  A coupled system of nonlinear mixed-type equations modeling early stages of
angiogenesis is analyzed in a bounded domain. The system consists of stochastic
differential equations describing the movement of the positions of the tip and
stalk endothelial cells, due to chemotaxis, durotaxis, and random motion;
ordinary differential equations for the volume fractions of the extracellular
fluid, basement membrane, and fibrin matrix; and reaction-diffusion equations
for the concentrations of several proteins involved in the angiogenesis
process. The drift terms of the stochastic differential equations involve the
gradients of the volume fractions and the concentrations, and the diffusivities
in the reaction-diffusion equations depend nonlocally on the volume fractions,
making the system highly nonlinear. The existence of a unique solution to this
system is proved by using fixed-point arguments and H\"older regularity theory.
Numerical experiments in two space dimensions illustrate the onset of formation
of vessels.
