Query to determine start and end dates based on time overlap
Due to the lack of clarity in the question, I came up with four different solutions. The solutions differ on: 


Whether you are to "cascade" per Chris's answer
When you have a closed date, whether you use the earliest date for that group or the started date for the record that is closed.


Please note this is done in SQL Server, not MySQL. Other than some very minor syntax changes, it should work the same.

Common setup and sample data for all four methods

CREATE TABLE #example 
(
    id int NOT NULL DEFAULT '0',
    borrower_id int NOT NULL,
    started datetime NULL DEFAULT NULL,
    closed datetime NULL DEFAULT NULL,
    dead datetime NULL DEFAULT '0000-00-00 00:00:00'
);

CREATE TABLE #result 
(   
    borrower_id int NOT NULL DEFAULT '0',    
    started datetime NULL DEFAULT NULL,    
    ended datetime NULL DEFAULT NULL 
);    

INSERT INTO #example 
    (id, borrower_id, started, closed, dead) 
VALUES 
    (7714,238846,'2015-01-27 15:14:50','2015-02-02 14:14:13',NULL), 
    (7882,238846,'2015-01-28 13:25:58',NULL,'2015-05-15 12:16:07'), 
    (13190,259140,'2015-03-17 10:11:44',NULL,'2015-03-18 07:31:57'), 
    (13192,259140,'2015-03-17 10:12:17',NULL,'2015-03-18 11:46:46'), 
    (13194,259140,'2015-03-17 10:12:53',NULL,'2015-03-18 11:46:36'), 
    (14020,259140,'2015-03-23 14:32:16','2015-03-24 15:57:32',NULL), 
    (17124,242650,'2015-04-16 16:19:08','2015-04-16 16:21:06',NULL), 
    (19690,238846,'2015-05-15 13:17:31',NULL,'2015-05-27 13:56:43'), 
    (20038,242650,'2015-05-19 15:38:17',NULL,NULL), 
    (20040,242650,'2015-05-19 15:39:58',NULL,'2015-05-21 12:01:02'), 
    (20302,242650,'2015-05-21 13:09:06',NULL,NULL), 
    (20304,242650,'2015-05-21 13:09:54',NULL,NULL), 
    (20306,242650,'2015-05-21 13:10:19',NULL,NULL), 
    (20308,242650,'2015-05-21 13:12:20',NULL,NULL), 
    (21202,238846,'2015-05-29 16:47:29',NULL,NULL), 
    (21204,238846,'2015-05-29 16:47:56',NULL,NULL), 
    (21208,238846,'2015-05-29 17:05:15',NULL,NULL), 
    (21210,238846,'2015-05-29 17:05:55',NULL,NULL), 
    (21918,242650,'2015-06-04 17:04:29',NULL,'2015-06-12 15:47:23'); 




1. CASCADING - USING CLOSED RECORD solution

This is the solution I believe the asker is looking for &amp; matches his results.

select *
into #temp1
from #example

while (select count(1) from #temp1)&gt;0
begin
    --Grab only one user's records and place into a temp table to work with
    declare @curUser int
    set @curUser=(select min(borrower_id) from #temp1)

    select * 
    into #temp2
    from #temp1 t1
    where t1.borrower_id=@curUser

    while(select count(1) from #temp2)&gt;0
    begin
        --Grab earliest start date and use as basis for 15 day window (#2 rule)
        --Use the record as basis for rules 3 and 4
        declare @minTime datetime
        set @minTime=(select min(started) from #temp2)

        declare @maxTime datetime
        set @maxTime=@minTime

        declare @curId int
        set @curId=(select min(id) from #temp2 where started=@minTime)

        select * 
        into #temp3
        from #temp2 t2
        where t2.id=@curId

        --Remove earliest record from pool of potential records to check rules against
        delete 
        from #temp2 
        where id=@curId

        --Insert all records within 15 days of start date, then remove record from pool
        while (select count(1) 
                from #temp2 t2 
                where t2.started&lt;=DATEADD(day,15,@maxTime) 
                    or t2.closed&lt;=DATEADD(day,15,@maxTime) 
                    or t2.dead&lt;=DATEADD(day,15,@maxTime)  )&gt;0
        begin
            insert into #temp3
            select *
            from #temp2 t2
            where t2.started&lt;=DATEADD(day,15,@maxTime)  or t2.closed&lt;=DATEADD(day,15,@maxTime)  or t2.dead&lt;=DATEADD(day,15,@maxTime) 

            delete
            from #temp2
            where started&lt;=DATEADD(day,15,@maxTime)  or closed&lt;=DATEADD(day,15,@maxTime)  or dead&lt;=DATEADD(day,15,@maxTime) 

