Placeholder. A time-indexed, causal, measure-valued stochastic process. As seen in state filters, optimal control, financial mathematics etc.
Cosma’s explanation of SDEs looks good for cannibalising for parts when I write my own.
Useful tools: infinitesimal generators, martingales.
One difficulty in this field is that 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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