This page is an index into the various stochastic calculus posts on the blog.

I decided to use this blog to post some notes on stochastic calculus, which I started writing some years ago while learning the subject myself. The aim was to introduce the theory of stochastic integration in as direct and natural way as possible, without losing any of the mathematical rigour. The required background for properly understanding these notes is measure theoretic probability theory. These notes are currently in progress, and are being updated regularly.

Filtrations and Processes

- Stochastic Processes, Indistinguishability and Modifications
- Filtrations and Adapted Processes
- Stopping Times and the Debut Theorem
- Sigma Algebras at a Stopping Time
- Predictable Stopping Times
- Martingales and Elementary Integrals
- Upcrossings, Downcrossings, and Martingale Convergence
- Cadlag Modifications
- Optional Sampling
- Martingale Convergence
- Martingale Inequalities
- U.C.P. and Semimartingale Convergence
- Class (D) Processes
- Localization
- Local Martingales
- The Kolmogorov Continuity Theorem

Stochastic Integration

- The Stochastic Integral
- Extending the Stochastic Integral
- Properties of the Stochastic Integral
- Quadratic Variations and Integration by Parts
- Properties of Quadratic Variations
- Ito’s Lemma
- The Generalized Ito Formula
- Existence of Solutions to Stochastic Differential Equations
- SDEs with Locally Lipschitz Coefficients
- Existence of the Stochastic Integral
- Existence of the Stochastic Integral 2 – Vector Valued Measures
- Further Properties of the Stochastic Integral
- Semimartingale Completeness
- The Stochastic Fubini Theorem

Martingales as Integrators

- Martingales are Integrators
- Preservation of the Local Martingale Property
- Quadratic Variations and the Ito Isometry
- Continuous Local Martingales
- The Burkholder-Davis-Gundy Inequality

- Special Processes
- Lévy’s Characterization of Brownian Motion
- Time-Changed Brownian Motion
- Girsanov Transformations
- SDEs Under Changes of Time and Measure
- The Martingale Representation Theorem
- Continuous Processes with Independent Increments
- Poisson Processes
- Markov Processes
- Feller Processes
- Properties of Feller Processes
- Bessel Processes
- Processes With Independent Increments
- Lévy Processes
- Properties of Lévy Processes
- Brownian Bridges
- Brownian Bridge Fourier Expansions

The General Theory of Semimartingales

- The General Theory of Semimartingales
- The Bichteler-Dellacherie Theorem
- Continuous Semimartingales
- Predictable Stopping Times
- Predictable FV Processes
- Special Semimartingales
- Compensators
- Compensators of Stopping Times
- Compensators of Counting Processes
- The Doob-Meyer Decomposition
- Quasimartingales
- Rao’s Quasimartingale Decomposition
- Properties of Quasimartingales
- The Doob-Meyer Decomposition for Quasimartingales
- Constructing Martingales with Prescribed Jumps
- Purely Discontinuous Local Martingales
- Purely Discontinuous Semimartingales

The Projection Theorems

- The Projection Theorems
- Projection in Discrete Time
- Optional Projection For Right-Continuous Processes
- Predictable Projection For Left-Continuous Processes
- Measurable Projection And The Debut Theorem
- Optional Processes
- Predictable Processes
- The Section Theorems
- Pathwise Regularity of Optional and Predictable Processes
- The Projection Theorems
- Properties of Optional and Predictable Projections
- Pathwise Properties of Optional and Predictable Projections
- Dual Projections
- Properties of the Dual Projections
- Proof of the Measurable Projection and Section Theorems
- Proof of Optional and Predictable Section

Point Processes

- Poisson Point Processes
- Criteria for Poisson Point Processes
- Drawdown Point Processes
- Brownian Drawdowns

Local Times

In addition to the notes listed above, I am also starting to post examples demonstrating the various results and techniques of stochastic calculus, together with counterexamples to show how they can fail if the necessary conditions are not met. In stochastic process theory, in particular, there are often measurability or integrability conditions required which, if they are not met, can cause the expected results to fail in quite subtle ways. The aim is to build up a collection of examples showing what can go wrong, and to help understand the limits of the standard theory.

