Analyze profits under random walk and Brownian motion

Consider a simple symmetric random walk (S_t) for t = 0, 1, ..., T with S_0 = 0 and increments X_t = S_t − S_{t−1} that are i.i.d. with P(X_t = + 1) = P(X_t = − 1) = 1/2. Trading strategy: for each t in {1, ..., T}, after observing the change from t−1 to t, open one unit position at price S_t: go long if X_t = +1 and short if X_t = −1. Hold all opened positions until time T and liquidate everything at price S_T. Let P_T be the total profit at time T (sum over all positions, where each long contributes S_T − S_t and each short contributes −(S_T − S_t)). ( 1) Compute E[P_T] and Var(P_T). ( 2) Now consider an alternative one-step strategy: at each t you take a unit position based on X_t but close it immediately at time t+1; compute the expectation and variance of the one-step profit and of the cumulative profit after T steps. ( 3) Generalize part ( 1) to a biased walk with P(X_t = + 1) = p ≠ 1/2. ( 4) Suppose the price follows standard Brownian motion B_t with B_0 = 0. Sample at discrete times t_k = kΔ, apply the same trading rule using increments B_{t_k} − B_{t_{k−1}}, and discuss the limits of E and Var of the profit as Δ → 0; state any additional assumptions you need and whether the limit is well-defined.