In this Chapter we show how general results in Chapters 3, 4 and 6 can sometimes be strengthened when symmetry is present. Many of the ideas are just simple observations. Since the topic has a \discrete math" avor our default convention is to work in discrete time, though as always the continuous-time case is similar. Note that we use the word \symmetry" in the sense of spatial symmetry (which ...

We introduce a multi-coin discrete quantum random walk where the amplitude for a coin flip depends upon previous tosses. Although the corresponding classical random walk is unbiased, a bias can be introduced into the quantum walk by varying the history dependence. By mixing the biased random walk with an unbiased one, the direction of the bias can be reversed leading to a new quantum version of...

Quantum walks have recently been introduced and investigated, with the hope that they may be useful in constructing new efficient quantum algorithms. For reviews of quantum walks, see Refs. [4, 16, 24]. There are two distinct types of the quantum walk: one is a discrete-time case [2, 5, 12, 14, 17, 18, 19, 22, 23], the other is a continuous-time case [1, 3, 7, 8, 10, 15, 20]. The quantum walk c...

We prove an invariance principle for the bridge of a random walk conditioned to stay positive, when the random walk is in the domain of attraction of a stable law, both in the discrete and in the absolutely continuous setting. This includes as a special case the convergence under diffusive rescaling of random walk excursions toward the normalized Brownian excursion, for zero mean, finite varian...

In this note we propose a numerical method to approximate the solution of a Backward Stochastic Differential Equations with Jumps (BSDEJ). This method is based on the construction of a discrete BSDEJ driven by a complete system of three orthogonal discrete time-space martingales, the first a random walk converging to a Brownian motion; the second, another random walk, independent of the first o...