I wonder how to solve the following quadratic optimization problem?

$\min_{f:[0,1]\to \mathbb{R}} \int_0^1 f(t)^2dt-2\int_0^1 f(t)dB_t$

where $B(t)$ is standard Brownian motion.

Intuitively, the optimization can be performed pointwise, i.e., for any $t\in[0,1]$, solve

$\min_{f(t)} f(t)^2dt-2f(t)dB_t$,

however the solution is $f(t)=dB_t/dt$ which doesn't make sense since $B_t$ is not differentiable.

I wonder how to solve the following quadratic optimization problem?

$$\min_{f:[0,1]\to \mathbb{R}} \int_0^1 f(t)^2dt-2\int_0^1 f(t)dB_t$$

where $B(t)$ is standard Brownian motion.

Intuitively, the optimization can be performed pointwise, i.e., for any $t\in[0,1]$, solve

$$\min_{f(t)} f(t)^2dt-2f(t)dB_t,$$

however the solution is $f(t)=dB_t/dt$ which doesn't make sense since $B_t$ is not differentiable.