            --set new max time from any column
            if (select max(started) from #temp3)&gt;@maxTime
                set @maxTime=(select max(started) from #temp3)
            if (select max(closed) from #temp3)&gt;@maxTime
                set @maxTime=(select max(started) from #temp3)
            if (select max(dead) from #temp3)&gt;@maxTime
                set @maxTime=(select max(started) from #temp3)

        end

        --Calculate end time according to rule #3
        declare @end datetime 
        set @end = null
        set @end=(select min(closed) from #temp3)

        if @end is not null
        begin
            set @minTime=(select started from #temp3 where closed=@end)
        end

        if @end is null
        begin
            if(select count(1) from #temp3 where dead is null)=0
            set @end= (select max(dead) from #temp3)
        end

        insert into #result (borrower_id,started,ended)
        values (@curUser,@minTime,@end)

        drop table #temp3
    end

    --Done with the one user, remove him from temp table and iterate thru to the next user
    delete  
    from #temp1 
    where borrower_id=@curUser    

    drop table #temp2

end

drop table #temp1

drop table #example

select * from #result order by started

drop table #result


2. NON-CASCADING - USING CLOSED RECORD solution

Start calculated by first closed date when available, then by earliest start date.

select *
into #temp1
from #example

while (select count(1) from #temp1)&gt;0
begin
    --Grab only one user's records and place into a temp table to work with
    declare @curUser int
    set @curUser=(select min(borrower_id) from #temp1)

    select * 
    into #temp2
    from #temp1 t1
    where t1.borrower_id=@curUser

    while(select count(1) from #temp2)&gt;0
    begin
        --Grab earliest start date and use as basis for 15 day window (#2 rule)
        --Use the record as basis for rules 3 and 4
        declare @minTime datetime
        set @minTime=(select min(started) from #temp2)

        declare @curId int
        set @curId=(select min(id) from #temp2 where started=@minTime)

        select * 
        into #temp3
        from #temp2 t2
        where t2.id=@curId

        --Remove earliest record from pool of potential records to check rules against
        delete 
        from #temp2 
        where id=@curId

        --Insert all records within 15 days of start date, then remove record from pool
        insert into #temp3
        select *
        from #temp2 t2
        where t2.started&lt;=DATEADD(day,15,@minTime)

        delete
        from #temp2
        where started&lt;=DATEADD(day,15,@minTime)

        --Insert all records within 15 days of closed, then remove record from pool
        insert into #temp3
        select *
        from #temp2 t2
        where t2.closed&lt;=DATEADD(day,15,@minTime)

        delete
        from #temp2
        where closed&lt;=DATEADD(day,15,@minTime)

        --Insert all records within 15 days of dead, then remove record from pool
        insert into #temp3
        select *
        from #temp2 t2
        where t2.dead&lt;=DATEADD(day,15,@minTime)

        delete
        from #temp2
        where dead&lt;=DATEADD(day,15,@minTime)

        --Calculate end time according to rule #3
        declare @end datetime 
        set @end = null
        set @end=(select min(closed) from #temp3)

        if @end is not null
        begin
            set @minTime=(select started from #temp3 where closed=@end)
        end

        if @end is null
        begin
            if(select count(1) from #temp3 where dead is null)=0
            set @end= (select max(dead) from #temp3)
        end

        insert into #result (borrower_id,started,ended)
        values (@curUser,@minTime,@end)

        drop table #temp3
    end

    --Done with the one user, remove him from temp table and iterate thru to the next user
    delete  
    from #temp1 
    where borrower_id=@curUser


    drop table #temp2

end

drop table #temp1

drop table #example

select * from #result

drop table #result


3. NON-CASCADING - USING EARLIEST DATE solution

Start calculated by earliest date only.

