DH 1st floor SR

We analyse the effect of a natural change to the time variable on the convergence of the Crank-Nicholson scheme when applied to the solution of the heat equation with Dirac delta function initial conditions. In the original variables, the scheme is known to diverge as the time step is reduced with the ratio (lambda) of the time step to space step held constant - the value of lambda controls how fast the divergence occurs. After introducing the square root of time variable we prove that the numerical scheme for the transformed PDE now always converges and that lambda controls the order of convergence, quadratic convergence being achieved for lambda below a critical value. Numerical results indicate that the time change used with an appropriate value of lambda also results in quadratic convergence for the calculation of gamma for a European call option without the need for Rannacher start-up steps. Finally, some results and analysis are presented for the effect of the time change on the calculation of the option value and greeks for the American put calculated by the penalty method with Crank-Nicholson time-stepping.

The industrial prilling process is amongst the most favourite technique employed in generating monodisperse droplets. In such a process long curved jets are generated from a rotating drum which in turn breakup and from droplets. In this talk we describe the experimental set-up and the theory to model this process. We will consider the effects of changing the rheology of the fluid as well as the addition of surface agents to modify breakup characterstics. Both temporal and spatial instability will be considered as well as nonlinear numerical simulations with comparisons between experiments.

We consider a portfolio optimisation problem on infinite horizon when

the investment policy satisfies the drawdown constraint, which is the

wealth process of an investor is always above a threshold given as a

function of the past maximum of the wealth process. The preferences are

given by a utility function and investor aims to maximise an asymptotic

growth rate of her expected utility of wealth. This problem was firstly

considered by Grossman and Zhou [3] and solved for a Black-Scholes

market and linear drawdown constraint.

The main contribution of the paper is an equivalence result: the

constrained problem with utility U and drawdown function w has the same

value function as the unconstrained problem with utility UoF, where

function F is given explicitly in terms of w. This work was inspired by

ideas from [2], whose results are a special case of our work. We show

that the connection between constrained and unconstrained problems holds

for a much more general setup than their paper, i.e. a general

semimartingale market, larger class of utility functions and drawdown

function which is not necessarily linear. The paper greatly simplifies

previous approaches using the tools of Azema-Yor processes developed in

[1]. In fact we show that the optimal wealth process for constrained

problem can be found as an explicit Azema-Yor transformation of the

optimal wealth process for the unconstrained problem.

We further provide examples with explicit solution for complete and

incomplete markets.

[1] Carraro, L., Karoui, N. E., and Obloj, J. On Azema-Yor processes,

their optimal properties and the Bachelier-Drawdown equation, to appear in

Annals of Probability, 2011.

[2] Cvitanic, J., and Karatzas, I. On portfolio optimization under

drawdown constraints. IMA Volumes in Mathematics and Its Applications

65(3), 1994, 35-45

[3] Grossman, S. J., and Zhou, Z. Optimal investment strategies for

controlling drawdowns. Mathematical Finance 3(3), 1993, 241-276

In this work, we study equilibrium solutions for a LQ

control problem with state-dependent terms in the objective, which

destroy the time-consisitence of a pre-commited optimal solution.

We get a sufficient condition for equilibrium by a system of

stochastic differential equations. When the coefficients in the

problem are all deterministic, we find an explicit equilibrium

for general LQ control problem. For the mean-variance portfolio

selection in a complete financial market, we also get an explicit

equilibrium with random coefficient of the financial.

Cubature on Wiener space" is a numerical method for the weak

approximation of SDEs. After an introduction to this method we present

some cases when the method is computationally expensive, and highlight

some techniques that improve the tractability. In particular, we adapt

the Multilevel Monte-Carlo framework and extend the Milstein-scheme

based version of Mike Giles to higher dimensional and higher degree cases.

: Recently strict local martingales have been used to model

exchange rates. In such models, put-call parity does not hold if one

assumes minimal superreplicating costs as contingent claim prices. I

will illustrate how put-call parity can be restored by changing the

definition of a contingent claim price.

More precisely, I will discuss a change of numeraire technique when the

underlying is only a local martingale. Then, the new measure is not

necessarily equivalent to the old measure. If one now defines the price

of a contingent claim as the minimal superreplicating costs under both

measures, then put-call parity holds. I will discuss properties of this

new pricing operator.

To illustrate this techniques, I will discuss the class of "Quadratic

Normal Volatility" models, which have drawn much attention in the

financial industry due to their analytic tractability and flexibility.

This talk is based on joint work with Peter Carr and Travis Fisher.

When modelling biochemical reactions within cells, it is vitally important to take into account the effect of intrinsic noise in the system, due to the small copy numbers of some of the chemical species. Deterministic systems can give vastly different types of behaviour for the same parameter sets of reaction rates as their stochastic analogues, giving us an incorrect view of the bifurcation behaviour.

The stochastic description of this problem gives rise to a multi-dimensional Markov jump process, which can be approximated by a system of stochastic differential equations. Long-time behaviour of the process can be better understood by looking at the steady-state solution of the corresponding Fokker-Planck equation.

In this talk we consider a new finite element method which uses simulated trajectories of the Markov-jump process to inform the choice of mesh in order to approximate this invariant distribution. The method has been implemented for systems in 3 dimensions, but we shall also consider systems of higher dimension.

1pm Kawei Wang

\newline Title: A Model of Behavioral Consumption in Contnuous Time

\newline Abstract: Inspired by Jin and Zhou (2008), we try to construct a model

of consumption within the framework of Prospect Theory and Cumulative

Prospect Theory in continuous time.

\newline

\newline

1.20 Rasmus Wissmann

\newline Title: A Principal Component Analysis-based Approach for High-Dimensional PDEs in Derivative Pricing

\newline Abstract: Complex derivatives, such as multi asset and path dependent options,

often lead to high-dimensional problems. These are generally hard to

tackle with numerical PDE methods, because the computational effort

necessary increases exponentially with the number of dimensions. We

investigate a Principal Component Analysis-based approach that aims to

make the high-dimensional problem tractable by splitting it into a

number of low-dimensional ones. This is done via a diagonalization of

the PDE according to the eigenvectors of the covariance matrix and a

subsequent Taylor-like approximation. This idea was first introduced by

Reisinger and Wittum for the basic case of a vanilla option on a basket

of stocks [1]. We aim to extend the approach to more complex derivatives

and markets as well as to develop higher order versions. In this talk we

will present the basic ideas, initial results for the example of a

ratchet cap under the LIBOR Market Model and the current plans for

further research.

[1] C. Reisinger and G. Wittum, Efficient Hierarchical Approximation of

High-Dimensional Option Pricing Problems, SIAM Journal of Scientific

Computing, 2007:29

\newline

\newline

1.40 Pedro Vitoria

\newline Title: Infinitesimal Mean-Variance and Forward Utility

\newline Abstract: Mean-Variance, introduced by Markowitz in his seminal paper of 1952, is

a classic criterion in Portfolio Theory that is still predominantly used

today in real investment practice. In the academic literature, a number of

interesting results have been produced in continuous-time version of this

model.

In my talk, I will establish a link between the multi-period

Mean-Variance model and its continuous-time limit. A key feature of the

results is that, under suitable but mild technical conditions, it

captures the results of Forward Utility, thus establishing an important

link between Mean-Variance and forward utility maximisation.