99 research outputs found
Utility Maximisation: Non-concave utility and non linear expectation
Since the birth of mathematical nance, portfolio selection has been one of the topics which have attracted a lot of interest, with models formulated in discrete and continuous time and developed in complete and incomplete markets. In conventional or neoclassical finance, many models are based off the assumption that agents make decisions by maximising their expected utility. Deviations between models and market observations have generated a recent field of study, behavioural finance, which incorporates psychology, sociology and finance together to resolve observed phenomenon like bubbles which conventional finance cannot explain. In this thesis, we will be restricting ourselves to the complete continuous market and look at a new formulation of expected utility maximisation with behavioural finance elements incorporated into it, namely S-shaped utilities and probability distortions. We consider the three general cases of expected utility maximisation: utility from terminal wealth, utility from consumption and utility from terminal wealth and consumption. We shall review the neoclassical problems and then explore the cases with behavioural elements installed. \ud
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Key Words: Portfolio Selection, continuous time, martingale approach, Sshaped function, probability distortion, cumulative prospect theor
Non-linear biases, stochastically-sampled effective Hamiltonians and spectral functions in quantum Monte Carlo methods
In this article we study examples of systematic biases that can occur in
quantum Monte Carlo methods due to the accumulation of non-linear expectation
values, and approaches by which these errors can be corrected. We begin with a
study of the Krylov-projected FCIQMC (KP-FCIQMC) approach, which was recently
introduced to allow efficient, stochastic calculation of dynamical properties.
This requires the solution of a sampled effective Hamiltonian, resulting in a
non-linear operation on these stochastic variables. We investigate the
probability distribution of this eigenvalue problem to study both stochastic
errors and systematic biases in the approach, and demonstrate that such errors
can be significantly corrected by moving to a more appropriate basis. This is
lastly expanded to include consideration of the correlation function QMC
approach of Ceperley and Bernu, showing how such an approach can be taken in
the FCIQMC framework.Comment: 12 pages, 7 figure
The full replica symmetry breaking in the Ising spin glass on random regular graph
In this paper, we extend the full replica symmetry breaking scheme to the
Ising spin glass on a random regular graph. We propose a new martingale
approach, that overcomes the limits of the Parisi-M\'ezard cavity method,
providing a well-defined formulation of the full replica symmetry breaking
problem in random regular graphs. Finally, we define the order parameters of
the system and get a set of self-consistency equations for the order parameters
and the free energy. We face up the problem only from a technical point of
view: the physical meaning of this approach and the quantitative evaluation of
the solution of the self-consistency equations will be discussed in next works.Comment: 23 page
Malliavin calculus method for asymptotic expansion of dual control problems
We develop a technique based on Malliavin-Bismut calculus ideas, for
asymptotic expansion of dual control problems arising in connection with
exponential indifference valuation of claims, and with minimisation of relative
entropy, in incomplete markets. The problems involve optimisation of a
functional of Brownian paths on Wiener space, with the paths perturbed by a
drift involving the control. In addition there is a penalty term in which the
control features quadratically. The drift perturbation is interpreted as a
measure change using the Girsanov theorem, leading to a form of the integration
by parts formula in which a directional derivative on Wiener space is computed.
This allows for asymptotic analysis of the control problem. Applications to
incomplete It\^o process markets are given, in which indifference prices are
approximated in the low risk aversion limit. We also give an application to
identifying the minimal entropy martingale measure as a perturbation to the
minimal martingale measure in stochastic volatility models
Representation Theorems for Quadratic -Consistent Nonlinear Expectations
In this paper we extend the notion of ``filtration-consistent nonlinear
expectation" (or "-consistent nonlinear expectation") to the case
when it is allowed to be dominated by a -expectation that may have a
quadratic growth. We show that for such a nonlinear expectation many
fundamental properties of a martingale can still make sense, including the
Doob-Meyer type decomposition theorem and the optional sampling theorem. More
importantly, we show that any quadratic -consistent nonlinear
expectation with a certain domination property must be a quadratic
-expectation. The main contribution of this paper is the finding of the
domination condition to replace the one used in all the previous works, which
is no longer valid in the quadratic case. We also show that the representation
generator must be deterministic, continuous, and actually must be of the simple
form
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