<apex:component controller="NewITAssetController">
    <apex:attribute name="con" type="NewITAssetController" description="page controller" required="false"/>
    <apex:outputPanel>
        <label>Markets</label>
        <apex:SelectList value="{!con.SelectedMarkets}" styleClass="form-control" html-data-id="markets" multiselect="true" html-aria-describedby="" size="5" html-onInput="setvalue();">
            <apex:selectOptions value="{!Markets}" />
        </apex:SelectList>
    </apex:outputPanel>
</apex:component>

<apex:page controller="NewITAssetController" standardStylesheets="false" showHeader="false" sidebar="false" doctype="html-5.0" title="New IT Item">
    <apex:form id="Form" styleClass="bg-light">
        <apex:actionFunction name="setvalue" action="{!SetValue}" rerender=""/>
        <c:AssetInput con="{!self}"/>
    </apex:form>
</apex:page>

public class NewITAssetController {
    public String[] SelectedMarkets {get;set;}
    public NewItAssetController getSelf() {
        return this;
    }
    public void SetValue(){
        system.debug('SelectedMarkets='+SelectedMarkets);
    }
}

SelectedMarkets=()