  This paper investigates the callable convertible bond problem in the presence
of a liquidity constraint modelled by Poisson signals. We assume that neither
the bondholder nor the firm has absolute priority when they stop the game
simultaneously, but instead, a proportion $m\in[0,1]$ of the bond is converted
to the firm's stock and the rest is called by the firm. The paper thus
generalizes the special case studied in [Liang and Sun, Dynkin games with
Poisson random intervention times, SIAM Journal on Control and Optimization, 57
(2019), 2962-2991] where the bondholder has priority ($m=1$), and presents a
complete solution to the callable convertible bond problem with liquidity
constraint. The callable convertible bond is an example of a Dynkin game, but
falls outside the standard paradigm since the payoffs do not depend in an
ordered way upon which agent stops the game. We show how to deal with this
non-ordered situation by introducing a new technique which may be of interest
in its own right, and then apply it to the bond problem.
