Approximations for Stochastic McKeanVlasov Equations with NonLipschitz Coefficients by an EulerMaruyama Scheme
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2023Copyright
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In this thesis we study stochastic McKeanVlasov equations. These are stochastic
differential equations where the coefficients depend also on the distribution of the
solution. This dependency adds to the complexity of the equation so in this thesis we
will study these equations using a discrete approximation.
We focus on considering the existence of a unique strong solution to stochastic
McKeanVlasov equations using a discrete and recursive EulerMaruyama approximation,
as well as the convergence rate of the approximation. Our main source is
the article EulerMaruyama Approximations for Stochastic McKeanVlasov Equations
with NonLipschitz Coefficients written by Xiaojie Ding and Huijie Qiao, which we
follow throughout this thesis.
In the thesis we recall some preliminary theory surrounding stochastic processes and
stochastic differential equations and introduce some results. We give the definitions
for weak and strong solutions for the McKeanVlasov equation as well as the definition
for the martingale problem. We also introduce some useful inequalities. We give
the assumptions under which we work in this thesis, such as the assumption that the
coefficients of the McKeanVlasov equations satisfy some nonLipschitz conditions.
One of the main results in this thesis is to show the existence of unique strong solutions.
We approach this in two steps: first, we show the recursive construction of the
EulerMaruyama approximation. With this approximation we show that there exists
a solution to the martingale problem and hence we get the existence of a weak solution.
Then, using Ito’s formula we prove that pathwise uniqueness holds under our assumptions.
After these two steps we show that the existence of a strong unique
solution can be proven. We also investigate with the help of Ito’s formula the convergence
rate of the EulerMaruyama approximation used to show the existence of the
solution.
...
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