  This paper first studies super linear G-expectation. Uniqueness and existence
theorem for backward stochastic differential equations (BSDEs) under super
linear expectation is established to provide probabilistic interpretation for
the viscosity solution of a class of Hamilton-Jacobi-Bellman equations,
including the well known Black-Scholes-Barrenblett equation, arising in the
uncertainty volatility model in mathematical finance. We also show that BSDEs
under super linear expectation could characterize a class of stochastic control
problems. A direct connection between recursive super (sub) strategies with
mutually singular probability measures and classical stochastic control
problems is provided. By this result we give representation for solutions of
Black-Scholes-Barrenblett equations and G-heat equations.
