Kuroda and Nagai \cite{KN} state that the factor process in the Risk Sensitive control Asset Management (RSCAM) is stable under the Föllmer-Schweizer minimal martingale measure . Fleming and Sheu \cite{FS} and more recently Föllmer and Schweizer \cite{FoS} have observed that the role of the minimal martingale measure in this portfolio optimization is yet to be established. In this article we aim to address this question by explicitly connecting the optimal wealth allocation to the minimal martingale measure. We achieve this by using a "trick" of observing this problem in the context of model uncertainty via a two person zero sum stochastic differential game between the investor and an antagonistic market that provides a probability measure. We obtain some startling insights. Firstly, if short-selling is not permitted and if the factor process evolves under the minimal martingale measure then the investor's optimal strategy can only be to invest in the riskless asset (i.e. the no-regret strategy). Secondly, if the factor process and the stock price process have independent noise, then even if the market allows short selling, the optimal strategy for the investor must be the no-regre
We have re-observed and re-analysed the optical spectrum of the Schweizer-Middleditch star, a hot subdwarf which lies along almost the same line-of-sight as the centre of the historic SN1006 supernova remnant (SNR). Although this object is itself unlikely to be the remnant of the star which exploded in 1006AD, Wellstein et al. (1999) have demonstrated that it could be the remnant of the donor star in a pre-supernova Type Ia interacting binary, if it possesses an unusually low mass. We show that, if it had a mass of 0.1-0.2 Msun, the SM star would lie at the same distance (~800pc) as the SNR as estimated by Willingale et al. (1995). However, most distance estimates to SN1006 are much larger than this, and there are other convincing arguments to suggest that the SM star lies behind this SNR. Assuming the canonical subdwarf mass of 0.5 Msun, we constrain the distance of the SM star as 1050 pc<d<2100 pc. This places the upper limit on the distance of SN1006 at 2.1 kpc.
In this paper we study the Föllmer-Schweizer decomposition of a square integrable random variable $ξ$ with respect to a given semimartingale $S$ under restricted information. Thanks to the relationship between this decomposition and that of the projection of $ξ$ with respect to the given information flow, we characterize the integrand appearing in the Föllmer-Schweizer decomposition under partial information in the general case where $ξ$ is not necessarily adapted to the available information level. For partially observable Markovian models where the dynamics of $S$ depends on an unobservable stochastic factor $X$, we show how to compute the decomposition by means of filtering problems involving functions defined on an infinite-dimensional space. Moreover, in the case of a partially observed jump-diffusion model where $X$ is described by a pure jump process taking values in a finite dimensional space, we compute explicitly the integrand in the Föllmer-Schweizer decomposition by working with finite dimensional filters.
We investigate two hedging problems in exponential Lévy models. First, we provide an explicit representation for the Föllmer--Schweizer decomposition of European type options under mild conditions, which implies a closed-form expression of the corresponding local risk-minimizing strategies. Secondly, we discretize stochastic integrals driven by an exponential Lévy process using a jump correction method. The convergence rate of the resulting discretization error as the expected number of discretization times increases is measured in weighted BMO spaces, implying also $L_p$-estimates, $p \in (2, \infty)$. Moreover, the effect of a change of measure satisfying a reverse Hölder inequality is addressed. As an application, the error caused by discretizing the local risk-minimizing strategies is investigated in dependence of properties of the Lévy measure, the regularity of the payoff function and the chosen random discretization times.
First, we consider the problem of hedging in complete binomial models. Using the discrete-time Föllmer-Schweizer decomposition, we demonstrate the equivalence of the backward induction and sequential regression approaches. Second, in incomplete trinomial models, we examine the extension of the sequential regression approach for approximation of contingent claims. Then, on a finite probability space, we investigate stability of the discrete-time Föllmer-Schweizer decomposition with respect to perturbations of the stock price dynamics and, finally, perform its asymptotic analysis under simultaneous perturbations of the drift and volatility of the underlying discounted stock price process, where we prove stability and obtain explicit formulas for the leading order correction terms.
