Finding rows with consecutive increase in the values of a column

In any case, it helps to put it in terms of increasing rows-per-stock (the actual quoteid value isn't really helpful here). Count of days captured (in this table) is easiest - if you want something else (like only business days, ignoring weekends/holidays, or whatever) it gets more involved; you'd probably need a calendar file. You're going to want an index over [stockid, createdate], if you don't have one already.

WITH StockRow AS (SELECT stockId, closePrice, createdDate,
                         ROW_NUMBER() OVER(PARTITION BY stockId 
                                           ORDER BY createdDate) rn
                  FROM Quote),

     RunGroup AS (SELECT Base.stockId, Base.createdDate,
                         MAX(Restart.rn) OVER(PARTITION BY Base.stockId
                                              ORDER BY Base.createdDate) groupingId
                  FROM StockRow Base
                  LEFT JOIN StockRow Restart
                         ON Restart.stockId = Base.stockId
                            AND Restart.rn = Base.rn - 1
                            AND Restart.closePrice > Base.closePrice)

SELECT stockId, 
       COUNT(*) AS consecutiveCount, 
       MIN(createdDate) AS startDate, MAX(createdDate) AS endDate
FROM RunGroup
GROUP BY stockId, groupingId
HAVING COUNT(*) >= 3
ORDER BY stockId, startDate

Which yields the following results from the provided data:

Increasing_Run
stockId   consecutiveCount  startDate    endDate
===================================================
1         5                 2012-01-01   2012-01-05
2         4                 2012-01-01   2012-01-04
3         3                 2012-01-02   2012-01-04

SQL Fiddle Example
(Fiddle also has an example for multiple runs)

This analysis will ignore all gaps, correctly matches all runs (the next time a positive run starts).


So what's going on here?

StockRow AS (SELECT stockId, closePrice, createdDate,
                    ROW_NUMBER() OVER(PARTITION BY stockId 
                                      ORDER BY createdDate) rn
             FROM Quote)

This CTE is being used for one purpose: we need a way to find the next/previous row, so first we number each row in order (of the date)...

RunGroup AS (SELECT Base.stockId, Base.createdDate,
                    MAX(Restart.rn) OVER(PARTITION BY Base.stockId
                                         ORDER BY Base.createdDate) groupingId
             FROM StockRow Base
             LEFT JOIN StockRow Restart
                    ON Restart.stockId = Base.stockId
                       AND Restart.rn = Base.rn - 1
                           AND Restart.closePrice > Base.closePrice)

... and then join them based on the index. If you end up on something that has LAG()/LEAD(), using those instead will almost certainly be a better option. There's one critical thing here though - matches are only if the row is out-of-sequence (less than the previous row). Otherwise, the value end up being null (with LAG(), you'd need to use something like CASE afterwards to pull this off). You get a temporary set that looks something like this:

B.rn   B.closePrice   B.createdDate  R.rn   R.closePrice   R.createdDate  groupingId
1      15             2012-01-01     -      -              -              -
2      13             2012-01-02     1      15             2012-01-01     1
3      17             2012-01-03     -      -              -              1
4      18             2012-01-04     -      -              -              1
5      10             2012-01-05     4      18             2012-01-04     4

... So there's values for Restart only when the previous was greater than the "current" row. The use of MAX() in the window function is being used to the greatest value seen so far... which because null is lowest, causes the row-index to be retained for all other rows until another mismatch occurs (which gives a new value). At this point, we essentially have the intermediate results of a gaps-and-islands query, ready for the final aggregation.

SELECT stockId, 
       COUNT(*) AS consecutiveCount, 
       MIN(createdDate) AS startDate, MAX(createdDate) AS endDate
FROM RunGroup
GROUP BY stockId, groupingId
HAVING COUNT(*) >= 3
ORDER BY stockId, startDate

The final part of the query is getting the start and end dates of the run, and counting the number of entries between those dates. If there was something more complicated for the date calculation, it probably needs to happen at this point. The GROUP BY is showing one of the few legitimate instances of not including a column in the SELECT clause. The HAVING clause is used to eliminate runs that are "too short".