Abstract:
The modelling power of Itˆo integrals has a far reaching impact on a spectrum of diverse fields. For
example, in mathematics of finance, its use has given insights into the relationship between call options
and their non-deterministic underlying stock prices; in the study of blood clotting dynamics, its utility
has helped provide an understanding of the behaviour of platelets in the blood stream; and in the investigation
of experimental psychology, it has been used to build random fluctuations into deterministic
models which model the dynamics of repetitive movements in humans.
Finding the quadrature for these integrals using continuous groups or Lie groups has to take families
of time indexed random variables, known as Wiener processes, into consideration. Adaptations of Sophus
Lie’s work to stochastic ordinary differential equations (SODEs) have been done by Gaeta and Quintero
[1], Wafo Soh and Mahomed [2], ¨Unal [3], Meleshko et al. [4], Fredericks and Mahomed [5], and Fredericks
and Mahomed [6]. The seminal work [1] was extended in Gaeta [7]; the differential methodology of [2]
and [3] were reconciled in [5]; and the integral methodology of [4] was corrected and reconciled in [5] via [6].
Symmetries of SODEs are analysed. This work focuses on maintaining the properties of the Weiner
processes after the application of infinitesimal transformations. The determining equations for first-order
SODEs are derived in an Itˆo calculus context. These determining equations are non-stochastic.
Many methods of deriving Lie point-symmetries for Itˆo SODEs have surfaced. In the Itˆo calculus context
both the formal and intuitive understanding of how to construct these symmetries has led to seemingly
disparate results. The impact of Lie point-symmetries on the stock market, population growth and
weather SODE models, for example, will not be understood until these different results are reconciled as
has been attempted here.
Extending the symmetry generator to include the infinitesimal transformation of the Wiener process
for Itˆo stochastic differential equations (SDEs), has successfully been done in this thesis. The impact of
this work leads to an intuitive understanding of the random time change formulae in the context of Lie
point symmetries without having to consult much of the intense Itˆo calculus theory needed to derive it
formerly (see Øksendal [8, 9]). Symmetries of nth-order SODEs are studied. The determining equations of
these SODEs are derived in an Itˆo calculus context. These determining equations are not stochastic in nature.
SODEs of this nature are normally used to model nature (e.g. earthquakes) or for testing the safety
and reliability of models in construction engineering when looking at the impact of random perturbations. The symmetries of high-order multi-dimensional SODEs are found using form invariance arguments on
both the instantaneous drift and diffusion properties of the SODEs. We then apply this to a generalised
approximation analysis algorithm. The determining equations of SODEs are derived in an It¨o calculus
context.
A methodology for constructing conserved quantities with Lie symmetry infinitesimals in an Itˆo integral
context is pursued as well. The basis of this construction relies on Lie bracket relations on both the
instantaneous drift and diffusion operators.