We treat shared value problems for rational functions $R(z)$ and their derivative $R'(z)$ in the plane and on the sphere. We also consider shared values for the pair $R(w)$ and $\partial_{z} R = λw \cdot R'(w)$ on ${\mathbb C} \setminus \{ 0 \}$ and $\widehat{\mathbb C}$, again with rational functions $R$. In ${\mathbb C} \setminus \{ 0 \}$ this is related to shared values of meromorphic functions $f : {\mathbb C} \to \widehat{\mathbb C}$ and $f'$ through $f(z)=R(w)$ with $w=\exp(λz)$, while on $\widehat{\mathbb C}$ this is connected to shared limit values in a similar fashion.
We consider uniqueness results for meromorphic functions $f:{\mathbb C} \to \widehat{\mathbb C}$ such that for certain values $a\in {\mathbb C}$ the implication $f(z)=a \Rightarrow f'(z)=a$ holds, i.e. that $f$ and $f'$ share values {\it partially}. In particular, we give a result for four partially shared values.
Let $A$ be the ring of elements in an algebraic function field $K$ over $\mathbb{F}_q$ which are integral outside a fixed place $\infty$. In contrast to the classical modular group $SL_2(\mathbb{Z})$ and the Bianchi groups, the {\it Drinfeld modular group} $G=GL_2(A)$ is not finitely generated and its automorphism group $\mathrm{Aut}(G)$ is uncountable. Except for the simplest case $A=\mathbb{F}_q[t]$ not much is known about the generators of $\mathrm{Aut}(G)$ or even its structure. We find a set of generators of $\mathrm{Aut}(G)$ for a new case. \par On the way, we show that {\it every} automorphism of $G$ acts on both, the {\it cusps} and the {\it elliptic points} of $G$. Generalizing a result of Reiner for $A=\mathbb{F}_q[t]$ we describe for each cusp an uncountable subgroup of $\mathrm{Aut}(G)$ whose action on $G$ is essentially defined on the stabilizer of that cusp. In the case where $δ$ (the degree of $\infty$) is $1$, the elliptic points are related to the isolated vertices of the quotient graph $G\setminus\mathcal{T}$ of the Bruhat-Tits tree. We construct an infinite group of automorphisms of $G$ which fully permutes the isolated vertices with cyclic stabilizer.
When it comes to structural estimation of risk preferences from data on choices, random utility models have long been one of the standard research tools in economics. A recent literature has challenged these models, pointing out some concerning monotonicity and, thus, identification problems. In this paper, we take a second look and point out that some of the criticism - while extremely valid - may have gone too far, demanding monotonicity of choice probabilities in decisions where it is not so clear whether it should be imposed. We introduce a new class of random utility models based on carefully constructed generalized risk premia which always satisfy our relaxed monotonicity criteria. Moreover, we show that some of the models used in applied research like the certainty-equivalent-based random utility model for CARA utility actually lie in this class of monotonic stochastic choice models. We conclude that not all random utility models are bad.
Let $N\geq 1$ be a non-square free integer and let $W_N$ be a non-trivial subgroup of the group of the Atkin-Lehner involutions of $X_0(N)$ such that the modular curve $X_0(N)/W_N$ has genus at least two. We determine all pairs $(N,W_N)$ such that $X_0(N)/W_N$ is a bielliptic curve and the pairs $(N,W_N)$ such that $X_0(N)/W_N$ has an infinite number of quadratic points over $\mathbb{Q}$.
We determine all entire functions $f$ such that for nonzero complex values $a eq b$ the implications $f=a \Rightarrow f' =a$ and $f' =b \Rightarrow f=b$ hold. This solves an open problem in uniqueness theory. In this context we give a normality criterion, which might be interesting in its own right.
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