The chordal Loewner equation and monotone probability theory

In Ref. 5, O. Bauer interpreted the chordal Loewner equation in terms of noncommutative probability theory. We follow this perspective and identify the chordal Loewner equations as the non-autonomous versions of evolution equations for semigroups in monotone and anti-monotone probability theory. We also look at the corresponding equation for free probability theory.

Pessimistic Portfolio Choice with One Safe and One Risky Asset and Right Monotone Probability Difference Order

As is well known, a first-order dominant deterioration in risk does not necessarily cause a risk-averse investor to reduce his holdings of that deteriorated asset under the expected utility framework, even in the simplest portfolio setting with one safe asset and one risky asset. The purpose of this paper is to derive conditions on shifts in the distribution of the risky asset under which the counterintuitive conclusion above can be overthrown under the rank-dependent expected utility framework, a more general and prominent alternative of the expected utility. Two new criterions of changes in risk, named the monotone probability difference (MPD) and the right monotone probability difference (RMPD) order, are proposed, which is a particular case of the first stochastic dominance. The relationship among MPD, RMPD, and the other two important stochastic orders, monotone likelihood ratio (MLR) and monotone probability ratio (MPR), is examined. A desired comparative statics result is obtained when a shift in the distribution of the risky asset satisfies the RMPD criterion.

NONCOMMUTATIVE BROWNIAN MOTIONS ASSOCIATED WITH KESTEN DISTRIBUTIONS AND RELATED POISSON PROCESSES

We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition P we assign the weight w(P) = pe(P)qe'(P), where e(P) and e'(P) are, respectively, the numbers of disorders and orders in P related to the natural partial order on the set of blocks of P implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney's numbers (related to inner blocks in non-crossing partitions) and generalized Euler's numbers (related to orders and disorders in ordered non-crossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes.

DISCRETE INTERPOLATION BETWEEN MONOTONE PROBABILITY AND FREE PROBABILITY

We construct a sequence of states called m-monotone product states which give a discrete interpolation between the monotone product of states of Muraki in monotone probability and the free product of states of Avitzour and Voiculescu in free probability. We derive the associated basic limit theorems and develop the combinatorics based on non-crossing ordered partitions with monotone order starting from depth m. We deduce an explicit formula for the Cauchy transforms of the m-monotone central limit measures and for the associated Jacobi coefficients. A new type of combinatorics of inner blocks in non-crossing partitions leads to explicit formulas for the mixed moments of m-monotone Gaussian operators, which are new even in the case of monotone independent Gaussian operators with arcsine distributions.