D'Math University | Education & Specialist
MSc Stochastic Processes
An advanced postgraduate programme in the theory and applications of stochastic processes — the mathematics of randomness in continuous time. From Brownian motion and martingales to Itô calculus and stochastic differential equations, graduates develop the specialist expertise required for careers in finance, insurance, data science, and mathematical research.
Postgraduate
1 Year
Online & Blended
Specialist Mathematics
12
Core Modules
£65k
Average Graduate Salary
50+
Partner Universities
1
Year Programme
Programme Overview
Programme Overview
- Rigorous postgraduate programme in the mathematics of random phenomena in continuous time
- Semester 1: Measure-theoretic probability, Markov chains, Poisson processes, and martingale theory
- Semester 2: Brownian motion, Itô stochastic calculus, SDEs, and applications in finance and biology
- Filtering theory, Markov decision processes, and stochastic optimal control
- Computational simulation of stochastic processes using Python and R
- Dissertation on a research or applied problem in stochastic processes
- Strong foundation for PhD study in probability, statistics, mathematical finance, or machine learning
Entry Requirements
- BSc in Mathematics or Statistics (2:1 or above)
- Strong background in probability theory, real analysis, and linear algebra
- Prior exposure to measure-theoretic probability advantageous
- Programming experience in Python or R preferred
- Applicants from physics, engineering, or quantitative finance welcome with equivalent preparation
- Two academic references
- English proficiency: IELTS 6.5+ or equivalent
Core Curriculum
Measure-Theoretic Probability
Probability spaces, random variables, expectations, conditional expectation, convergence theorems, and independence.
Markov Chains
Discrete and continuous-time Markov chains, classification of states, stationary distributions, and ergodic theorems.
Martingale Theory
Filtrations, stopping times, optional sampling theorem, Doob's martingale convergence theorem, and applications.
Brownian Motion
Construction and properties of Brownian motion, quadratic variation, reflection principle, and Donsker's theorem.
Itô Stochastic Calculus
Itô integral, Itô's formula, Girsanov's theorem, Feynman-Kac formula, and representations of martingales.
Stochastic Differential Equations
Existence and uniqueness of SDE solutions, numerical methods for SDEs, diffusion processes, and applications.
Applications: Finance & Biology
Black-Scholes model, interest rate models, population dynamics, and epidemic modelling via stochastic processes.
Dissertation
Original research project on a theoretical or applied aspect of stochastic processes, supervised by a specialist faculty member.
Course Catalogue
Click any course to view its objective and learning outcomes.
STP 501 Probability Theory +
Objective
To establish measure-theoretic probability rigorously.
Learning Outcomes
- Apply Lebesgue measure and integration.
- Use convergence theorems.
- Apply Radon-Nikodym theorem.
- Use Lp spaces.
- Apply almost sure convergence.
Interactive Activity — Distribution Plotter
Pick a distribution and adjust its parameters. Read off mean and variance directly from the plot.
Distribution:
p1 =
0.0
p2 =
1.0
Interactive Activity — Central Limit Theorem Simulator
Sample n values, take their average, repeat. The histogram of averages converges to a normal distribution — CLT in action.
Source:
Sample size n =
10
Total sample means: 0
STP 502 Discrete-Time Markov Chains +
Objective
To analyse Markov chains in discrete time.
Learning Outcomes
- Apply transition matrices.
- Compute stationary distributions.
- Use ergodic theorems.
- Apply MCMC.
- Analyse hidden Markov models.
Interactive Activity — Distribution Plotter
Pick a distribution and adjust its parameters. Read off mean and variance directly from the plot.
Distribution:
p1 =
0.0
p2 =
1.0
Interactive Activity — Central Limit Theorem Simulator
Sample n values, take their average, repeat. The histogram of averages converges to a normal distribution — CLT in action.
