• You say "recursively resolve", but in fact resolve is not recursive. It returns Prop A is Prop B is ${prop_c} instead of Prop A is Prop B is Prop C, if you call it with Prop A is ${prop_b}. If you call it with prop_a, as advertised, it returns prop_a.

• You allow nested ${} in you syntax. What is your semantics of it?

• Do you have any test cases for this. The bug and Unclear semantics of nested parens for instance, strongly indicate you should have.

• Since you access the super class through its public interface, namely the the singe call to the super.getProperty you could instead pass the properties as a parameter. It will improve usability. Then you can make the static, if you wish. Since the method does not refer to any instance members otherwise.

• StringBuffer and Stack have built in synchronization, whose overhead you can avoid by using StringBuilder and ArrayList.

Here is the test code to save others some effort:

import java.util.Properties;
import java.util.Stack;


public class Main {
    public static String resolve(Properties props,  String s ) {
          StringBuffer sb = new StringBuffer( 256 );
          Stack<StringBuffer> stack = new Stack<StringBuffer>();
          int len = s.length();

          for( int i = 0; i < len; i++ ) {
            char c = s.charAt( i );

            switch( c ) {
              case '$': {
                if( i + 1 < len && s.charAt( i + 1 ) == '{' ) {
                  stack.push( sb );
                  sb = new StringBuffer( 256 );
                  i++;
                }
                break;
              }

              case '}': {
                if( stack.isEmpty() ) {
                  throw new IllegalArgumentException( "unexpected '}'" );
                }

                String name = sb.toString();

                sb = stack.pop();
                sb.append( props.get(name) );
                break;
              }

              default: {
                sb.append( c );
                break;
              }
            }
          }

          if( !stack.isEmpty() ) {
            throw new IllegalArgumentException( "missing '}'" );
          }

          return sb.toString();
        }

    public static void main(String[] args) {
        Properties props = new Properties();

        props.setProperty("prop_a", "Prop A is ${prop_b}"); 
        props.setProperty("prop_b", "Prop B is ${prop_c}"); 
        props.setProperty("prop_c", "Prop C.");

        System.out.println(resolve(props, "Prop A is ${prop_b}"));
        System.out.println(resolve(props, "prop_a"));
        System.out.println(resolve(props, "${prop_a}"));
        System.out.println(resolve(props, "${${prop_b}}"));
    }
}