The solution of the perturbed Tanaka-equation is pathwise unique

The Tanaka equation $dX_t={\operatorname{sign}}(X_t)\,dB_t$ is an example of a stochastic differential equation (SDE) without strong solution. Hence pathwise uniqueness does not hold for this equation. In this note we prove that if we modify the right-hand side of the equation, roughly speaking, with a strong enough additive noise, independent of the Brownian motion B, then the solution of the obtained equation is pathwise unique.Comment: Published in at http://dx.doi.org/10.1214/11-AOP716 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Detecting Curved Objects Against Cluttered Backgrounds

Detecting curved objects against cluttered backgrounds is a hard problem in computer vision. We present new low-level and mid-level features to function in these environments. The low-level features are fast to compute, because they employ an integral image approach, which makes them especially useful in real-time applications. The mid-level features are built from low-level features, and are optimized for curved object detection. The usefulness of these features is tested by designing an object detection algorithm using these features. Object detection is accomplished by transforming the mid-level features into weak classifiers, which then produce a strong classifier using AdaBoost. The resulting strong classifier is then tested on the problem of detecting heads with shoulders. On a database of over 500 images of people, cropped to contain head and shoulders, and with a diverse set of backgrounds, the detection rate is 90% while the false positive rate on a database of 500 negative images is less than 2%

Skew-Unfolding the Skorokhod Reflection of a Continuous Semimartingale

The Skorokhod reflection of a continuous semimartingale is unfolded, in a possibly skewed manner, into another continuous semimartingale on an enlarged probability space according to the excursion-theoretic methodology of Prokaj (2009). This is done in terms of a skew version of the Tanaka equation, whose properties are studied in some detail. The result is used to construct a system of two diffusive particles with rank-based characteristics and skew-elastic collisions. Unfoldings of conventional reflections are also discussed, as are examples involving skew Brownian Motions and skew Bessel processes.Comment: 20 pages. typos corrected, added a remark after Proposition 2.3, simplified the last part of Example 2.

Shadow price in the power utility case

We consider the problem of maximizing expected power utility from consumption over an infinite horizon in the Black-Scholes model with proportional transaction costs, as studied in Shreve and Soner [Ann. Appl. Probab. 4 (1994) 609-692]. Similar to Kallsen and Muhle-Karbe [Ann. Appl. Probab. 20 (2010) 1341-1358], we derive a shadow price, that is, a frictionless price process with values in the bid-ask spread which leads to the same optimal policy.Comment: Published at http://dx.doi.org/10.1214/14-AAP1058 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org