Source:
Sample size n =
10
Total sample means: 0
Interactive Activity — Bayesian Coin Update
Beta(α, β) prior on a coin's bias. Click "Toss" to flip a coin (true bias hidden) and watch the posterior update via Bayes' rule.
prior α =
1.0
prior β =
1.0
true p =
Toss the coin to start updating the posterior.
STP 503 Continuous-Time Markov Chains +
Objective
To analyse Markov chains in continuous time.
Learning Outcomes
- Apply Q-matrices.
- Compute holding times.
- Use Kolmogorov equations.
- Apply queueing models.
- Analyse birth-death processes.
Interactive Activity — Distribution Plotter
Pick a distribution and adjust its parameters. Read off mean and variance directly from the plot.
Distribution:
p1 =
0.0
p2 =
1.0
Interactive Activity — Central Limit Theorem Simulator
Sample n values, take their average, repeat. The histogram of averages converges to a normal distribution — CLT in action.
Source:
Sample size n =
10
Total sample means: 0
Interactive Activity — Bayesian Coin Update
Beta(α, β) prior on a coin's bias. Click "Toss" to flip a coin (true bias hidden) and watch the posterior update via Bayes' rule.
prior α =
1.0
prior β =
1.0
true p =
Toss the coin to start updating the posterior.
STP 504 Brownian Motion +
Objective
To study Brownian motion and its properties.
Learning Outcomes
- Construct Brownian motion.
- Apply martingale properties.
- Use reflection principle.
- Apply scaling and Lévy's theorem.
- Analyse local time.
Interactive Activity — Distribution Plotter
Pick a distribution and adjust its parameters. Read off mean and variance directly from the plot.
Distribution:
p1 =
0.0
p2 =
1.0
Interactive Activity — Central Limit Theorem Simulator
Sample n values, take their average, repeat. The histogram of averages converges to a normal distribution — CLT in action.
Source:
Sample size n =
10
Total sample means: 0
Interactive Activity — Bayesian Coin Update
Beta(α, β) prior on a coin's bias. Click "Toss" to flip a coin (true bias hidden) and watch the posterior update via Bayes' rule.
prior α =
1.0
prior β =
1.0
true p =
Toss the coin to start updating the posterior.
Interactive Activity — Random Walk & Brownian Motion
Run a 1D or 2D random walk, or simulate continuous Brownian motion. Observe diffusion behaviour and the √n scaling of displacement.
Type:
Steps:
500
STP 505 Itô Calculus +
Objective
To develop stochastic calculus for Brownian-driven processes.
Learning Outcomes
- Apply Itô's lemma.
- Solve SDEs.
- Apply Girsanov's theorem.
- Use Feynman-Kac.
- Apply martingale representation.
Interactive Activity — Derivative as Slope of Tangent
Drag the slider to move point P along the curve. The tangent line updates — its slope is the derivative.
f(x):
x =
1.00
Interactive Activity — Riemann Sum Approximation
Drag the slider to add more rectangles. Watch the approximation converge to the true integral.
Rectangles n =
8
Interactive Activity — Bayesian Coin Update
Beta(α, β) prior on a coin's bias. Click "Toss" to flip a coin (true bias hidden) and watch the posterior update via Bayes' rule.
prior α =
1.0
prior β =
1.0
true p =
Toss the coin to start updating the posterior.
STP 506 Lévy Processes +
Objective
To study processes with stationary independent increments.
Learning Outcomes
- Apply Lévy-Khintchine formula.
- Classify Lévy processes.
- Use jump processes.
- Apply stable processes.
- Use compound Poisson.
STP 507 Martingale Theory +
Objective
To apply martingale methods rigorously.
Learning Outcomes
- Apply martingale convergence.
- Use Doob's inequalities.
- Apply optional stopping.
- Use Burkholder-Davis-Gundy.
- Apply martingale CLT.
STP 508 Ergodic Theory +
Objective
To study long-time average behaviour of dynamical systems.
Learning Outcomes
- Apply Birkhoff ergodic theorem.
- Use mixing properties.
- Apply entropy of dynamical systems.
- Use Kolmogorov-Sinai entropy.
