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An optimal consumption problem of maximizing the expected discounted value of utility is discussed for an economic growth model with the random technological shocks. Applying the technique of dynamic programming principle, we derive the Hamilton‐Jacobi‐Bellman equation corresponding to the optimization problem and prove that the value function is a unique viscosity solution to the equation. Moreover, the optimal consumption policy is given in a feedback form under weak assumptions.

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Liquidity financial markets, whose risks are caused by uncertain volatilities, are investigated. We suppose that the price process of risky asset satisfies a mean reversion model which contains a G-Brownian motion instead of the classical Brownian motion. Under the assumption of no arbitrage, employing the concept of arbitrage and the properties of G-expectation, an interval of no-arbitrage price for the general European contingent claims is deduced.

In financial markets with volatility uncertainty, we assume that their risks are caused by uncertain volatilities and their assets are effectively allocated in the risk-free asset and a risky stock, whose price process is supposed to follow a geometric -Brownian motion rather than a classical Brownian motion. The concept of arbitrage is used to deal with this complex situation and we consider stock price dynamics with no-arbitrage opportunities. For general European contingent claims, we deduce the interval of no-arbitrage price and the clear results are derived in the Markovian case.