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Find $E(X_t)$ for

a) $dX_t = (X_t + W_t) \,dW_t$ with $X_0=1$

b) $dX_t = (W_t^2 + t) \,dW_t + 2 \,dW_t$ with $X_0=0$

c) $dX_t = (X_t + t) \,dW_t + W_t^2 \,dW_t$ with $X_0=1$

I know that I have to use the known relationship of:

$$\frac{dE(S_t)}{dt} = \mu E(S_t)$$

Which stems from the stochastic differential equation for the asset model process:

$$ dS_t = \mu S_t\, dt + \sigma S_T \, dW_t$$

However I'm not sure how to use this to find $E(X_t)$, what would be mu and what is sigma?

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  • $\begingroup$ How did you calculate the expected value for the process $S_t$ (known as geometric Brownian motion)? Maybe you could try the same technique. $\endgroup$ – Harto Saarinen Mar 25 '18 at 18:43
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You have made in a mistake in writing down the SDE for the asset price, it should be

$$ dS_t = \mu S_t dt + \sigma S_t dW_t $$

Define the stochastic exponential by $dz_t=z_t dw_t$, or simply $z_t=1\cdot \exp(w_t - 0.5t)$.

Let us solve $dx_t = (x_t+w_t) dw_t, \, x_0=1 $.

Define $y_t = x_t+w_t$. See that $dx_t=y_t dw_t$ and that

$dy_t = d(y_t+1) = (y_t+1)dw_t \Rightarrow y_t+1=\left( y_0+1 \right) z_t. $

Therefore, $x_t = -(1+w_t) +2 z_t $. Let us verify that this is correct,

\begin{align} dx_t =& -dw_t + 2 dz_t \\= & \left( 2z_t -1 \right) dw_t \\ = &\left( 2z_t -1-w_t + w_t \right) dw_t \\ = & \left(x_t + w_t \right) dw_t \end{align}

It follows that $\mathbb{E}(x_t)=-1+2\mathbb{E}(z_t)=-1+2=1$.

You can use the same idea for the second and third SDE. For example, if $dx_t=(w_t^2+t+2) dw_t$, define $y_t=w_t^2+t+2$, find $dy_t$, then deduce $x_t$.

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