Narrow Results

Let ${𝒩}\subset {ℳ}$ be a unital inclusion of arbitrary von Neumann algebras. We give a 2-$C^{\ast }$-categorical/planar algebraic description of normal faithful conditional expectations $E:{ℳ}\to {𝒩}\subset {ℳ}$ with finite index and their duals $E^{\prime }:{{𝒩}}^{\prime }\to {{ℳ}}^{\prime }\subset {{𝒩}}^{\prime }$ by means of the solutions of the conjugate equations for the inclusion morphism $\iota :{𝒩}\to {ℳ}$ and its conjugate morphism $\overline{\iota }:{ℳ}\to {𝒩}$. In particular, the theory of index for conditional expectations admits a 2-$C^{\ast }$-categorical formulation in full generality. Moreover, we show that a pair $({𝒩}\subset {ℳ},E)$ as above can be described by a Q-system, and vice versa. These results are due to Longo in the subfactor/simple tensor unit case [R. Longo, Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial, Comm. Math. Phys. 130(1990) 285–309; A duality for Hopf algebras and for subfactors. I, Comm. Math. Phys. 159 (1994) 133–150].

Average options are path-dependent and have payoffs which depend on the average price over a fixed period leading up to the maturity date. This option is of interest and important for thinly-traded assets since price manipulation is prohibited, and both the investor and issuer may enjoy a certain degree of protection from the caprice of the market. However, to deal with unexpected situations incurred the usual simple average options may not be sufficient. Therefore, in this paper, we consider to propose a more general weight instead of the simple average, for which it may be possible to control the weight in the light of the unexpected circumstances. Further, we derive approximate solutions for the weighted sums of asset prices, and in order for these formulae to be applicable some adjustment must be taken into account along with Monte Carlo simulations. Finally, some comparisons for these results are made.

In this paper we give a purely noncommutative criterion for the characterization of two-state normal distribution. We prove that families of two-state normal distribution can be described by relations which is similar to the conditional expectation in free probability, but has no classical analogue. We also show a generalization of Bożejko, Leinert and Speichers formula from Ref. 10 (relating moments and noncommutative cumulants).

We first develop a theory of conditional expectations for random variables with values in a complete metric space $M$ equipped with a contractive barycentric map $\beta$, and then give convergence theorems for martingales of $\beta$-conditional expectations. We give the Birkhoff ergodic theorem for $\beta$-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the $\beta$-conditional expectation. Moreover, we prove the continuity property of the ergodic limit function by finding a complete metric between contractive barycentric maps on the Wasserstein space of Borel probability measures on $M$. Finally, the large deviation property of $\beta$-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.

In this paper, we discuss the measure theoretic composition operators on $L^2(\mu )$ in some operator classes, such as class $p$-$wA(s,t)$, class $p$-$A(s,t)$ and class $p$-$A$. We also study some spectral properties of class $p$-$wA(s,t)$ composition operators on $L^2(\mu )$.

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation ${𝔼}_{\phi }\left[{(z-X-f(X)Y{f}^{\ast }(X))}^{-1}|X\right]$ for free random variables $X,Y$ and a Borel function $f$ is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form $X+f(X)Y{f}^{\ast }(X)$. The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.

Independent of the other communities who have developed theories of wavelets over the last twenty years, we developed over the same period a view of wavelets seen as stochastic processes. That context arose naturally from our theory of Time operators in statistical mechanics. Essential ingredients in our theory included Kolmogorov dynamical systems and conditional expectations. The purpose of the present paper is to come up-to-date on the relationship of our theory to the general theory of wavelets.

The isometry formula for the Itô integral is generalized to a new stochastic integral involving both adapted and instantly independent stochastic processes. The proof is rather elementary. An example is given to ilustrate the isometry formula.