The Multifractal Nature of Volterra-L\'{e}vy Processes

We consider the regularity of sample paths of Volterra-L\'{e}vy processes. These processes are defined as stochastic integrals M(t)=\int_{0}^{t}F(t,r)dX(r), \ \ t \in \mathds{R}_{+}, where $X$ is a L\'{e}vy process and $F$ is a deterministic real-valued function. We derive the spectrum of singularities and a result on the 2-microlocal frontier of $\{M(t)\}_{t\in [0,1]}$, under regularity assumptions on the function $F$.Comment: 21 pages, Stochastic Processes and their Applications, 201

On the Maximal Displacement of Subcritical Branching Random Walks

We study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin. For each $n\in\mathbb{N},$ let $M_{n}$ be the rightmost position reached by the branching random walk up to generation $n$. Under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function, we show that there exists $\rho>1$ such that the function g(c,n):=\rho ^{cn} P(M_{n}\geq cn), \quad \mbox{for each }c>0 \mbox{ and } n\in\mathbb{N}, satisfies the following properties: there exist $0<\underline{\delta}\leq \overline{\delta} < {\infty}$ such that if $c<\underline{\delta}$, then $$0<\liminf_{n\rightarrow\infty} g (c,n)\leq \limsup_{n\rightarrow\infty} g (c,n) {\leq 1},$$ while if $c>\overline{\delta}$, then $$\lim_{n\rightarrow\infty} g (c,n)=0.$$ Moreover, if the jump distribution has a finite right range $R$, then $\overline{\delta} < R$. If furthermore the jump distribution is "nearly right-continuous", then there exists $\kappa\in (0,1]$ such that $\lim_{n\rightarrow \infty}g(c,n)=\kappa$ for all $c<\underline{\delta}$. We also show that the tail distribution of $M:=\sup_{n\geq 0}M_{n}$, namely, the rightmost position ever reached by the branching random walk, has a similar exponential decay (without the cutoff at $\underline{\delta}$). Finally, by duality, these results imply that the maximal displacement of supercritical branching random walks conditional on extinction has a similar tail behavior.Comment: 29 page

Optimal Portfolio Liquidation in Target Zone Models and Catalytic Superprocesses

We study optimal buying and selling strategies in target zone models. In these models the price is modeled by a diffusion process which is reflected at one or more barriers. Such models arise for example when a currency exchange rate is kept above a certain threshold due to central bank intervention. We consider the optimal portfolio liquidation problem for an investor for whom prices are optimal at the barrier and who creates temporary price impact. This problem will be formulated as the minimization of a cost-risk functional over strategies that only trade when the price process is located at the barrier. We solve the corresponding singular stochastic control problem by means of a scaling limit of critical branching particle systems, which is known as a catalytic superprocess. In this setting the catalyst is a set of points which is given by the barriers of the price process. For the cases in which the unaffected price process is a reflected arithmetic or geometric Brownian motion with drift, we moreover give a detailed financial justification of our cost functional by means of an approximation with discrete-time models.Comment: 16 pages, 2 figure