Itrel Monroe

Bio: Itrel Monroe is an academic researcher. The author has contributed to research in topics: Section (fiber bundle). The author has an hindex of 2, co-authored 2 publications receiving 153 citations.

On Embedding Right Continuous Martingales in Brownian Motion

Abstract: A stopping time $T$ for the Wiener process $W(t)$ is called minimal if there is no stopping time $S \leqq T$ such that $W(S)$ and $W(T)$ have the same distribution. In the first section, it is shown that if $E\{W(T)\} = 0$, then $T$ is minimal if and only if the process $W(t \wedge T)$ is uniformly integrable. Also, if $T$ is minimal and $E\{W(T)\} = 0$ then $E\{T\} = E\{W(T)^2\}$. In the second section, these ideas are used to show that for any right continuous martingale $M(t)$, there is a right continuous family of minimal stopping times $T(t)$ such that $W(T(t))$ has the same finite joint distributions as $M(t)$. In the last section it is shown that if $T$ is defined in the manner proposed by Skorokhod (and therefore minimal) such that $W(T)$ has a stable distribution of index $\alpha > 1$ then $T$ is in the domain of attraction of a stable distribution of index $\alpha/2$.

On the $\gamma$-Variation of Processes with Stationary Independent Increments

Abstract: Let $\{X_t; t \geqq 0\}$ be a stochastic process in $R^N$ defined on the probability space $(\Omega, \mathscr{F}, \mathbf{P})$ which has stationary independent increments. Let $ u$ be the Levy measure for $X_t$ and let $\beta = \inf\{\alpha > 0: \int_{|x| \beta, \mathbf{P}\{\mathbf{V}_\gamma(\mathbf{X}(\bullet, \omega); a, b) < \infty\} = 1$.

The Skorokhod Embedding Problem and Model-Independent Bounds for Option Prices

Abstract: This set of lecture notes is concerned with the following pair of ideas and concepts: 1. The Skorokhod Embedding problem (SEP) is, given a stochastic process X=(X t ) t≥0 and a measure μ on the state space of X, to find a stopping time τ such that the stopped process X τ has law μ. Most often we take the process X to be Brownian motion, and μ to be a centred probability measure. 2. The standard approach for the pricing of financial options is to postulate a model and then to calculate the price of a contingent claim as the suitably discounted, risk-neutral expectation of the payoff under that model. In practice we can observe traded option prices, but know little or nothing about the model. Hence the question arises, if we know vanilla option prices, what can we infer about the underlying model?

La variation d'ordre p des semi-martingales

Abstract: Soient X une semi-martingale, p un nombre reel positif, S = (t i) une subdivision de l'intervalle $$[0,t],\sum {\left| {X_{t_i + 1} - X_{t_i } } \right|} ^p $$ la somme variationnelle correspondante. Le but de ce travail est double: etudier la finitude de la borne superieure de ces sommes variationnelles lorsque S varie dans l'ensemble des subdivisions de [0, t]; en tirer des consequences quant a la convergence en probabilite, en moyenne et presque sure de ces sommes lorsque le pas des subdivisions S tend vers zero, sans qu'elles soient necessairement emboItees. Les methodes font appel aux inegalites de Burkholder a la place des techniques de fonctions caracteristiques employees jusqu'a present pour traiter ces problemes lorsque X est un processus a accroissements independants stationnaires.

The Skorokhod embedding problem and its offspring

Abstract: This is a survey about the Skorokhod embedding problem. It presents all known solutions together with their properties and some applications. Some of the solutions are just described, while others are studied in detail and their proofs are presented. A certain unification of proofs, thanks to real potential theory, is made. Some new facts which appeared in a natural way when different solutions were cross-examined, are reported. Azema and Yor's and Root's solutions are studied extensively. A possible use of the latter is suggested together with a conjecture.

Fluctuation Theory for Lévy Processes

Abstract: Recently there has been renewed interest in fluctuation theory for Levy processes. Inthis brief survey we describe several aspects of this topic, including Wiener-Hopf factorisation,the ladder processes, Spitzer’s condition, the asymptotic behaviour of Levy processes at zero and infinity, and other path properties. Some open problems are also presented.

Martingales in Banach Spaces

Abstract: Introduction Description of the contents 1. Banach space valued martingales 2. Radon Nikodym property 3. Harmonic functions and RNP 4. Analytic functions and ARNP 5. The UMD property for Banach spaces 6. Hilbert transform and UMD Banach spaces 7. Banach space valued H1 and BMO 8. Interpolation methods 9. The strong p-variation of martingales 10. Uniformly convex of martingales 11. Super-reflexivity 12. Interpolation and strong p-variation 13. Martingales and metric spaces 14. Martingales in non-commutative LP *.