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My objective is to find the dynamics of the discounted prices, given by $\mathbf{y}_{t} = \mathbf{P}_{t}\mathrm{e}^{-\int^{t}_{0} r_{s} ds}$. I know the dynamics should be $d\mathbf{y}_{t} = \mathrm{diag}(\mathbf{y}_{t})[(\mu_{t} - r_{t}\mathbf{1})dt + \sigma_{t}d\mathbf{W}_{t}]$, where $\mu_{t}$ is a vector, $\mathbf{1}$ is a vector of ones, and $\sigma_{t}$ is a matrix.

My approach is as follows. Define $f(t,\mathbf{y}_{t},D_{t}) = \mathbf{P}_{t}D_{t}$, where $D_{t} = \mathrm{e}^{-\int^{t}_{0}r_{s}ds}$. The dynamics of $D_{t}$ is: $dD_{t} = -D_{t}r_{t}dt$. Using Itô's Lemma, we get $$\begin{align*} d\mathbf{y_{t}} &= 0dt + D_{t}d\mathbf{P}_{t} + \mathbf{P}_{t}dD_{t} + \frac{1}{2} \cdot 0(d\mathbf{P}_{t})^{2} + \frac{1}{2} \cdot 0 (dD_{t})^{2} + \mathbf{1} (d\mathbf{P}_{t})(dD_{t})\\ &= D_{t}d\mathbf{P}_{t} + \mathbf{P}_{t}dD_{t} \end{align*}.$$

It is given that the dynamics of $\mathbf{P}_{t}$ is: $d\mathbf{P}_{t} = \mathrm{diag}(\mathbf{P}_{t})[\mu_{t}dt + \sigma_{t}d\mathbf{W}_{t}]$, again $\mu_{t}$ is a vector and $\sigma_{t}$ is a matrix. Inserting the $\mathbf{P}_{t}$ and $D_{t}$ dynamics we get

$$\begin{align*} d\mathbf{y}_{t} &= D_{t}d\mathbf{P}_{t} + \mathbf{P}_{t}dD_{t}\\ &= D_{t}\left[\mathrm{diag}(\mathbf{P}_{t})[\mu_{t}dt + \sigma_{t}d\mathbf{W}_{t}]\right] + \mathbf{P}_{t}\left[-D_{t}r_{t}dt\right]\\ &= D_{t}\mathrm{diag}(\mathbf{P}_{t})\mu_{t}dt - \mathbf{P}_{t}D_{t}r_{t}dt + \mathrm{diag}(\mathbf{P}_{t})D_{t}\sigma_{t}d\mathbf{W}_{t}\\ &= \mathrm{diag}(\mathbf{y}_{t})\mu_{t}dt - \mathbf{y}_{t}r_{t}dt + \mathrm{diag}(\mathbf{y}_{t})\sigma_{t}d\mathbf{W}_{t}. \end{align*}$$ This is where I'm stuck and it is probably some linear algebra rule that I'm missing, when deriving the relevant derivatives. To get the right result I need $\mathbf{y}_{t}r_{t}dt$ to be $\mathrm{diag}(\mathbf{y}_{t})r_{t}\mathbf{1}dt$.

Any help would be appreciated.

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1 Answer 1

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The realisation we need to make is that $\mathbf{P}_{t} = \mathrm{diag}(\mathbf{P}_{t})\mathbf{1}$. Doing this, we get

$$\begin{align*} d\mathbf{y}_{t} &= D_{t}d\mathbf{P}_{t} + \mathbf{P}_{t}dD_{t}\\ &= D_{t}\left[\mathrm{diag}(\mathbf{P}_{t})[\mu_{t}dt + \sigma_{t}d\mathbf{W}_{t}]\right] + \mathbf{P}_{t}\left[-D_{t}r_{t}dt\right]\\ &= D_{t}\mathrm{diag}(\mathbf{P}_{t})\mu_{t}dt - \mathbf{P}_{t}D_{t}r_{t}dt + \mathrm{diag}(\mathbf{P}_{t})D_{t}\sigma_{t}d\mathbf{W}_{t}\\ &= D_{t}\mathrm{diag}(\mathbf{P}_{t})\mu_{t}dt - \mathrm{diag}(\mathbf{P}_{t})\mathbf{1}D_{t}r_{t}dt + \mathrm{diag}(\mathbf{P}_{t})D_{t}\sigma_{t}d\mathbf{W}_{t}\\ &= \mathrm{diag}(\mathbf{P}_{t}D_{t})\mu_{t}dt - \mathrm{diag}(\mathbf{P}_{t}D_{t})\mathbf{1}r_{t}dt + \mathrm{diag}(\mathbf{P}_{t}D_{t})\sigma_{t}d\mathbf{W}_{t}\\ &= \mathrm{diag}(\mathbf{y}_{t})\mu_{t}dt - \mathrm{diag}(\mathbf{y}_{t})r_{t}\mathbf{1}dt + \mathrm{diag}(\mathbf{y}_{t})\sigma_{t}d\mathbf{W}_{t}\\ &= \mathrm{diag}(\mathbf{y}_{t})[(\mu_{t} - r_{t}\mathbf{1})dt + \sigma_{t}d\mathbf{W}_{t}] \end{align*}$$

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