We study two stochastic process models for loss-carried-forward taxation on the wealth of a large company. The models are based upon a Lévy process $X$, which represents the wealth of the company without taxation, and both are defined such that tax is applied only as the wealth of the company increases. In one process, denoted $U$, the tax rate depends on the value of the underlying Lévy process $X$ at the time that tax is taken out, whereas in the other process, say $V$, the tax rate depends on the value of the taxed process $V$ itself. We show the existence of the process $V$, under simple conditions, and demonstrate that there is a correspondence between the two types of taxation. Using this, we find simple expression for the net present value of tax paid until ruin of the company, among other functionals. We also give some applications to our results, which help to clarify the existing literature on processes with tax.

We are interested in the following singular forward stochastic differential equation (SDE) $$ d X_t = b(t , X_t, \omega ) d t + \sigma dBt, ~~ 0 \leq t \leq T, ~~ X_0 = x \in \mathbb{R},$$ where the coeffcient $b : [0, T]\times\mathbb{R}\times\Omega \to \mathbb{R}$ is Borel measurable and of linear growth in the second variable and is adapted and $\sigma \in \mathbb{R}^d$. The driving noise $B_t$ is a $d$-dimensional Brownian motion. We obtain the existence and uniqueness of a strong solution in the present situation where the drift is not necessarily deterministic. The method is purely probabilitic and relies on Malliavin calculus. As a byproduct, we obtain Malliavin differentiability of the solutions and provide an explicit representation and moment estimates for the Malliavin derivative.

This talk is based on join works with Ludovic Tangpi, University of Vienna.

We bring a probabilistic approach to a particular question in the Ising model. We fix the total magnetisation of the model, and look at the distribution of the resulting shape of the boundary between areas with positive and negative spins. The Law of Large Numbers and results from random graph models tell us that, on the $n^{-1}$ scale, we will see one giant component, with an approximately circular shape. We use the theory of random walks to find an equivalent to the Central Limit Theorem describing the fluctuations, on an $n^{-1/2}$ scale, away from this limiting shape.

The model is the Shepp statistics, one-dimensional version of which is defined as the maximium of increments of Brownian motion within finite time interval. The distribution of a finite time horizon maximum of a Brownian motion is known explicitly while the distribution of the Shepp statistics is not available in a closed form. In our talk we first discuss how to approximate the tail distribution of the Shepp statistics in one-dimensional case and then we proceed to multi-dimensional processes. As a by-product we derive a new inequality for the `supremum’ of vector-valued Brownian motion.

Mostly based on a joint work with K. Debicki, E. Hashorva and L. Wang.