= Solution
The arbitrage-free value of the discounted claim is the <conditional expectation>
$$
V_t=\mathbb E_Q[H\mid\mathcal F_t].
$$
Conditional on the observed path $(S_0^1,\ldots,S_t^1)=(x_0,\ldots,x_t)$, the future multiplicative increments are independent of $\mathcal F_t$ and have the same law as a fresh stock path divided by its initial value. Therefore
$$
\boxed{V_t(\omega)=v_t(S_0^1,S_1^1(\omega),\ldots,S_t^1(\omega)),}
$$
where
$$
\boxed{
v_t(x_0,\ldots,x_t)=
\mathbb E_Q\left[h\left(x_0,\ldots,x_t,
x_t\frac{S_1^1}{S_0^1},\ldots,
x_t\frac{S_{T-t}^1}{S_0^1}\right)\right].}
$$
This is <Risk-neutral valuation in the Cox--Ross--Rubinstein model> for a path-dependent claim.
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