Brownian Super-exponents

We introduce a transform on the class of stochastic exponentials for d-dimensional Brownian motions. Each stochastic exponential generates another stochastic exponential under the transform. The new exponential process is often merely a supermartingale even in cases where the original process is a martingale. We determine a necessary and sufficient condition for the transform to be a martingale process. The condition links expected values of the transformed stochastic exponential to the distribution function of certain time-integrals.Comment: 10 page

Expert Opinions and Logarithmic Utility Maximization in a Market with Gaussian Drift

This paper investigates optimal portfolio strategies in a financial market where the drift of the stock returns is driven by an unobserved Gaussian mean reverting process. Information on this process is obtained from observing stock returns and expert opinions. The latter provide at discrete time points an unbiased estimate of the current state of the drift. Nevertheless, the drift can only be observed partially and the best estimate is given by the conditional expectation given the available information, i.e., by the filter. We provide the filter equations in the model with expert opinion and derive in detail properties of the conditional variance. For an investor who maximizes expected logarithmic utility of his portfolio, we derive the optimal strategy explicitly in different settings for the available information. The optimal expected utility, the value function of the control problem, depends on the conditional variance. The bounds and asymptotic results for the conditional variances are used to derive bounds and asymptotic properties for the value functions. The results are illustrated with numerical examples.Comment: 21 page

On the Kolmogorov--Wiener--Masani spectrum of a multi-mode weakly stationary quantum process

We introduce the notion of a $k$-mode weakly stationary quantum process $\varrho$ based on the canonical Schr\"odinger pairs of position and momentum observables in copies of $L^2(\mathbb{R}^k)$, indexed by an additive abelian group $D$ of countable cardinality. Such observables admit an autocovariance map $\widetilde{K}$ from $D$ into the space of real $2k \times 2k$ matrices. The map $\widetilde{K}$ on the discrete group $D$ admits a spectral representation as the Fourier transform of a $2k \times 2k$ complex Hermitain matrix-valued totally finite measure $\Phi$ on the compact character group $\widehat{D}$, called the Kolmogorov-Wiener-Masani (KWM) spectrum of the process $\varrho$. Necessary and sufficient conditions on a $2k \times 2k$ complex Hermitian matrix-valued measure $\Phi$ on $\widehat{D}$ to be the KWM spectrum of a process $\varrho$ are obtained. This enables the construction of examples. Our theorem reveals the dramatic influence of the uncertainty relations among the position and momentum observables on the KWM spectrum of the process $\varrho$. In particular, KWM spectrum cannot admit a gap of positive Haar measure in $\widehat{D}$. The relationship between the number of photons in a particular mode at any site of the process and its KWM spectrum needs further investigation.Comment: 17 pages, added Theorem 4.2 and some remarks. Comments welcome. Keywords: Weakly stationary quantum process, Kolmogorov-Wiener-Masani spectrum, autocovariance map, spectral representation, uncertainty relation

Comparing the GG-Normal Distribution to its Classical Counterpart

In one dimension, the theory of the $G$-normal distribution is well-developed, and many results from the classical setting have a nonlinear counterpart. Significant challenges remain in multiple dimensions, and some of what has already been discovered is quite nonintuitive. By answering several classically-inspired questions concerning independence, covariance uncertainty, and behavior under certain linear operations, we continue to highlight the fascinating range of unexpected attributes of the multidimensional $G$-normal distribution.Comment: Final version. To appear in Communications on Stochastic Analysis. Title has changed. Keywords: sublinear expectation, multidimensional $G$-normal distribution, independenc

Solutions of semilinear wave equation via stochastic cascades

We introduce a probabilistic representation for solutions of quasilinear wave equation with analytic nonlinearities. We use stochastic cascades to prove existence and uniqueness of the solution

Mathematical Formulation of an Optimal Execution Problem with Uncertain Market Impact

We study an optimal execution problem with uncertain market impact to derive a more realistic market model. We construct a discrete-time model as a value function for optimal execution. Market impact is formulated as the product of a deterministic part increasing with execution volume and a positive stochastic noise part. Then, we derive a continuous-time model as a limit of a discrete-time value function. We find that the continuous-time value function is characterized by a stochastic control problem with a Levy process.Comment: 17 pages. Forthcoming in "Communications on Stochastic Analysis.

On the adjoint Markov policies in stochastic differential games

We consider time-homogeneous uniformly nondegenerate stochastic differential games in domains and propose constructing $\varepsilon$-optimal strategies and policies by using adjoint Markov strategies and adjoint Markov policies which are actually time-homogeneous Markov, however, relative not to the original process but to a couple of processes governed by a system consisting of the main original equation and of an adjoint stochastic equations of the same type as the main one. We show how to find $\varepsilon$-optimal strategies and policies in these classes by using the solvability in Sobolev spaces of not the original Isaacs equation but of its appropriate modification. We also give an example of a uniformly nondegenerate game where our assumptions are not satisfied and where we conjecture that there are no not only optimal Markov but even $\varepsilon$-optimal adjoint (time-homogeneous) Markov strategies for one of the players.Comment: 22 page

How to differentiate a quantum stochastic cocycle.

Two new approaches to the infinitesimal characterisation of quantum stochastic cocycles are reviewed. The first concerns mapping cocycles on an operator space and demonstrates the role of H\"older continuity; the second concerns contraction operator cocycles on a Hilbert space and shows how holomorphic assumptions yield cocycles enjoying an infinitesimal characterisation which goes beyond the scope of quantum stochastic differential equations

Stochastic integral characterizations of semi-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes

In this paper, three topics on semi-selfdecomposable distributions are studied. The first one is to characterize semi-selfdecomposable distributions by stochastic integrals with respect to Levy processes. This characterization defines a mapping from an infinitely divisible distribution with finite log-moment to a semi-selfdecomposable distribution. The second one is to introduce and study a Langevin type equation and the corresponding Ornstein-Uhlenbecktype process whose limiting distribution is semi-selfdecomposable. Also, semi-stationary Ornstein-Uhlenbeck type processes with semi-selfdecomposable distributions are constructed. The third one is to study the iteration of the mapping above. The iterated mapping is expressed as a single mapping with a different integrand. Also, nested subclasses of the class of semi-selfdecomposable distributions are considered, andit is shown that the limit of these nested subclasses is the closure of the class of semi-stable distributions

The subcritical phase for a homopolymer model

We study a model of continuous-time nearest-neighbor random walk on $\mathbb{Z}^d$ penalized by its occupation time at the origin, also known as a homopolymer. For a fixed real parameter $\beta$ and time $t>0$, we consider the probability measure on paths of the random walk starting from the origin whose Radon-Nikodym derivative is proportional to the exponent of the product $\beta$ times the occupation time at the origin up to time $t$. The case $\beta>0$ was studied previously by Cranston and Molchanov arXiv:1508.06915. We consider the case $\beta<0$, which is intrinsically different only when the underlying walk is recurrent, that is $d=1,2$. Our main result is a scaling limit for the distribution of the homopolymer on the time interval $[0,t]$, as $t\to\infty$, a result that coincides with the scaling limit for penalized Brownian motion due to Roynette and Yor. In two dimensions, the penalizing effect is asymptotically diminished, and the homopolymer scales to standard Brownian motion. Our approach is based on potential analytic and martingale approximation for the model. We also apply our main result to recover a scaling limit for a wetting model. We study the model through analysis of resolvents.Comment: 32 page