(1 point) Let f(t, x) be a smooth function of variables t and x, and let ✗+ be a process satisfying (1) dX+udt + odz = where z is a Weiner (standard Brownian) process. According to Ito calculus, (2) (dz)² = dt (3) dt dz = 0 (4) (dt)² = 0 The goal is to find a formula for df for f(t, Xt). This will be a version of Ito's Lemma sufficiently general for use in deriving the Black-Scholes equation. Taylor expansion of f about (t, x) gives (5) Aƒ = f(t + At, x + Ax) − f(t, x) = = ft At+ fx Ax+ (At)²+ ftx (At)(Ax)+ (Ax)² + ... Before entering your answers, write partials using subscripts: write as fta, and type as ftx. Ətəx Replacing a with Xt, At with dt and Ax with dX+ gives (6) df = f(t + dt, Xt + dxt) ft dt+ fx - f(t, Xt) = = dx++ (dt)²+ ftx (dt) (dX+)+ (dX+)² + ... Before entering answers, rewrite partials using subscripts: type frx (x, X+) as fxx. From (1)-(4), after simplification (7) (dX+)² (8) dt dXt = 0 dt (Type σ as sigma) Now using (1), (4), (7) and (8), equation (6) reduces to (9) df = Type μ as mu, σ as sigma. dt+ dz