This post introduces the Ito rule of stochastic calculus. This rule provides the semimartingale decomposition of a nonlinear (this being the non trivial case) function of given semimartingales. Its simplest incarnation is the product rule of stochastic calculus which gives the semimartingale decomposition of the product of two given semimartingales.
The previous posts introduced stochastic integration of an integrand \(H\) against an integrator \(M\) which was chosen to be a continuous martingale. Continuous semimartingales are now defined as continuous processes that can be decomposed into a continuous martingale and a continuous finite variation process. Because ‘standard’ integration theory allows to integrate against finite variation processes, one can easily extend stochastic integration from continuous martingale integrators to continuous semimartingale integrators.
As explained in this post, stochastic integration has initially been developed under the condition \(E\left[\int_{0}^{T}H^{2}_{s}d[M]_{s}\right]<\infty\). This condition has then been replaced by the more general condition \(P\left(\int_{0}^{T}H^{2}_{s}d[M]_{s}<\infty \right)=1\). This turns out to be the most general integrability condition under which stochastic integration can be defined. This post introduces the idea behind this generalization. The process whereby stochastic integration is thus generalized is called localization. In words, the above condition means that the set of trajectories for which integrated variance remains bounded should have probability one.
Stochastic integration was first developed in the context of the Brownian motion and the theory was then extended to martingales with continuous paths, which we will call continuous martingales in what follows. Continuous martingales have a lot of structure which eases the corresponding stochastic integration theory. The quadratic variation of a continuous martingale is the central concept in this theory. The purpose of this note is to provide an easy introduction to this subject before presenting Ito calculus in a later post.
The purpose of this note is to introduce diffusions which are made up of a drift and a martingale component. I start from the elementary discrete time setup where the drift of the process is most easily understood. I then explain how the specialized decomposition of a diffusion into a drift and a Brownian integral can arise as the limit of the decompositions obtained on the discretized process.
Building on the portfolio return post, this article describes the stochastic inregral \(\int_{0}^{\cdot}H_{u}dM_{u}\) of an integrand \(H\) against a martingale \(M\) as a martingale transform. The integral re-weights the increments of \(M\) using a system of ‘predetermined’ weights in such a way that the resulting process remains a martingale. The preservation of the martingale property is a key requirement of the standard stochastic integration theory. The sensitivity of the resulting martingale \(\int_{0}^{\cdot}H_{u}dM_{u}\) with respect to the infinitesimal increments of \(M\) is an instanciation of the concept of Malliavin derivative. The post ends by considering the martingale representation property which plays a key role in financial theory.
(Warning: this post was written when Wordpress was used to generate the blog. Quicklatex is no longer used. MathJax remains the tool used for math rendering). This post gives a few recipes to integrate mathematical formulas in html pages. This site uses MathJax, a javascript implementation of Latex.
This post introduces the notion of a martingale. The concept is key to finance, but it is also central in stochastic analysis. A martingale is a process which, at any given date, is expected to remain at its current level in the future. In other words, its future increments (called martingale differences) are expected to be equal to zero. Picking a variable \(X_{T}\), the process \((E_{t}[X_{T}])_{t \in [0,T]}\) is a martingale which is ‘closed’ by \(X_{T}\). The concept is illustrated using the discretized version of a martingale, namely a random walk with centered increments.Continuous time martingales with (loosely speaking) independent identically distributed increments are called Levy martingales. The post introduces two Levy martingales, the Brownian motion and the compensated Poisson proces.
This post introduces the dynamics of the value of a portfolio. It starts with an unfrequently rebalanced portfolio: portfolio weights are assumed ‘simple’, i.e. piecewise constant. In this case, the portfolio value dynamics is trivial. More complex rebalancing schemes can be approximated using sequences of simple weighting schemes, just as a continuous function can be approximated using peacewise constant functions. The dynamics of portfolio value is then given by a stochastic integral. This example stresses an essential feature of stochastic integrals: portfolio weights (integrands) are forced to lag prices (integrators).
This post takes another stab at return compounding, when price trajectories potentially are of infinite variation. Starting from first principles, it shows that the differential equation followed by the price can either be a standard differential equation or a stochastic differential equation. The key concept is the quadratic variation of trajectories.
