Given:
Consider a two-asset, continuous time model (B,S) where $$dB_t = B_t r dt, \quad dS_t = S_t ( \mu dt + \sigma dW_t)$$ Clearly, the martingale deflator is: $$Y_t = e^{(-r - \frac{\lambda^2}{2})t - \lambda W_t}$$
There is a theorem that states the following:
For a claim with payout $\xi_T$, $T>0$, if $\xi_T>0$, $\mathcal{F}_T$-measurable and such that $\xi_TY_T$ is integrable, then there exists an admissible strategy $(\pi_t)_{t \in [0,T]}$ that replicates the European claim with payout $\xi_T$.
Problem:
Suppose that $\xi_T = g(X_T)$, where $g$ is a bounded smooth Borel function. (Then the hypothesis for $\xi_t$ in the above theorem are satisfied.)
Suppose further that $g$ has a bounded derivative and is an increasing function.
Show that there exists an admissible replicating strategy $(\pi_t)_{t \in [0,T]}$ such that $\pi_t \geq 0$ a.s. for all $t \geq 0$.