In Hui-Hsiung Kuo's Introduction to Stochastic Integration, lemma 4.3.3, equations (4.3.11) says that since $f$ is assumed to be bounded, we have \begin{equation} \int_a^b |f(t,\cdot)-f(t-n^{-1}\tau,\cdot)|^2\ dt\to 0,\quad\text{almost surely}, \end{equation} as $n\to\infty$. I have question about how this convergence holds, do we need to require that $f$ is left continuous?
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No here we only assume $f$ is bounded and in $L^{2}$, he just skipped a few steps. In particular, he skipped the proof that
$$g(y):=\left(\int_{R} |f(x-y)-f(x)|^{p}dx\right)^{1/p}$$
is a continuous function of $y$.
As the author mentioned this proof is actually based on the original proof by Itô. In Øksendal's SDE book in " Construction of the Itô Integral" he goes over the same proof.
The particular step you are studying was about proving that: if $g_{n}(t):=f\ast \psi_{n}(t)$, where $\psi_{n}(t)$ is an approximation to the identity mollifier, then $$E[\int |f(s)-g_{n}(s)|^{2}ds]\to 0.$$
And the logic is indeed just that $f\ast \psi_{n}\stackrel{L^{p}}{\to} f$. For $p=2$ case see Approximate $L^2$ function by convolving with mollifiers. For general $L^{p}$ see mollifiers. The logic is
- Upper bound $$\|f-f\ast \psi_{n}\|_{p}^{p}\leq \int \|f(\cdot-y)-f\|_{p}\psi_{n}(y)dy\leq \int g(\epsilon y)\psi(y)dy,$$where $g(y):=\|f(\cdot-y)-f\|_{p}$
- Prove that $g(y)$ is continuous and bounded. For continuous approximate $f\in L^{p}$ by some continuous function $h$ with compact support such that $\|f-h\|_{L^{p}}\leq \delta/3$.
- Conclude by applying dominated convergence theorem.
Thomas Kojar
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