This page lists various projects during my Masters degrees (2010-2012) at University of Warwick.

### RSK correspondence and Pitman’s Theorem

My MASDOC MSc dissertation, supervised by Professor Neil O’Connell, is a survey focused on a variety of RSK correspondences in combinatorics and Pitman’s theorems in probability and their interconnections. The dissertation can be downloaded here.

### Simulating spiral waves on spherical surfaces using surface finite element method

This is the RSG implementation stage project I worked on. This is joint work with Amal Alphonse and Simon Bignold supervised by Professor Dwight Barkley and Dr Andreas Dedner. The objective is to simulate the following equation:

\begin{align*} \dot u + u\nabla_\Gamma\cdot\mathbf{v} - a\Delta_\Gamma u &= \frac{1}{\epsilon}u(1-u)(u-\frac{v+b}{c}); \text{ in }\mathcal{G}_t:= \bigcup _{t\in[0,T]}\Gamma_t\times\{t\}\\ \dot v+v\nabla_\Gamma\cdot\mathbf{v}&=u-v, \text{ in }\mathcal{G}_t \\ u(\cdot,0)&=u_0,\quad v(\cdot,0)=v_0,\text{ in }\Gamma_0.\end{align*}

where \(\dot u\) and \(\dot v\) denotes the material derivative of \(u\) and \(v\) and \(\mathbf{v}\) is the surface velocity. We implement (using Distributed and Unified Numerics Environment, DUNE) this equation on both fixed and deforming surfaces. Below is a simulation on a deforming surface and a simulation of pulse,

The details of these simulation can be found on the project webpage. You can also download the research report here.

### Path-dependent option pricing using Monte Carlo method

This is the RSG (Research Study Group) proposal stage topic I worked on. For definition and description of RSG see here and here (Warwick login required). This is joint work with Owen Daniel and Barnaby Garrod, supervised by Dr Aleksandar Mijatović. The objective is to compute the price \(V(0)\) of European path-dependent options like Asian, lookback or Parisian options:

\begin{equation*} V(0)=\mathbb{E}_xe^{-rT}V(X,T),\end{equation*}

where the \(V(\cdot,T)\) is a functional on \(C([0,T])\) and \(\mathbb{E}_x\) is the risk-neutral measure starting with \(x\). In finance, \(X\) stands for underlying price of the option, \(V(X,T)\) is the payoff of the option at expiry \(T\). We proposed direct sampling and an importance sampling method to approximate the law of \((X_{t})_{0\le t\le T}\), and hence to approximate the expectation. The proposal can be downloaded here, whose implementation was passed to another group whose members are Chin Lun, Abhishek Shukla and Matthew Thorpe.

### Stochastic travelling wave solutions of stochastic reaction-diffusion equations

This is my first masters project, supervised by Dr Roger Tribe. Consider two similar 1-dimensional stochastic parabolic equations:

\[ du_{t}=\frac{d^2u_{t}}{dx^2}dt+f(u_{t})dt+g(u_{t})\circ dB_{t}.\qquad(1) \] \[ du_{t}=\frac{d^2u_{t}}{dx^2}dt+f(u_{t})dt+g(u_{t})dW_{t}(x).\qquad(2) \]

Where \(f,g\in C^3[0,1]\) with \(f(0)=f(1)=g(0)=g(1)=0\), the noise \(dB_{t}\) is a Brownian motion noise and \(dW_{t}(x)\) is space-time white noise. Equation (1) has stochastic travelling wave solutions (Tribe and Woodward 2011) when \(f\) is stable(which means \(f>0\) on \((0,1)\)), bistable(\(f<0\) on \((0,a)\) and \(f>0\) on \((a,1)\) for some \(a\)) or unstable(\(f>0\) on \((0,a)\) and \(f<0\) on \((a,1)\) for some \(a\)). My objective was to find the asymptotic wave speeds of stochastic travelling wave solutions of (1) and (2) and to prove they are independent of \(g\).