60H15 Stochastic partial differential equations [See also 35R60]
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This thesis concerns itself with the long-term behavior of generalized Langevin dynamics with multiplicative noise,
i.e. the solutions to a class of two-component stochastic differential equations in \( \mathbb{R}^{d_1}\times\mathbb{R}^{d_2} \)
subject to outer influence induced by potentials \( \Phi \) and \( \Psi \),
where the stochastic term is only present in the second component, on which it is dependent.
In particular, convergence to an equilibrium defined by an invariant initial distribution \( \mu \) is shown
for weak solutions to the generalized Langevin equation obtained via generalized Dirichlet forms,
and the convergence rate is estimated by applying hypocoercivity methods relying on weak or classical Poincaré inequalities.
As a prerequisite, the space of compactly supported smooth functions is proven to be a domain of essential m-dissipativity
for the associated Kolmogorov backward operator on \(L^2(\mu)\).
In the second part of the thesis, similar Langevin dynamics are considered, however defined on a product of infinite-dimensional separable Hilbert spaces.
The set of finitely based smooth bounded functions is shown to be a domain of essential m-dissipativity for the corresponding Kolmogorov operator \( L \) on \( L^2(\mu) \)
for a Gaussian measure \( \mu \), by applying the previous finite-dimensional result to appropriate restrictions of \( L \).
Under further bounding conditions on the diffusion coefficient relative to the covariance operators of \( \mu \),
hypocoercivity of the generated semigroup is proved, as well as the existence of an associated weakly continuous Markov process
which analytically weakly provides a weak solution to the considered Langevin equation.
In traditional portfolio optimization under the threat of a crash the investment horizon or time to maturity is neglected. Developing the so-called crash hedging strategies (which are portfolio strategies which make an investor indifferent to the occurrence of an uncertain (down) jumps of the price of the risky asset) the time to maturity turns out to be essential. The crash hedging strategies are derived as solutions of non-linear differential equations which itself are consequences of an equilibrium strategy. Hereby the situation of changing market coefficients after a possible crash is considered for the case of logarithmic utility as well as for the case of general utility functions. A benefit-cost analysis of the crash hedging strategy is done as well as a comparison of the crash hedging strategy with the optimal portfolio strategies given in traditional crash models. Moreover, it will be shown that the crash hedging strategies optimize the worst-case bound for the expected utility from final wealth subject to some restrictions. Another application is to model crash hedging strategies in situations where both the number and the height of the crash are uncertain but bounded. Taking the additional information of the probability of a possible crash happening into account leads to the development of the q-quantile crash hedging strategy.