Fudan-Warwick Spring School

Course 1: Simulation of Levy processes (Aleksandar Mijatovic) 15-19 April, 2019

Time and Place:

04.15 Monday 13:30-16:00 HGX304

04.16 Tuesday 13:30-16:00 Guanghua Main Building 1501

04.18 Thursday 9:00-11:30 Guanghua Main Building 1501

04.19 Friday 13:30-16:00 Guanghua Main Building 1801

Abstract: These lectures describe a novel Monte Carlo algorithm for the simulation from the joint law of the position, the running supremum and the time of the supremum of a general Levy process at an arbitrary finite time. We will show that the bias decays geometrically, in contrast to the power law for the random walk approximation (RWA). We will identify the law of the error and, inspired by the recent work of Ivanovs \cite{MR3784492} on RWA, characterise its asymptotic behaviour. We will establish a central limit theorem, construct non-asymptotic and asymptotic confidence intervals and prove that the multilevel Monte Carlo (MLMC) estimator has optimal computational complexity (i.e. of order $\epsilon^{-2}$ if the $L^2$-norm of the error is at most $\epsilon$) for locally Lipschitz and barrier-type functionals of the triplet. If the increments of the L\'{e}vy process cannot be sampled directly, we will combine our algorithm with the Asmussen-Rosi\'nski approximation by choosing the rate of decay of the cutoff level for small jumps so that the corresponding MC and MLMC estimators have minimal computational complexity. We will also describe an unbiased version of our estimator using ideas from Rhee and Glynn. New research directions based on this algorithm will be discussed.

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Course 2: Skorokhod embeddings (David Hobson) 22-26 April, 2019

Time and Place:

04.22 Monday 9:00-11:30 Guanghua Main Building 1501

04.22 Monday 13:30-16:00 HGX304

04.23 Tuesday 9:00-11:30 Guanghua Main Building 1501

04.23 Tuesday 13:30-16:00 Guanghua Main Building 1501

Abstract: Let $X= (X_t)_{t \geq 0}$ be a stochastic process on a state space $E$ and let $\mu$ be a measure on $E$. The Skorokhod embedding problem (SEP) is to construct a stopping time (where possible) such that the law of $X_\tau$ is $\mu$. The classical case is when $X$ is Brownian motion, null at zero, $E$ is the real numbers and $\mu$ is a centred, square integrable probability measure. In this set of talks we will discuss some classical solutions to the SEP and extensions to other processes, other classes of target laws, and processes with non-trivial starting law. In addition we will discuss applications including applications to mathematical finance.

Course schedule:

Lecture 1: Introduction to Skorokhod embeddings:

Statement of the general problem, review of properties of Brownian motion, some first embeddings and properties of good solutions.

Lecture 2: Classical Skorokhod embeddings