- Apply ergodic theorems to MCMC.
STP 509 Queueing Theory +
Objective
To model service systems with random arrivals.
Learning Outcomes
- Apply M/M/1 and M/M/c queues.
- Use Erlang formulas.
- Apply queueing networks.
- Use Jackson networks.
- Apply queues to systems design.
Interactive Activity — 2×2 Matrix Transformation
Set the entries of a 2×2 matrix. Watch how it transforms the unit square. Determinant = signed area of the transformed square.
a =
1.0
b =
0.5
c =
-0.3
d =
1.0
Interactive Activity — Bayesian Coin Update
Beta(α, β) prior on a coin's bias. Click "Toss" to flip a coin (true bias hidden) and watch the posterior update via Bayes' rule.
prior α =
1.0
prior β =
1.0
true p =
Toss the coin to start updating the posterior.
Interactive Activity — Random Walk & Brownian Motion
Run a 1D or 2D random walk, or simulate continuous Brownian motion. Observe diffusion behaviour and the √n scaling of displacement.
Type:
Steps:
500
STP 510 Renewal Theory +
Objective
To study renewal processes and their applications.
Learning Outcomes
- Apply renewal equation.
- Use Blackwell's theorem.
- Apply key renewal theorem.
- Use regenerative processes.
- Apply renewal-reward processes.
STP 511 Stochastic Differential Equations +
Objective
To solve and analyse SDEs rigorously.
Learning Outcomes
- Apply existence-uniqueness theorems.
- Solve linear SDEs.
- Use Euler-Maruyama scheme.
- Apply Milstein scheme.
- Analyse SDE stability.
Interactive Activity — Distribution Plotter
Pick a distribution and adjust its parameters. Read off mean and variance directly from the plot.
Distribution:
p1 =
0.0
p2 =
1.0
Interactive Activity — Central Limit Theorem Simulator
Sample n values, take their average, repeat. The histogram of averages converges to a normal distribution — CLT in action.
Source:
Sample size n =
10
Total sample means: 0
STP 512 Master's Dissertation +
Objective
To complete an original stochastic processes research project.
Learning Outcomes
- Identify a research-quality problem.
- Apply rigorous methodology.
- Survey relevant literature.
- Write a 15,000-word dissertation.
- Defend orally.
Career Pathways
Quantitative Analyst
Apply Itô calculus and SDE models to derivatives pricing, volatility modelling, and risk management at investment banks.
ML Research Scientist
Use stochastic process theory to develop diffusion models, Gaussian processes, and probabilistic machine learning systems.
Actuarial Analyst
Model insurance claim processes, ruin theory, and reserve calculations using advanced stochastic modelling techniques.
Biostatistician
Apply stochastic models to survival analysis, disease dynamics, genetic drift, and longitudinal clinical trial data.
Academic Probabilist
Pursue doctoral research and an academic career advancing the theory of stochastic processes at leading mathematics departments.
Data Science Researcher
Develop principled probabilistic models for uncertainty quantification, Bayesian deep learning, and Gaussian process regression.
Top Global Universities
ETH Zürich
University of Cambridge
University of Oxford
Courant Institute (NYU)
Imperial College London
University of Warwick
Paris VI (Sorbonne)
Stanford University
University of Toronto
University of Edinburgh
Why D'Math University
STEP 01
Full Mathematical Rigour
We teach stochastic processes from measure-theoretic foundations — the level expected for research, financial mathematics, and advanced data science.
STEP 02
Broad Applications
The curriculum covers applications in quantitative finance, mathematical biology, machine learning, and physics — preparing graduates for diverse careers.
STEP 03
Computational Component
Simulation of stochastic processes, Monte Carlo methods, and numerical SDE solvers are integrated throughout — not just theory but practice.
STEP 04
PhD Pathway
This MSc is an ideal stepping stone to doctoral research in probability theory, mathematical finance, statistics, or machine learning theory.
Applications open year-round — join the next cohort today.