Let $f: [0, 1] \to \mathbb R$ be a bounded, continuous function, and $W$ a standard Brownian motion.

Denote $Y := \int_0^1 f(t) \, dW_t$.

For each $\varepsilon > 0$, consider the conditioned random variable $Y_\varepsilon := \varepsilon Y | \{W_1 \geq \frac{1}{\epsilon}\}.$

Let $B$ be a Brownian bridge on $[0, 1]$ conditioned to start at $0$ and end at $1$.

Write $Z := \int_0^1 f(t) \, dB_t$ for the stochastic integral of $f$ with respect to the Brownian bridge $B$.

Question: Is it true that $Y_\varepsilon$ converges in law to $Z$ as $\varepsilon \to 0$?

Let $f: [0, 1] \to \mathbb R$ be a bounded, continuous function, and $W$ a standard Brownian motion.

Denote $Y := \int_0^1 f(t) \, dW_t$.

For each $\varepsilon > 0$, consider the conditioned random variable $Y_\varepsilon := \varepsilon Y | \{W_1 \geq \frac{1}{\epsilon}\}.$

Question: Is it true that $Y_\varepsilon$ converges in law to the deterministic random variable $\int_0^1 f(t) \, dt$ as $\varepsilon \to 0$?