Embedding laws in diffusions by functions of time

We present a constructive probabilistic proof of the fact that if $B=(B_t)_{t\ge0}$ is standard Brownian motion started at $0$, and $\mu$ is a given probability measure on $\mathbb{R}$ such that $\mu(\{0\})=0$, then there exists a unique left-continuous increasing function $b:(0,\infty)\rightarrow\mathbb{R}\cup\{+\infty\}$ and a unique left-continuous decreasing function $c:(0,\infty)\rightarrow\mathbb{R}\cup\{-\infty\}$ such that $B$ stopped at $\tau_{b,c}=\inf\{t>0\vert B_t\ge b(t)$ or $B_t\le c(t)\}$ has the law $\mu$. The method of proof relies upon weak convergence arguments arising from Helly's selection theorem and makes use of the L\'{e}vy metric which appears to be novel in the context of embedding theorems. We show that $\tau_{b,c}$ is minimal in the sense of Monroe so that the stopped process $B^{\tau_{b,c}}=(B_{t\wedge\tau_{b,c}})_{t\ge0}$ satisfies natural uniform integrability conditions expressed in terms of $\mu$. We also show that $\tau_{b,c}$ has the smallest truncated expectation among all stopping times that embed $\mu$ into $B$. The main results extend from standard Brownian motion to all recurrent diffusion processes on the real line.Comment: Published at http://dx.doi.org/10.1214/14-AOP941 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

An Optimal Skorokhod Embedding for Diffusions

Given a Brownian motion $B_t$ and a general target law $\mu$ (not necessarily centered or even integrable) we show how to construct an embedding of $\mu$ in $B$. This embedding is an extension of an embedding due to Perkins, and is optimal in the sense that it simultaneously minimises the distribution of the maximum and maximises the distribution of the minimum among all embeddings of $\mu$. The embedding is then applied to regular diffusions, and used to characterise the target laws for which a $H^p$-embedding may be found.Comment: 22 pages, 4 figure

Classes of Skorokhod Embeddings for the Simple Symmetric Random Walk

The Skorokhod Embedding problem is well understood when the underlying process is a Brownian motion. We examine the problem when the underlying is the simple symmetric random walk and when no external randomisation is allowed. We prove that any measure on Z can be embedded by means of a minimal stopping time. However, in sharp contrast to the Brownian setting, we show that the set of measures which can be embedded in a uniformly integrable way is strictly smaller then the set of centered probability measures: specifically it is a fractal set which we characterise as an iterated function system. Finally, we define the natural extension of several known constructions from the Brownian setting and show that these constructions require us to further restrict the sets of target laws

Model-independent pricing with insider information: a Skorokhod embedding approach

In this paper, we consider the pricing and hedging of a financial derivative for an insider trader, in a model-independent setting. In particular, we suppose that the insider wants to act in a way which is independent of any modelling assumptions, but that she observes market information in the form of the prices of vanilla call options on the asset. We also assume that both the insider's information, which takes the form of a set of impossible paths, and the payoff of the derivative are time-invariant. This setup allows us to adapt recent work of Beiglboeck, Cox and Huesmann (2016) to prove duality results and a monotonicity principle, which enables us to determine geometric properties of the optimal models. Moreover, we show that this setup is powerful, in that we are able to find analytic and numerical solutions to certain pricing and hedging problems

Theory of adiabatic Hexaamminecobalt-Self-Exchange

We have reexamined the thermally induced Co(NH_3)_6^{2+/3+} [Co(II/III)] redox reaction using the first principles density-functional-theory method, semiclassical Marcus theory, and known charge transfer parameters. We confirm a previously suggested mechanism involving excited state (^2E_g) of Co(II) which becomes lower than the ground state (^4T_1g) in the transition state region. This lowers the transition state barrier considerably by about 6.9 kcal/mol and leads to a spin-allowed and adiabatic electron exchange process. Our calculations are consistent with previous experimental results regarding the spin-excitation energy (^3T_1g) of Co(III), and the fact that an optical absorption peak (^2E_g) of the Co(II) species could not be found experimentally. Our rate is of order 6 10^{-3} 1/Ms and hence 2 orders of magnitude faster than determined previously by experiments.Comment: 10 pages, 5 figures, 4 tables; submitted to J.Chem.Phy

The southern dust pillars of the Carina Nebula

We present preliminary results from a detailed study towards four previously detected bright mid-infrared sources in the southern part of the Carina Nebula: G287.73--0.92, G287.84--0.82, G287.93--0.99 and G288.07--0.80. All of these sources are located at the heads of giant dust pillars that point towards the nearby massive star cluster, Trumpler 16. It is unclear if these pillars are the prime sites for a new generation of triggered star formation or if instead they are the only remaining parts of the nebula where ongoing star fromation can take place.Comment: 2 pages, to appear in the proceedings of "Hot Star Workshop III: The Earliest Phases of Massive Star Birth" (ed. P.A. Crowther