How to handle error while executing Flux.map()

You need flatMap instead which let's you return an empty sequence if the processing failed:

myflux.flatMap(v -> {
    try {
        return Flux.just(converter.convertHistoricalCSVToStockQuotation(stock));
    } catch (IllegalArgumentException ex) {
        return Flux.empty();
    }
});

If you want to use Reactor 3's methods for dealing with exceptions, you can use Mono.fromCallable.

flatMap(x -> 
    Mono.fromCallable(() -> converter.convertHistoricalCSVToStockQuotation(x))
        .flux()
        .flatMap(Flux::fromIterable)
        .onErrorResume(Flux::empty)
)

Unfortunately there is no Flux.fromCallable, so assuming the callable returns a list, you have to convert it to a Flux manually.