Let 1 , 2 , be i i d r v s with zero mean and variance 2 0, S 0 0, S t 1 t for t 0,1, Show that X t S 2 t 2 t is a martingale with respect to t

Question: Let 1 , 2 , ... be i.i.d. r.v.s with zero mean and variance 2 > 0, S 0 = 0, S

Let ξ₁, ξ₂, ... be i.i.d. r.v.’s with zero mean and variance σ² > 0, S₀ = 0, S_t = ξ₁+...+ξ_t for t = 0,1, .... Show that X_t = S²_t −σ²_t is a martingale with respect to ξ_t.

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