  The topics treated in this thesis are inherently two-fold. The first part
considers the problem of a market maker optimally setting bid/ask quotes over a
finite time horizon, to maximize her expected utility. The intensities of the
orders she receives depend not only on the spreads she quotes, but also on
unobservable factors modelled by a hidden Markov chain. This stochastic control
problem under partial information is solved by means of stochastic filtering,
control and PDMPs theory. The value function is characterized as the unique
continuous viscosity solution of its dynamic programming equation and
numerically compared with its full information counterpart. The optimal full
information spreads are shown to be biased when the exact market regime is
unknown, as the market maker needs to adjust for additional regime uncertainty
in terms of PnL sensitivity and observable order flow volatility.
  The second part deals with numerically solving nonzero-sum stochastic impulse
control games. These offer a realistic and far-reaching modelling framework,
but the difficulty in solving such problems has hindered their proliferation. A
policy-iteration-type solver is proposed to solve an underlying system of
quasi-variational inequalities, and it is validated numerically with reassuring
results.
  Eventually, the focus is put on games with a symmetric structure and an
improved algorithm is put forward. A rigorous convergence analysis is
undertaken with natural assumptions on the players strategies, which admit
graph-theoretic interpretations in the context of weakly chained diagonally
dominant matrices. The algorithm is used to compute with high precision
equilibrium payoffs and Nash equilibria of otherwise too challenging problems,
and even some for which results go beyond the scope of the currently available
theory.
