Question

1. Use Ito's formula to compute the stochastic differentials dZ_t when: (W_t is a Brownian motion) (a) Z_t = W_t^3, (b) Z_t = int_0^t W_u^2 dW_u + int_0^t W_u du, (c) Z_t = S_t^2 where dS_t = mu S_t dt + sigma S_t dW_t (mu and sigma are constants), (d) Z_t = exp(4S_t) where dS_t = 2dt + 3dW_t,

          1. Use Ito's formula to compute the stochastic differentials dZ_t when:
(W_t is a Brownian motion)

(a) Z_t = W_t^3,
(b) Z_t = int_0^t W_u^2 dW_u + int_0^t W_u du,
(c) Z_t = S_t^2 where dS_t = mu S_t dt + sigma S_t dW_t (mu and sigma are constants),
(d) Z_t = exp(4S_t) where dS_t = 2dt + 3dW_t,

1. Use Ito's formula to compute the stochastic differentials dZt when:
(Wt is a Brownian motion)

(a) Zt = Wt^3,
(b) Zt = int0^t Wu^2 dWu + int0^t Wu du,
(c) Zt = St^2 where dSt = mu St dt + sigma St dWt (mu and sigma are constants),
(d) Zt = exp(4St) where dSt = 2dt + 3dWt,

Added by Victoria S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Dominador Tan

08/03/2023

Step 1

For Zt = WB, we have dZt = dWtB + WtdB. Since B is a constant, dB = 0. Therefore, dZt = dWtB. Show more…

Show all steps

Thanks for your feedback!

Use Ito's formula to compute the stochastic differentials dZ_t when: (W_t is a Brownian motion) (a) Z_t = W_t^3, (b) Z_t = int_0^t W_u^2 dW_u + int_0^t W_u du, (c) Z_t = S_t^2 where dS_t = mu S_t dt + sigma S_t dW_t (mu and sigma are constants), (d) Z_t = exp(4S_t) where dS_t = 2dt + 3dW_t,