select *
into #temp1
from #example

while (select count(1) from #temp1)&gt;0
begin
    --Grab only one user's records and place into a temp table to work with
    declare @curUser int
    set @curUser=(select min(borrower_id) from #temp1)

    select * 
    into #temp2
    from #temp1 t1
    where t1.borrower_id=@curUser

    while(select count(1) from #temp2)&gt;0
    begin
        --Grab earliest start date and use as basis for 15 day window (#2 rule)
        --Use the record as basis for rules 3 and 4
        declare @minTime datetime
        set @minTime=(select min(started) from #temp2)

        declare @curId int
        set @curId=(select min(id) from #temp2 where started=@minTime)

        select * 
        into #temp3
        from #temp2 t2
        where t2.id=@curId

        --Remove earliest record from pool of potential records to check rules against
        delete 
        from #temp2 
        where id=@curId

        --Insert all records within 15 days of start date, then remove record from pool
        insert into #temp3
        select *
        from #temp2 t2
        where t2.started&lt;=DATEADD(day,15,@minTime) or t2.closed&lt;=DATEADD(day,15,@minTime) or t2.dead&lt;=DATEADD(day,15,@minTime)

        delete
        from #temp2
        where started&lt;=DATEADD(day,15,@minTime) or closed&lt;=DATEADD(day,15,@minTime) or dead&lt;=DATEADD(day,15,@minTime)

        --Calculate end time according to rule #3
        declare @end datetime 
        set @end = null

        set @end=(select min(closed) from #temp3)

        if @end is null
        begin
            if(select count(1) from #temp3 where dead is null)=0
            set @end= (select max(dead) from #temp3)
        end

        insert into #result (borrower_id,started,ended)
        values (@curUser,@minTime,@end)

        drop table #temp3
    end

    --Done with the one user, remove him from temp table and itterate thru to the next user
    delete  
    from #temp1 
    where borrower_id=@curUser    

    drop table #temp2

end

drop table #temp1

drop table #example

select * from #result

drop table #result


4. CASCADING - USING EARLIEST DATE solution

Start calculated by earliest date only.

select *
into #temp1
from #example

while (select count(1) from #temp1)&gt;0
begin
--Grab only one user's records and place into a temp table to work with
declare @curUser int
set @curUser=(select min(borrower_id) from #temp1)

select * 
into #temp2
from #temp1 t1
where t1.borrower_id=@curUser

while(select count(1) from #temp2)&gt;0
begin
    --Grab earliest start date and use as basis for 15 day window (#2 rule)
    --Use the record as basis for rules 3 and 4
        declare @minTime datetime
    set @minTime=(select min(started) from #temp2)


    declare @maxTime datetime
    set @maxTime=@minTime

    declare @curId int
    set @curId=(select min(id) from #temp2 where started=@minTime)

    select * 
    into #temp3
    from #temp2 t2
    where t2.id=@curId

    --Remove earliest record from pool of potential records to check rules against
    delete 
    from #temp2 
    where id=@curId

    --Insert all records within 15 days of start date, then remove record from pool
    while (select count(1) 
            from #temp2 t2 
            where t2.started&lt;=DATEADD(day,15,@maxTime) 
                or t2.closed&lt;=DATEADD(day,15,@maxTime) 
                or t2.dead&lt;=DATEADD(day,15,@maxTime)  )&gt;0
    begin
        insert into #temp3
        select *
        from #temp2 t2
        where t2.started&lt;=DATEADD(day,15,@maxTime)  or t2.closed&lt;=DATEADD(day,15,@maxTime)  or t2.dead&lt;=DATEADD(day,15,@maxTime) 

        delete
        from #temp2
        where started&lt;=DATEADD(day,15,@maxTime)  or closed&lt;=DATEADD(day,15,@maxTime)  or dead&lt;=DATEADD(day,15,@maxTime) 

        --set new max time from any column
        if (select max(started) from #temp3)&gt;@maxTime
            set @maxTime=(select max(started) from #temp3)
        if (select max(closed) from #temp3)&gt;@maxTime
            set @maxTime=(select max(started) from #temp3)
        if (select max(dead) from #temp3)&gt;@maxTime
            set @maxTime=(select max(started) from #temp3)

    end

    --Calculate end time according to rule #3
    declare @end datetime 
    set @end = null

    set @end=(select min(closed) from #temp3)

    if @end is null
    begin
        if(select count(1) from #temp3 where dead is null)=0
        set @end= (select max(dead) from #temp3)
    end

    insert into #result (borrower_id,started,ended)
    values (@curUser,@minTime,@end)

    drop table #temp3
end

--Done with the one user, remove him from temp table and iterate thru to the next user
delete  
from #temp1 
where borrower_id=@curUser

drop table #temp2

end

drop table #temp1

drop table #example

select * from #result order by started

drop table #result