Supermartingale control of deflated consumption gains ($M_t=Y_t{X}_t-Y_0{X}_0+{\int }_0^t{Y}_s{c}_{s}ds$)

If wealth and consumption are nonnegative, this local martingale is bounded below by minus the finite initial deflated capital. Adding that capital gives a nonnegative local martingale, hence a supermartingale by the Conditional Fatou lemma. Consequently expected discounted consumption cannot exceed initial deflated wealth. The consumption sign and predictable stochastic integrability are part of the hypotheses.