Stochastic differential equations

Itō and friends


SDEs are time-indexed, causal stochastic processes which notionally integrate an ordinary differential equation over some driving noise. As seen in state filters, optimal control, financial mathematics etc.

Terminology problem: when people talk about these they often mean, conceptually, stochastic integral equations, in the sense that the driving noise process is an integrator. When you differentiate the noise process, it leads, AFAICT to Malliavin calculus.

Cosma’s explanation of SDEs looks good for cannibalising for parts when I write my own.

Useful tools: infinitesimal generators, martingales, Dale Robert’s cheat sheet, Itō-Taylor expansions

Terminology problem: Many references take SDEs to be synonymous with Itō processes, whose driving noise is Brownian. In full generality, e.g. (Kallenberg 2002) they are a lot more general than that.

One confusion is that Itō’s formula, which is an important tool here, is applicable more broadly than to Brownian-type diffusions

Let \(X=\left(X^{1}, \ldots, X^{n}\right)\) be an n-tuple of semimartingales and let \(f: \mathbb{R}^{n} \rightarrow\) R have continuous second order partial derivatives. Then \(f(X)\) is again a semimartingale and the following formula holds:

\[ \begin{aligned} f\left(X_{t}\right)=& f\left(X_{0}\right)+\sum_{i=1}^{n} \int_{0+}^{t} \frac{\partial f}{\partial x_{i}}\left(X_{s-}\right) d X_{s}^{i} \\ &+\frac{1}{2} \sum_{1 \leq i, j \leq n} \int_{0+}^{t} \frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}\left(X_{s-}\right) d\left[X^{i}, X^{j}\right]_{s}^{c} \\ &+\sum_{0<s \leq t}\left(f\left(X_{s}\right)-f\left(X_{s-}\right)-\sum_{i=1}^{n} \frac{\partial f}{\partial x_{i}}\left(X_{s-}\right) \Delta X_{s}^{i}\right) \end{aligned} \]

This does get messy for non-Brownian processes, however. Schoutens, Leuven, and Studer (2001) give a tractable example for Poisson processes.


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