Notation in Kuo's Introduction to Stochastic Integration

Question

I am reading this book by Kuo (really like it so far) and I really don't understand the notation used in section 10.5. In particular, when he writes $(X_{t_2} \in dx_2)$. I can't find anywhere else... Is someone familiar with it?

Answer 1

Note, that for fixed $x_1$ we can define the measure $\mu_{x_1}(A) := \mathbb{P}(X_{t_2} \in A \: | \: X_{t_1}=x_1)$ and it is thus natural to use the notation $$\int_\mathbb{R} f(x_2) \: \mu_{x_1}(dx_2) = \int_{\mathbb{R}} f(x_2) \: \mathbb{P}(X_{t_2}\in dx_2 \: | \: X_{t_1}=x_1),$$ when we integrate with respect to the conditional distribution of $X_{t_2}$ given $X_{t_1}=x_1$. Similarly it is common to use the notation $$\int_{\mathbb{R}} f(x) \: \mathbb{P}_X(dx) = \int_{\mathbb{R}} f(x) \: \mathbb{P}(X\in dx)$$ when integrating with respect to the distribution of $X$.

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Apart from being a convenient notation, it also has a theoritical grounding. Similarly to how you can approximate a Riemann integral with Riemann sums $$\int_a^b f(x) \: dx \approx \sum_{i=1}^n f(x_i)(x_i - x_{i-1})$$ where $\{x_0,\dots,x_n\}$ is a partition of $[a,b]$. You can aproximate integrals with respect to a probability distribution $\mathbb{P}_X$ as $$\int_a^b f(x) \: \mathbb{P}_X(dx) \approx \sum_{i=1}^n f(x_i) \mathbb{P}_X((x_{i-1},x_i]) = \sum_{i=1}^n f(x_i)\mathbb{P}(X \in (x_{i-1},x_i]).$$ Under suitable conditions on $f$ (for instance $f$ being continuous), this approximation will indeed approach the correct value of the integral when $\sup|x_i-x_{i-1}| \rightarrow 0$ and it is thus natural to replace $X \in (x_{i-1},x_i]$ with $X\in dx$ in the limit.