Suppose $X=(X^0,\dots, X^n)$ is a continuous time financial market on $[0,T]$ with $X^1,\dots, X^n$ risky assets and $X^0$ the bank account, and the source of randomness is a $d$-dimensional Brownian motion on a filtered probability space $(\Omega,\mathcal{F}, (\mathcal{F}_t) P)$ where the filtration is the natural filtration of the brownian motion, and $X$ is an $n+1$-dimensional vector Ito-process
Let $Q$ be an equivalent local martingale measure. Suppose $H$ is a $Q$-integrable european contingent claim, where $H=h(X^1_T,\dots, X^n_T)$ where $h$ is smooth. The risk-neutral price of $H$ at time $t$ w.r.t to $Q$ is $$ \pi_t := X^0_t E^Q\left(H/X_T^0|\mathcal{F}_t\right).$$
Now consider the discounted market $\bar{X}$. How is $\pi_t$ related to $\bar{\pi}_t$ where $\bar{\pi}_t$ is the risk-neutral price of the claim $\bar{H}:=h(\bar{X}^1,\dots,\bar{X}^n)$ w.r.t to $Q$ in the discounted market, i.e. $$ \bar{\pi}_t = \bar{X}^0_t E^Q\left(\bar{H}/\bar{X}_T^0|\mathcal{F}_t\right) = E^Q(\bar{H}|\mathcal{F}_t)? $$
If $h$ is a linear form, this is easy as $\bar{H} = H/X^0_T$. What can be said if $h$ is non-linear? What, if furthermore $X^0$ is deterministic?
I.e., how can the risk-neutral price of a claim given by a formula of the final asset-prices be computed by looking at the claim given by the same formula in the discounted market?