(1 pt) Let f (t, x) be a smooth function of variables t and x, and let W₂ be a Wiener process. According to Ito calculus, (1) (dW)² = dt (2) dt dW₁ = 0 (3) (dt)² = 0 The goal is to find a formula for df for f (t, W₁), which will be a version of Ito's Lemma. Taylor expansion of f about (t, x) gives (4) f(t+At, x+Ax) − f(t, x) = - At+ Ax+ Use subscript notation for partial derivatives: type (At)²+ a²ƒ Ətəx as ftx. Replacing with Wt. At with dt and Ax with dW gives (5) df = f(t+dt, W₁ + dW₁) − f(t,W₁) = - dt+ dW₁+ (dt)2+ (At)(Ax)+ (Ax)²+... (dt) (dW)+ (dW)²+ Use subscript notation for partials: type fxx(x, W₁) as just fxx. Now using (1)-(3) and neglecting higher order terms, equation (5) reduces to (6) df = dt+ dWt