Problem 193 of 333MediumStochastic CalculusP193

Left versus right Riemann sums for W dW

Split $[0, T]$ into $n$ equal steps $t_i = iT/n$ and let $\Delta W_i = W_{t_{i+1}} - W_{t_i}$ for a standard Brownian motion $W$ . Define
$L_n = \sum_{i=0}^{n-1} W_{t_i}\,\Delta W_i, \qquad R_n = \sum_{i=0}^{n-1} W_{t_{i+1}}\,\Delta W_i.$
Find $E[L_n]$ and $E[R_n]$ , and the limits of $L_n$ and $R_n$ as $n \to \infty$ . Which one defines the Itô integral $\int_0^T W\,dW$ , and why?
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