Question

52. Consider the SDE in the interval [0, 1[: dXt = -1/2 * Xt/(1-t) dt + sqrt(1-t) dWt, X0 = x (a) Solve the equation above and show that the solution Xt is a Gaussian process. (b) Compare the variance of Xt with the corresponding variance of the Brownian bridge (exercise 33). Is X a Brownian bridge?

          52. Consider the SDE in the interval [0, 1[:

dXt = -1/2 * Xt/(1-t) dt + sqrt(1-t) dWt, X0 = x

(a) Solve the equation above and show that the solution Xt is a Gaussian process.
(b) Compare the variance of Xt with the corresponding variance of the Brownian bridge (exercise 33). Is X a Brownian bridge?

52. Consider the SDE in the interval [0, 1[:

dXt = -1/2 * Xt/(1-t) dt + sqrt(1-t) dWt, X0 = x

(a) Solve the equation above and show that the solution Xt is a Gaussian process.
(b) Compare the variance of Xt with the corresponding variance of the Brownian bridge (exercise 33). Is X a Brownian bridge?

Added by Aaron W.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

04/11/2023

Step 1

First, we rewrite the given SDE as follows: $$dX_t = \frac{1}{X_t} dt + (1 - t) dW_t$$ Now, we apply the Ito's formula to the function $f(X_t, t) = \ln(X_t)$: $$df(X_t, t) = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial X_t} dX_t + \frac{1}{2} Show more…

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Consider the SDE in the interval [0, 1[: dXt = -1/2 * Xt/(1-t) dt + sqrt(1-t) dWt, X0 = x (a) Solve the equation above and show that the solution Xt is a Gaussian process. (b) Compare the variance of Xt with the corresponding variance of the Brownian bridge (exercise 33). Is X a Brownian bridge?