Strategically-Timed Actions in Stochastic Differential Games

Financial systems are rich in interactions amenable to description by stochastic control theory. Optimal stochastic control theory is an elegant mathematical framework in which a controller, profitably alters the dynamics of a stochastic system by exercising costly control inputs. If the system includes more than one agent, the appropriate modelling framework is stochastic differential game theory — a multiplayer generalisation of stochastic control theory. There are numerous environments in which financial agents incur fixed minimal costs when adjusting their investment positions; trading environments with transaction costs and real options pricing are important examples. The presence of fixed minimal adjustment costs produces adjustment stickiness as agents now enact their investment adjustments over a sequence of discrete points. Despite the fundamental relevance of adjustment stickiness within economic theory, in stochastic differential game theory, the set of players’ modifications to the system dynamics is mainly restricted to a continuous class of controls. Under this assumption, players modify their positions through infinitesimally fine adjustments over the problem horizon. This renders such models unsuitable for modelling systems with fixed minimal adjustment costs. To this end, we present a detailed study of strategic interactions with fixed minimal adjustment costs. We perform a comprehensive study of a new stochastic differential game of impulse control and stopping on a jump-diffusion process and, conduct a detailed investigation of two-player impulse control stochastic differential games. We establish the existence of a value of the games and show that the value is a unique (viscosity) solution to a double obstacle problem which is characterised in terms of a solution to a non-linear partial differential equation (PDE). The study is contextualised within two new models of investment that tackle a dynamic duopoly investment problem and an optimal liquidity control and lifetime ruin problem. It is then shown that each optimal investment strategy can be recovered from the equilibrium strategies of the corresponding stochastic differential game. Lastly, we introduce a dynamic principal-agent model with a self-interested agent that faces minimally bounded adjustment costs. For this setting, we show for the first time that the principal can sufficiently distort that agent’s preferences so that the agent finds it optimal to execute policies that maximise the principal’s payoff in the presence of fixed minimal costs

Problems in Mathematical Finance Related to Transaction Costs and Model Uncertainty.

This thesis is devoted to the study of three problems in mathematical finance which involve either transaction costs or model uncertainty or both. In Chapter II, we investigate the Fundamental Theorem of Asset Pricing (FTAP) under both transaction costs and model uncertainty, where model uncertainty is described by a family of probability measures, possibly non-dominated. We first show that the recent results on the FTAP and the super-hedging theorem in the context of model uncertainty can be extended to the case where only options available for static hedging (hedging options) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of robust no-arbitrage which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are non-redundant. Next, we look at the more difficult case where the market consists of a money market and a dynamically traded stock with bid-ask spread. Under a continuity assumption, we prove using a backward-forward scheme that no-arbitrage is equivalent to the existence of a suitable family of consistent price systems. In Chapter III, we study the problem where an individual targets at a given consumption rate, invests in a risky financial market, and seeks to minimize the probability of lifetime ruin under drift uncertainty. Using stochastic control, we characterize the value function as the unique classical solution of an associated Hamilton-Jacobi-Bellman (HJB) equation, obtain feedback forms for the optimal investment and drift distortion, and discuss their dependence on various model parameters. In analyzing the HJB equation, we establish the existence and uniqueness of viscosity solution using Perron's method, and then upgrade regularity by working with an equivalent convex problem obtained via the Cole-Hopf transformation. In Chapter IV, we adapt stochastic Perron's method to the lifetime ruin problem under proportional transaction costs which can be formulated as a singular stochastic control problem. Without relying on the Dynamic Programming Principle, we characterize the value function as the unique viscosity solution of an associated variational inequality. We also provide a complete proof of the comparison principle which is the main assumption of stochastic Perron's method.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111560/1/yuchong_1.pd