For testing purposes, I used the following setup script:

CREATE TABLE inventory (
    initial int,
    date date PRIMARY KEY,
    recsold int,
    final int
)
INSERT INTO inventory VALUES
    (20, '1/1/2023', -1, 19),
    (20, '1/2/2023', 0, 20),
    (20, '1/3/2023', 4, 24),
    (20, '1/4/2023', 2, 22),
    (20, '1/5/2023', -2, 18)

Output of SELECT * FROM inventory :

initial date        recsold  final
20      2023-01-01  -1       19
20      2023-01-02   0       20
20      2023-01-03   4       24
20      2023-01-04   2       22
20      2023-01-05  -2       18

Updating columns initial and final can be done in various ways. A simple solution is an UPDATE statement with a subquery. For each row being updated, the subquery calculates the sum of all preceding rows.

DECLARE @start int = (SELECT TOP 1 initial FROM inventory ORDER BY date)

UPDATE i
SET initial = @start + acc.recsold
    , final = @start + acc.recsold + i.recsold
FROM inventory i
CROSS APPLY (SELECT ISNULL(SUM(recsold), 0) AS recsold FROM inventory WHERE date < i.date) acc

Expect this to run in O(n²) time. That might be OK for a small table, but for a large number of rows, it may be better to use a cursor. The following loop will update the rows one by one. Performance-wise, that is still far from ideal, but its time complexity can be close to O(n). Just try and see what works best for you.

DECLARE @initial int, @date date, @recsold int, @stock int
DECLARE cur CURSOR FOR SELECT initial, date, recsold FROM inventory ORDER BY date
OPEN cur
FETCH NEXT FROM cur INTO @initial, @date, @recsold
WHILE @@FETCH_STATUS = 0
BEGIN
    IF @stock IS NULL SET @stock = @initial
    UPDATE inventory SET initial = @stock, final = @stock + @recsold WHERE date = @date
    SET @stock += @recsold
    FETCH NEXT FROM cur INTO @initial, @date, @recsold
END
CLOSE cur
DEALLOCATE cur

Output of SELECT * FROM inventory :

initial date        recsold  final
20      2023-01-01  -1       19
19      2023-01-02   0       19
19      2023-01-03   4       23
23      2023-01-04   2       25
25      2023-01-05  -2       23

You can do it using row_number() to get the first row, and first_value() to get the initial final_inventory. the with sum over() we get the expected data :

with cte as (
  select *, row_number() over (order by thedate ) as id,
  first_value(final_inventory) over (order by thedate ) as first_final_inventory
  from mytable
),
cte2 as (
  select initial_inventory, thedate, received_sold, 
  sum(case when id = 1 then first_final_inventory else received_sold end) over (order by thedate) as 'new_final_inventory'
  from cte
)
select lag(new_final_inventory, 1, initial_inventory) over (order by thedate) as 'new_initial_inventory',
    thedate, received_sold, new_final_inventory
from cte2;

Without WITH :

select * from mytable t
inner join (
  select lag(new_final_inventory, 1, initial_inventory) over (order by thedate) as 'new_initial_inventory',
    thedate, received_sold, new_final_inventory
  from (
    select initial_inventory, thedate, received_sold, 
    sum(case when id = 1 then first_final_inventory else received_sold end) over (order by thedate) as 'new_final_inventory'
    from (
      select *, row_number() over (order by thedate ) as id,
      first_value(final_inventory) over (order by thedate ) as first_final_inventory
      from mytable
    ) as s2
  ) as s1
) as s on s.thedate = t.thedate

The Update query can be as follows :

update mytable
set initial_inventory = s.new_initial_inventory,
    final_inventory = s.new_final_inventory
from mytable t
inner join (
  select lag(new_final_inventory, 1, initial_inventory) over (order by thedate) as 'new_initial_inventory',
    thedate, received_sold, new_final_inventory
  from (
    select initial_inventory, thedate, received_sold, 
    sum(case when id = 1 then first_final_inventory else received_sold end) over (order by thedate) as 'new_final_inventory'
    from (
      select *, row_number() over (order by thedate ) as id,
      first_value(final_inventory) over (order by thedate ) as first_final_inventory
      from mytable
    ) as s2
  ) as s1
) as s on s.thedate = t.thedate

Demo here