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# 金融代写|固定收益与信贷代写Fixed Income and Credit代考|FINA406 Self-financing Portfolios in Discrete Time

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## 金融代写|固定收益与信贷代写Fixed Income and Credit代考|Self-financing Portfolios in Discrete Time

We consider a financial market living in discrete time on a filtered probability space $(\Omega, \mathcal{F}, P, \mathbf{F})$, so we are allowed trade at discrete points in time $t=0,1,2, \ldots$.

Basic Definitions
On the market we can trade in $N$ different assets with (adapted) price processes $S^1, \ldots, S^N$. We will use the following notation.
Definition $6.1$
\begin{aligned} S_n^i & =\text { the price of one unit of asset No. } i \text { at time } n, \ h_n^i & =\text { number of units of asset No. } i \text { bought at time } n, \ d_n^i & =\text { dividends from asset No. } i \text { at time } n, \ h_n & =\text { the portfolio }\left[h_n^1, \ldots, h_n^N\right], \ c_n & =\text { consumption at time } n, \ V_n & =\text { the value of the portfolio } h_{n-1} \text { at time } n .\end{aligned}
The interpretation of the dividend process $d$ is that if you are holding one unit of asset No. $i$ during the interval $[n-1, n]$, then you obtain the amount $d_n$ at time $n$.

$$d_{n+1}^i=\Delta D_n^i . \quad i=1, \ldots, N$$
The point of all this is that the concept of a cumulative dividend process can easily be extended to continuous time, and we now reformulate Proposition $6.5$ in the new terms as follows.

## 金融代写|固定收益与信贷代写Fixed Income and Credit代考|Self-financing Portfolios in Continuous Time

We now move to continuous time and consider a financial market with $N$ assets.
Definition $6.9$
$S_t^i=$ the price of one unit of asset No. $i$ at time $t$,
$h_t^i=$ number of units of asset No. $i$ held at time $t$,
$h_t=$ the portfolio $\left[h_t^1, \ldots, h_t^N\right]$,
$D_t^i=$ the cumulative dividend process for asset No. $i$,
$c_t=$ consumption rate at time $n$,
$V_t=$ the value of the portfolio $h_t$ at time tn.
Remark 6.3.1 There are two important differences from the discrete time definitions, and these concern consumption and dividends:

In discrete time, $c_n$ denotes consumption at time $n$, so it is measured in dollars. The consumption rate $c_t$ above, on the other hand, is measured in dollars per unit time, so in continuous time we have a continuous flow of consumption, where the dollar value of consumption over an infinitesimal interval $[t, t+d t]$ is given by $c_t d t$. It is possible to include “impulse consumption” where you get a lump sum of money at a specific point in time, but this would require a more general stochastic integration theory, so we do not include this.

We assume that the cumulative dividend process $D^i$ has a stochastic Itô differential, so in particular it has continuous trajectories. We thus exclude discrete dividends. To include discrete dividends would require a more general stochastic integration theory involving jump processes, and we would also have to change some of the definitions above.

## 金融代写|固定收益与信贷代写Fixed Income and Credit代考|Self-financing Portfolios in Discrete Time

$S_n^i=$ the price of one unit of asset No. $i$ at time $n, h_n^i \quad=$ number of units of asset No. $i$ bought at time $n, d_n^i=$ dividends from asset No. $i$ at ti:

$$d_{n+1}^i=\Delta D_n^i . \quad i=1, \ldots, N$$

## 金融代写|固定收益与信贷代写Fixed Income and Credit代考|Self-financing Portfolios in Continuous Time

$S_t^i=$ 一单位资产编号的价格 $i$ 在时间 $t$,
$h_t^i=$ 资产编号单位数 $i$ 在时间举行 $t$,
$h_t=$ 投资组合 $\left[h_t^1, \ldots, h_t^N\right]$ ，
$D_t^i=$ 痹产昊甸计分红过程 $i$,
$c_t=$ 时间消岪率 $n$,
$V_t=$ 投资组合的价值 $h_t$ 在时间 $\mathrm{tn}$ 。

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