solve step by step clearly showing the formulas used.....(b). Let Z = Z(t);≥ 0 be a standard Brownian motion. Define the stochastic process X as X(t) = Z(t)3 + ctZ(t),t ≥ 0 for some constant c.Find the value of the constant c such that the stochastic process X has zero drift.

To solve this problem, we need to find the value of the constant \( c \) such that the stochastic process \( X(t) = Z(t)^3 + ctZ(t) \) has zero drift, where \( Z(t) \) represents a standard Brownian motion.

### Step-by-Step Solution

1. **Identify the form of \( X(t) \):**
   Given:
   \[
   X(t) = Z(t)^3 + ctZ(t)
   \]
   where \( Z(t) \) is a standard Brownian motion.

2. **Compute \( dX(t) \) using Ito's Lemma:**
   Since \( X(t) \) is a function of \( Z(t) \), we will apply Ito's Lemma to find the differential \( dX(t) \). Recall that Ito's Lemma for a function \( f(Z(t), t) \) where \( f(Z(t), t) = Z(t)^3 + ctZ(t) \) is given by:
   \[
   df(Z(t), t) = \frac{\partial f}{\partial t} \, dt + \frac{\partial f}{\partial Z} \, dZ + \frac{1}{2} \frac{\partial^2 f}{\partial Z^2} \, dZ^2
   \]
   For a standard Brownian motion \( Z(t) \), we know:
   \[
   dZ^2 = dt
   \]

3. **Calculate the partial derivatives of \( f(Z, t) = Z^3 + ctZ \):**

   - **First partial derivative with respect to \( t \):**
     \[
     \frac{\partial f}{\partial t} = cZ
     \]

   - **First partial derivative with respect to \( Z \):**
     \[
     \frac{\partial f}{\partial Z} = 3Z^2 + ct
     \]

   - **Second partial derivative with respect to \( Z \):**
     \[
     \frac{\partial^2 f}{\partial Z^2} = 6Z
     \]

4. **Substitute into Ito's Lemma:**
   Now, we substitute the derivatives into Ito's formula to find \( dX(t) \):
   \[
   dX(t) = \left(cZ\right) dt + \left(3Z^2 + ct\right) dZ + \frac{1}{2} \left(6Z\right) dt
   \]

5. **Simplify the expression:**
   Combine terms involving \( dt \) and \( dZ \):
   \[
   dX(t) = \left(cZ + 3Z\right) dt + \left(3Z^2 + ct\right) dZ
   \]
   Simplify further:


Let Z = Z(t); ≥ 0 be a standard Brownian motion. Define the stochastic process X as X(t) = Z(t)^3 + ctZ(t), t ≥ 0 for some constant c. Find the value of the constant c such that the stochastic process X has zero drift.

Learn how to determine the constant c for which the stochastic process X(t) = Z(t)^3 + ctZ(t) has zero drift, using Brownian motion and Ito's Lemma. This solution provides a step-by-step approach for advanced calculus and probability students.

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