Try this.

data a;
infile datalines missover;
input ID $ A_date : $10. A  B_date : $10. B;
datalines;
A 2011/1/1 10 2011/1/1         10       
A 2011/2/1 20 2011/2/3         20
A 2011/3/1 40 2011/3/6         30
A 2011/4/1 40 2011/4/20 50
A 2011/5/1 50 2011/5/23 60
A 2011/6/1 60 2011/7/10 70
A 2011/7/1 70 2011/8/11 80
A 2011/8/1 80
B 2010/3/4 100 2010/3/3 100
B 2010/4/4 200 2010/4/3 100
B 2010/5/4 300 2010/5/3 300
B 2010/6/4 400 2010/8/2 400
B 2010/7/4 500 2010/8/9 500
B 2010/8/4 600 2010/9/1 600
; run;

proc sort data=a; by id A_date; run;

data b;
    set a;
        if A <= B then do;    /* the condition */
              result_A=A_date;
              result_B=b_date;
                    end;
           else result_A = A_date;
       if missing(b_date) then Result_B=a_date;
       by id;
run;

proc sort data=b;
     by id descending a_date; /* reverse the order */
run;

data c;           /* fill the missing value */
   set b;
   lag_Result_b = lag(result_b);
   if missing(result_b) then result_b=lag_result_b;
   drop lag_result_b;
run;

proc sort data=c out=result;   /* back to normal */
    by id a_date;
run;

proc print data=result; title 'result'; run;