Black-Scholes option valuation for scientific computing students

Mathematical finance forms a modern, attractive source of examples and case studies for classes in scientific computation. I will show here how the Nobel Prize winning Black-Scholes option valuation theory can be used to motivate exercises in Monte Carlo simulation, matrix computation and numerical methods for partial differential equations

A matrix perturbation view of the small world phenomenon

We use techniques from applied matrix analysis to study small world cutoff in a Markov chain. Our model consists of a periodic random walk plus uniform jumps. This has a direct interpretation as a teleporting random walk, of the type used by search engines to locate web pages, on a simple ring network. More loosely, the model may be regarded as an analogue of the original small world network of Watts and Strogatz [Nature, 393 (1998), pp. 440-442]. We measure the small world property by expressing the mean hitting time, averaged over all states, in terms of the expected number of shortcuts per random walk. This average mean hitting time is equivalent to the expected number of steps between a pair of states chosen uniformly at random. The analysis involves nonstandard matrix perturbation theory and the results come with rigorous and sharp asymptotic error estimates. Although developed in a different context, the resulting cutoff diagram agrees closely with that arising from the mean-field network theory of Newman, Moore, and Watts [Phys. Rev. Lett., 84 (2000), pp. 3201-3204]

Chemical master versus chemical langevin for first-order reaction networks

Markov jump processes are widely used to model interacting species in circumstances where discreteness and stochasticity are relevant. Such models have been particularly successful in computational cell biology, and in this case, the interactions are typically rst-order. The Chemical Langevin Equation is a stochastic dierential equation that can be regarded as an approximation to the underlying jump process. In particular, the Chemical Langevin Equation allows simulations to be performed more eectively. In this work, we obtain expressions for the rst and second moments of the Chemical Langevin Equation for a generic rst-order reaction network. Moreover, we show that these moments exactly match those of the under-lying jump process. Hence, in terms of means, variances and correlations, the Chemical Langevin Equation is an excellent proxy for the Chemical Master Equation. Our work assumes that a unique solution exists for the Chemical Langevin Equation. We also show that the moment matching re- sult extends to the case where a gene regulation model of Raser and O'Shea (Science, 2004) is replaced by a hybrid model that mixes elements of the Master and Langevin equations. We nish with numerical experiments on a dimerization model that involves second order reactions, showing that the two regimes continue to give similar results

Mathematical and computational modelling of post-transcriptional gene relation by micro-RNA

Mathematical models and computational simulations have proved valuable in many areas of cell biology, including gene regulatory networks. When properly calibrated against experimental data, kinetic models can be used to describe how the concentrations of key species evolve over time. A reliable model allows ‘what if’ scenarios to be investigated quantitatively in silico, and also provides a means to compare competing hypotheses about the underlying biological mechanisms at work. Moreover, models at different scales of resolution can be merged into a bigger picture ‘systems’ level description. In the case where gene regulation is post-transcriptionally affected by microRNAs, biological understanding and experimental techniques have only recently matured to the extent that we can postulate and test kinetic models. In this chapter, we summarize some recent work that takes the first steps towards realistic modelling, focusing on the contributions of the authors. Using a deterministic ordinary differential equation framework, we derive models from first principles and test them for consistency with recent experimental data, including microarray and mass spectrometry measurements. We first consider typical mis-expression experiments, where the microRNA level is instantaneously boosted or depleted and thereafter remains at a fixed level. We then move on to a more general setting where the microRNA is simply treated as another species in the reaction network, with microRNA-mRNA binding forming the basis for the post-transcriptional repression. We include some speculative comments about the potential for kinetic modelling to contribute to the more widespread sequence and network based approaches in the qualitative investigation of microRNA based gene regulation. We also consider what new combinations of experimental data will be needed in order to make sense of the increased systems-level complexity introduced by microRNAs

The sleekest link algorithm

How does Google decide which web sites are important? It uses an ingenious algorithm that exploits the structure of the web and is resistant to hacking. Here, we describe this PageRank algorithm, illustrate it by example, and show how it can be interpreted as a Jacobi iteration and a teleporting random walk. We also ask the algorithm to rank the undergraduate mathematics classes offered at the University of Strathclyde. PageRank draws upon ideas from linear algebra, graph theory and stochastic processes, and it throws up research-level challenges in scientific computing. It thus forms an exciting and modern application area that could brighten up many a mathematics class syllabus

Zero, one and two-switch models of gene regulation

We compare a hierarchy of three stochastic models in gene regulation. In each case, genes produce mRNA molecules which in turn produce protein. The simplest model, as described by Thattai and Van Oudenaarden (Proc. Nat. Acad. Sci., 2001), assumes that a gene is always active, and uses a first-order chemical kinetics framework in the continuous-time, discrete-space Markov jump (Gillespie) setting. The second model, proposed by Raser and O'Shea (Science, 2004), generalizes the first by allowing the gene to switch randomly between active and inactive states. Our third model accounts for other effects, such as the binding/unbinding of a transcription factor, by using two independent on/off switches operating in AND mode. We focus first on the noise strength, which has been defined in the biological literature as the ratio of the variance to the mean at steady state. We show that the steady state variance in the mRNA and protein for the three models can either increase or decrease when switches are incorporated, depending on the rate constants and initial conditions. Despite this, we also find that the overall noise strength is always greater when switches are added, in the sense that one or two switches are always noisier than none. On the other hand, moving from one to two switches may either increase or decrease the noise strength

Modeling and simulating chemical reactions

Many students are familiar with the idea of modeling chemical reactions in terms of ordinary differential equations. However, these deterministic reaction rate equations are really a certain large-scale limit of a sequence of finer-scale probabilistic models. In studying this hierarchy of models, students can be exposed to a range of modern ideas in applied and computational mathematics. This article introduces some of the basic concepts in an accessible manner and points to some challenges that currently occupy researchers in this area. Short, downloadable MATLAB codes are listed and described

Multi-level Monte Carlo for continuous time Markov chains, with applications in biochemical kinetics

We show how to extend a recently proposed multi-level Monte Carlo approach to the continuous time Markov chain setting, thereby greatly lowering the computational complexity needed to compute expected values of functions of the state of the system to a specified accuracy. The extension is non-trivial, exploiting a coupling of the requisite processes that is easy to simulate while providing a small variance for the estimator. Further, and in a stark departure from other implementations of multi-level Monte Carlo, we show how to produce an unbiased estimator that is significantly less computationally expensive than the usual unbiased estimator arising from exact algorithms in conjunction with crude Monte Carlo. We thereby dramatically improve, in a quantifiable manner, the basic computational complexity of current approaches that have many names and variants across the scientific literature, including the Bortz-Kalos-Lebowitz algorithm, discrete event simulation, dynamic Monte Carlo, kinetic Monte Carlo, the n-fold way, the next reaction method,the residence-time algorithm, the stochastic simulation algorithm, Gillespie's algorithm, and tau-leaping. The new algorithm applies generically, but we also give an example where the coupling idea alone, even without a multi-level discretization, can be used to improve efficiency by exploiting system structure. Stochastically modeled chemical reaction networks provide a very important application for this work. Hence, we use this context for our notation, terminology, natural scalings, and computational examples.Comment: Improved description of the constants in statement of Theorem