Google-sheets – Get column number by referencing column title

I'd use this formula in E3 and copy down:

=IF(AND(E2<>"",OR(C3<>"",D3<>"")),E2-C3+D3,"")

EXPLANATION:
The IF statement contains 3 comma separated parameters:

1. logical_expression
2. value_if_true
3. value_if_false

In this case the logical expression is AND(E2<>"",OR(C3<>"",D3<>"")). AND takes multiple logical expressions as comma separated parameters, and returns TRUE only if all expressions are true. In this case these two:

1. E2<>""
2. OR(C3<>"",D3<>"")

Number one checks if there is anything in the cell above (if it's not equal to nothing). Number two is an OR function, works as AND but returns TRUE if only one or more of the expressions are true. In this case:

1. C3 (the Debit value on this row) is not empty
2. D3 (the Credit value on this row) is not empty

So to summarize If the cell above has a value, and Debit or Credit has a value. The value_if_true is used. In the example: E2-C3+D3 (the cell above minus Debit plus Credit).

If the expression evaluates to false, the value_if_false is used, in this case "" (an empty string).

If you did not insist on array-based formula, then

sumifs(C:C, A:A, A2, B:B, "<="&B2)

entered in D2 and copied around the D column, would do the job.

I don't think that an arrayformula-based solution exists, because arrayformula runs through an array once, performing some element-wise operations. You want to compare each element with all others, which is one step further in complexity (a double for loop).

But you can use a custom function, something like =runningtotal(A2:A, B2:B, C2:C) which processes the columns and returns an array of running totals. The script implementing this function is below; add it via Tools > Script Editor. The logic is quite straightforward: as said above, it's a double for loop.

function runningTotal(names, dates, amounts) {
  var sum, totals = [], n = names.length;
  if (dates.length != n || amounts.length != n) {
    return 'Error: need three columns of equal length'; 
  }
  for (var i = 0; i < n; i++) {
    if (names[i][0]) {
      sum = 0;
      for (var j = 0; j < n; j++) {
        if (names[j][0] == names[i][0] && dates[j][0] <= dates[i][0]) {
          sum = sum + amounts[j][0];
        }
      }
    }
    else {
      sum = '';
    }
    totals.push([sum]);
  }
  return totals;
}