Book excerpt: In this thesis, we consider stochastic control problems and stochastic differential games in which controllers or players control the diffusion intensities of their individual state processes. The thesis consists of four parts. In the first part, we study two-player stochastic differential games with diffusion control on finite time horizons. The players' state processes are described in terms of controlled martingale problems. We allow for correlation between the players' states, described by some correlation coefficient. The reward of the game is zero-sum and depends only on the difference between the players' state processes at the finite time horizon. We explicitly compute saddle points of the game depending on the correlation coefficient. In particular, for correlation coefficients smaller than some explicit threshold, the game has a value and a saddle point (in strict controls). For correlation coefficients exceeding the threshold, we introduce the larger class of relaxed controls that can be interpreted as mixed strategies. We show that the game has a value and a saddle point in relaxed controls for all correlation coefficients. The second part treats a stochastic differential game of n players on deterministic and random time horizons. The players whose states at the finite time horizon are among the best p 2 (0; 1) of all states receive a fixed prize. Within the mean field limit version of the game, we compute an explicit equilibrium, i.e., a threshold strategy that consists in choosing the maximal fluctuation intensity when the state is below a given threshold and the minimal intensity above. We show that for large n, the symmetric n-tuple of the threshold strategy provides an approximate Nash equilibrium of the n-player game. We also derive the rate at which the approximate equilibrium reward and the best response reward converge against each other, as the number of players n tends to infinity. The third part presents a two-player stochastic differential game in which players cannot observe their opponents and the reward is zero-sum. The players are compared at an individual and independent random time horizon that is exponentially distributed, and only the leading player receives a fixed reward. We derive sufficient and necessary conditions for symmetric couples consisting of threshold strategies to be saddle points. We present an explicit saddle point. In the last part, we consider Brownian integral processes with integrands that are bounded and bounded away from zero. We provide an upper estimate for the expected occupation time in intervals. The estimate does not depend on the integrand but only on its bounds. We derive the estimate by solving stochastic control problems that consist in maximizing the expected occupation time in intervals.