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You have given the value of portfolio at time $t$ is \begin{equation} V_t=\phi_t S_t + \psi_t A_t \quad \cdots \cdots (1) \end{equation} where $\phi_t$and $\psi_t$ denote the number of units of the security and cash account respectively that is held in a portfolio at time t. So, the value of portfolio at time $t+dt$ would be $$V_{t+dt}=(\phi_t + ...


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For Q2, let $\lambda = \mu/\sigma$. Moreover, we define the measure $Q$ on $(\Omega, \mathcal{F})$ such that \begin{align*} \frac{dQ}{dP}\big|_{\mathcal{F}_t} = \exp\Big(-\frac{1}{2}\lambda^2 t - \lambda W_t\Big), \mbox{ for } t \ge 0. \end{align*} Then, by Girsanov theorem, $W^*$, where \begin{align*} W_t^* = \lambda + W_t, \end{align*} is a standard ...



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