.(2016) under only globally Lipschitz continuous assumptions.