How to persist bash history?

I don't know if this is expected, but my history is not saved across sessions. This is to say, if I close the window, then when I open it again, the history is empty. How can I persist it across sessions ?

Here are the outputs of the commands you asked:

 set -o | grep history
history         on

$ grep -i history ~/.bashrc ~/.bash_profile ~/etc/bash.bashrc ~/etc/profile ~/.profile
/cygdrive/c/cygwin/home/car/.bashrc:# Make bash append rather than overwrite the history on disk
/cygdrive/c/cygwin/home/car/.bashrc:# History Options
/cygdrive/c/cygwin/home/car/.bashrc:# Don't put duplicate lines in the history.
/cygdrive/c/cygwin/home/car/.bashrc:# export PROMPT_COMMAND="history -a"
grep: /cygdrive/c/cygwin/home/car/etc/bash.bashrc: No such file or directory
grep: /cygdrive/c/cygwin/home/car/etc/profile: No such file or directory
/cygdrive/c/cygwin/home/car/.profile:if [ "x$HISTFILE" == "x/.bash_history" ]; then
/cygdrive/c/cygwin/home/car/.profile:  HISTFILE=$HOME/.bash_history

$ ls -la ~/ | grep history -> no output

$ echo $HISTFILE 
~/.bash_history
$ echo $HISTSIZE
500
$ echo $HISTFILESIZE 
500

After the edits described in the answer below, I now get:

grep -i hist .bashrc
# Make bash append rather than overwrite the history on disk
shopt -s histappend
# History Options
# Don't put duplicate lines in the history.
export HISTCONTROL="ignoredups"
# (added) A new shell gets the history lines from all previous shells
PROMPT_COMMAND='history -a'
# HISTIGNORE is a colon-delimited list of patterns which should be excluded.

I am still unable to have a history saved across sessions. I read the following questions:

• Bash history loss when using histappend
• How to prevent Bash from altering history?
• What determines what shows up in the bash history command?
• How do I keep my bash history across sessions?
• save bash history, regularly

None seemed to address my issue, including the answer below from the very person which had their question answered from the supposed duplicate.

OK I found out what's wrong. I can't close the window, I have to type 'exit' for it to close gracefully.

Well, it looks like your ~/.bashrc does not have the necessary options. Make sure these lines are in your ~/.bashrc:

# Make Bash append rather than overwrite the history on disk:
shopt -s histappend
# A new shell gets the history lines from all previous shells
PROMPT_COMMAND='history -a'
# Don't put duplicate lines in the history.
export HISTCONTROL=ignoredups