- Stochastic Calculus Examples and Counterexamples
- Failure of Pathwise Integration for FV Processes
- Failure of the Martingale Property
- The Optimality of Doob’s Maximal Inequality
- Martingales with Non-Integrable Maximum
- Failure of the Martingale Property For Stochastic Integration
- A Martingale Which Moves Along a Deterministic Path
- Do Convex and Decreasing Functions Preserve the Semimartingale Property — A Possible Counterexample
- A Process With Hidden Drift

**Brownian Motion**

**Other Stochastic Calculus Posts**

Posts on stochastic calculus which do not fit into the categories above are listed here.

- Zero-Hitting and Failure of the Martingale Property
- The Maximum Maximum of Martingales with Known Terminal Distribution
- Do Convex and Decreasing Functions Preserve the Semimartingale Property?
- Analytic Sets
- Choquet’s Capacitability Theorem and Measurable Projection
- Proof of Measurable Section
- Pathwise Martingale Inequalities
- Pathwise Burkholder-Davis-Gundy Inequalities

Dear George Lowther,

I was wondering what your next posts will be talking about. Anything about Local Times, Malliavin Calculus, Quasi-sure Analysis, Backward SDEs, Large Deviations Theory, Stochastic Control Theory ?

Anyway whatever the subject you may pick I’ll be delighted to read more on your blog,

Best Regards

Hi. I already started writing something on quasimartingales, which is a continuation of the “General Theory of Semimartingales”. Actually, I started this before the new year but was busy with other stuff so didn’t get it completed an ready to post. Should post that in a few days. After the general theory then that finishes everything that was originally intended for these notes, although I’ll probably add some stuff such as local times. I’ll also give some thought to the other things you mention, which are quite interesting.

Dear Sir,

First of all, thank you so much for making this blog freely available for all; it will surely help me in my autodidactic pursuit re stochastic mathematics.

Secondly, I was wondering whether you might help me with something that I have been stuck with and unable to grasp whilst reading the book by Prof Salih Neftci called “An Introduction to the Mathematics of Financial Derivatives”. In there, he gives an equation but I do not know how he got the result. If you can help, then either a) you have the book – I can specify precisely which equation and which page, or b) you don’t have to book: I will handwrite the the equation and set up the problem, scan it and post an upload link as a comment here?

Kr,

WKW

Hi WKW,

Let me recommand you to rather ask this kind of question at the QuantSE forum “http://quant.stackexchange.com/questions” ( or alternatively at Wilmott.com forum), where might find a lot of skillfull people on those matters.

Best regards

TheBridge

I don’t have that book, but you can try posting it here and I’ll let you know if its something I know. But, as TheBridge mentions, using the QuantSE forum might be a better bet as it will reach a wider audience.

I thank u all you all great minds. pls, Ihave to sde model to solve by Ito formula. I will be grateful if any can help with the solution the model is: dXt =[rXt(1-Xt/k)-qzt]dt+gXtdWt

dXt=(rx[1-Xt/k])-qzt)dt+gXt[1-Xt/k]dWt

Notify me of new posts via email.

Hi there, guys I need an urgent help for this exercice. May you help me ont this ?

Thanks in advance ?

A continuous random variable Y with a support[0,∞) is said to be exponentially distributed with a parameterλ>0 if its cumulative distributive function is

ℙ(Y<y)=1−exp(−λy)

.

Compute the expectation and variance of an exponentially distributed random variable.

Prove that an exponentially distributed radom variable Y belongs to the functional space

Ln(Ω,A,ℙ)

,i.e., that E[∣∣Yn∣∣]<∞. Give a expression for E[Yn].

Hi Sacha,

There is a well known result tha says the following:

For every nonnegative random variable defined on a probability space and a real number the next equality is true, .

With this at hand and the usual formula you can answer every thing that you ask.

Is it possible to create the capability of compiling the updated post into a pdf file?