We introduce a general framework for continuous-time betting markets, in which a bookmaker can dynamically control the prices of bets on outcomes of random events. In turn, the prices set by the bookmaker affect the rate or intensity of bets placed by gamblers. The bookmaker seeks a price...

We determine the optimal investment strategy in a Black-Scholes financial market to minimize the so-called probability of drawdown, namely, the probability that the value of an investment portfolio reaches some fixed proportion of its maximum value to date. We assume that the portfolio is...

We show that the value function of a stochastic control problem is the unique solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, completely avoiding the proof of the so-called dynamic programming principle (DPP). Using Stochastic Perron's method we construct a super-solution...

We reveal an interesting convex duality relationship between two problems: (a) minimizing the probability of lifetime ruin when the rate of consumption is stochastic and when the individual can invest in a Black-Scholes financial market; (b) a controller-and-stopper problem, in which the...