Devido à falta de clareza na questão, cheguei a quatro soluções diferentes. As soluções diferem em:

1. Se você deve "cascatar" de acordo com a resposta de Chris
2. Quando você tiver uma data fechada, use a data mais antiga para esse grupo ou a data inicial para o registro que está fechado.

Observe que isso é feito no SQL Server, não no MySQL. Além de algumas alterações de sintaxe muito pequenas, deve funcionar da mesma forma.

Configuração comum e dados de amostra para todos os quatro métodos

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 - USANDO A SOLUÇÃO DE REGISTRO FECHADO

Esta é a solução que acredito que o solicitante está procurando e corresponde aos seus resultados.

select *
into #temp1
from #example

while (select count(1) from #temp1)>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)>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<=DATEADD(day,15,@maxTime) 
                    or t2.closed<=DATEADD(day,15,@maxTime) 
                    or t2.dead<=DATEADD(day,15,@maxTime)  )>0
        begin
            insert into #temp3
            select *
            from #temp2 t2
            where t2.started<=DATEADD(day,15,@maxTime)  or t2.closed<=DATEADD(day,15,@maxTime)  or t2.dead<=DATEADD(day,15,@maxTime) 

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

            --set new max time from any column
            if (select max(started) from #temp3)>@maxTime
                set @maxTime=(select max(started) from #temp3)
            if (select max(closed) from #temp3)>@maxTime
                set @maxTime=(select max(started) from #temp3)
            if (select max(dead) from #temp3)>@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. SEM CASCATA - USANDO solução de REGISTRO FECHADO

Início calculado pela primeira data de fechamento quando disponível e, em seguida, pela primeira data de início.

select *
into #temp1
from #example

while (select count(1) from #temp1)>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)>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<=DATEADD(day,15,@minTime)

        delete
        from #temp2
        where started<=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<=DATEADD(day,15,@minTime)

        delete
        from #temp2
        where closed<=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<=DATEADD(day,15,@minTime)

        delete
        from #temp2
        where dead<=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. SEM CASCATA - USANDO a solução EARLIEST DATE

Início calculado apenas pela data mais antiga.

select *
into #temp1
from #example

while (select count(1) from #temp1)>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)>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<=DATEADD(day,15,@minTime) or t2.closed<=DATEADD(day,15,@minTime) or t2.dead<=DATEADD(day,15,@minTime)

        delete
        from #temp2
        where started<=DATEADD(day,15,@minTime) or closed<=DATEADD(day,15,@minTime) or dead<=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)>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)>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<=DATEADD(day,15,@maxTime) 
                or t2.closed<=DATEADD(day,15,@maxTime) 
                or t2.dead<=DATEADD(day,15,@maxTime)  )>0
    begin
        insert into #temp3
        select *
        from #temp2 t2
        where t2.started<=DATEADD(day,15,@maxTime)  or t2.closed<=DATEADD(day,15,@maxTime)  or t2.dead<=DATEADD(day,15,@maxTime) 

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

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

Preocupa-me que não tenhamos uma imagem clara de como um grupo é definido. Digo isso apenas porque, dependendo de algumas condições não declaradas, as datas acima formarão um único grupo gigante ou 3 grupos onde um grupo domina o conjunto.

Faltando condições de agrupamento?

1) Essa regra de 15 dias é cascata? Se um registro Ycomeça 10 dias após outro registro Xe, em seguida, há outro registro Ziniciado 10 dias depois disso, isso forma um grupo de três registros X,Y,Zou dois grupos, cada um contendo dois registros X,Ye Y,Z? Eu fiz a suposição de que as regras de 15 dias se propagam para formar grupos maiores.

2) As datas são inclusivas? Por exemplo, se um registro tiver uma data inicial e uma data final muitos meses depois, todos os dias dentro desse intervalo serão mesclados no grupo? Trato ambas as possibilidades em minha análise rápida abaixo.

Agrupamentos potenciais

Portanto, se começarmos com id 7714, veremos que a data de início é 27/01. Claramente, a próxima entrada 7882começando em 28/01 cai neste grupo. Observe, no entanto, que 7882termina em 15/05, portanto, qualquer coisa que comece dentro de 15 dias a partir de 15/05 deve ser adicionada ao grupo.

Assim, 19690por meio 21210da inclusão no grupo, que por meio de cascata leva à 21918inclusão posterior no grupo. A cascata consumiu quase todas as entradas do conjunto. Chame isso GROUP A.

Se, no entanto, o agrupamento também incluir a data, todas as entradas de 13190até 17124também devem pertencer a GROUP A, e agora todos os ids estão em um único grupo.

Se as datas de GROUP Anão forem inclusivas, mas na verdade aderirem estritamente à regra '15 dias depois' com cascata, então, em vez disso, você teria um segundo grupo composto por 13190até 14020, e um terceiro grupo com uma única entrada, 17124.

Basicamente, minha pergunta é: algum deles corresponde ao agrupamento pretendido ou há alguma outra informação que está faltando na definição do grupo? Sinto muito por uma resposta tão prolixa, mas não parece que sua saída solicitada provisória atenda à sua definição de agrupamento.

Com esclarecimentos, tenho certeza de que podemos resolver esse problema.