随机微积分作业代写stochastic calculus代考| THE GENERAL MARKET MODEL

随机微积分作业代写stochastic calculus代考|  THE GENERAL MARKET MODEL

随机微积分(stochastic calculus),数学概念,是高等数学中研究函数的微分(Differentiation)、积分(Integration)以及有关概念和应用的数学分支。它是数学的一个基础学科,内容主要包括极限、微分学、积分学及其应用。微分学包括求导数的运算,是一套关于变化率的理论。它使得函数、速度、加速度和曲线的斜率等均可用一套通用的符号进行讨论。积分学,包括求积分的运算,为定义和计算面积、体积等提供一套通用的方法

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随机微积分作业代写stochastic calculus代考

微积分作业代写calclulus代考|Absence of arbitrage and existence of a local spot martingale measure

3.e Absence of arbitrage and existence of a local spot martingale measure. Let $B=\left(B_{0}, B_{1}, \ldots, B_{n}\right)$ be a finite market. From 3.d.5 we know that the existence of an $A$-local martingale measure $Q$ for $B$ rules out arbitrage strategies in the space $S\left(B^{A}\right)$. Let us now investigate the converse: Does the absence of arbitrage in $S\left(B^{A}\right)$ imply the existence of an A-local martingale measure?

We will assume that the numeraire $A$ is a security in the market $B$. Then the market $B^{A}$ contains the riskless bond $A^{A}(t)=1$. Equivalently, replacing the market $B$ with $B^{A}$, we may assume that
$$
B_{0}(t)=A(t)=1
$$
and seek an equivalent probability $Q$ on $\mathcal{F}=\mathcal{F}{T}$. such that all assets $B{i}$ in $B$ are $Q$-local martingales. Let us call such $Q$ a local martingale measure for $B$ (a local 1-martingale measure in previous terminology).

It turns out that it suffices to have no arbitrage in a much smaller space of trading strategies than $S(B)$. Indeed, let $\mathcal{V}$ be the space of all trading strategies $\theta \in S(B)$ such that the process $V_{t}(\theta)$ is uniformly bounded, that is, $\left|V_{t}(\theta)\right| \leq C$ for some constant $C$ and all $t \in\left[0, T^{}\right]$ and let $$ V=\left{V_{T^{}}(\theta) \mid \theta \in \mathcal{V}\right} \subseteq L^{\infty}(\Omega, \mathcal{F}, P)
$$
be the space of all terminal payoffs $h=V_{T^{*}}(\theta)$ of the trading strategies $\theta \in \mathcal{V}$. Write $L^{\infty}=L^{\infty}(\Omega, \mathcal{F}, P), L^{p}=L^{p}(\Omega, \mathcal{F}, P)$ and $L^{0}=L^{0}(\Omega, \mathcal{F}, P)$. Recall that $L^{0}$ is simply the space of all $\mathcal{F}$-measurable random variables.

The space $\mathcal{V}$ contains sufficiently many trading strategies to imply properties for the asset processes $B_{i}$ themselves. Consider an asset $B_{i}$ in $B$. Since the process $B_{i}$ is not necessarily bounded, $\mathcal{V}$ will not in general contain the buy and hold strategy investing in $B_{i}$. However, fixing $k>\left|B_{i}(0)\right|$ and setting
$$
\tau_{k}=\inf \left{t>0:\left|B_{i}(t)\right|>k\right},
$$

we have $\left|B_{i}\left(t \wedge \tau_{k}\right)\right| \leq k, t \in\left[0, T^{}\right]$, by continuity of $B_{i}$. If $\tau$ is any optional time with values in $\left[0, T^{}\right]$, let $\rho_{k}=\tau \wedge \tau_{k}$ and consider the strategy $\theta=\left(\theta_{0}, \ldots, \theta_{n}\right)$ investing only in $B_{0}=1$ and $B_{i}$, buying one share of $B_{i}$ at time zero, selling this at time $\rho_{k}$, with the proceeds buying shares of $B_{0}$ and holding these until time $T^{}$. More formally $$ \theta_{j}=0, j \neq 0, i, \quad \theta_{i}=1_{\left[0, \rho_{k}\right]}, \quad \theta_{0}=1_{\beth \rho_{k}, T^{} \rrbracket} B_{i}\left(\rho_{k}\right) .
$$
We claim that $\theta \in \mathcal{V}$. Clearly $V_{t}(\theta)=B_{i}\left(t \wedge \rho_{k}\right)$ is uniformly bounded. Let us now verify that $\theta$ is self-financing. Indeed, since $d B_{0}(s)=0$, we have
$$
\begin{aligned}
V_{t}(\theta) &=B_{i}\left(t \wedge \rho_{k}\right)=B_{i}(0)+\int_{0}^{t \wedge \rho_{k}} 1 d B_{i}(s) \
&=V_{0}(\theta)+\int_{0}^{t}\left{1_{\left[0, \rho_{k}\right]} d B_{i}(s)+1_{] \rho_{k}, T^{}\right]} B_{i}\left(\rho_{k}\right) d B_{0}(s)\right} \ &=V_{0}(\theta)+\int_{0}^{t} \theta(s) \cdot d B(s), \quad \text { as desired. } \end{aligned} $$ Price functional. Assume that there is no arbitrage in $\mathcal{V}$. If $\theta \in \mathcal{V}$, the element $h=V_{T^{}}(\theta) \in V$ can be thought of as a random payoff (claim) at time $T^{}$ which can be implemented at time zero at cost $V_{0}(\theta)$ by trading according to the strategy $\theta$. It is now tempting to define the price $\pi_{0}(h)$ at time zero of $h$ as $\pi_{0}(h)=V_{0}(\theta)$. To see that this price is well defined, we must show that $\theta, \phi \in \mathcal{V}$ and $V_{T^{}}(\theta)=h=V_{T^{}}(\phi)$ implies that $V_{0}(\theta)=V_{0}(\phi)$. To this end it will suffice to show that $\theta \in \mathcal{V}$ and $V_{T^{}}(\theta) \geq 0$ implies that $V_{0}(\theta) \geq 0$. The proof of this is very similar to the proof of the following $\quad \theta \in \mathcal{V}, V_{T^{}}(\theta) \geq 0, V_{T^{}}(\theta) \neq 0 \quad \Rightarrow \quad V_{0}(\theta)>0$,
which is needed below and which follows easily from the absence of arbitrage in $\mathcal{V}$. Note first that $\mathcal{V}$ contains the strategy $\rho$ which buys one share of the asset $B_{0}=1$ at time zero and holds until time $T^{}$. Assume now that (contrary to (2)) $\theta \in \mathcal{V}, V_{T^{}}(\theta) \geq 0, V_{T^{}}(\theta) \neq 0$ (hence $P\left[V_{T^{}}(\theta)>0\right]>0$ ) but $V_{0}(\theta) \leq 0$. Set $a=-V_{0}(\theta) \geq 0$ and consider the strategy $\chi=\theta+a \rho \in \mathcal{V}$ going long $\theta$ at time zero and using the surplus $a$ to buy and hold shares of $B_{0}$. Then $\chi$ has zero cost at time zero and payoff $V_{T^{}}(\chi)=V_{T^{}}(\theta)+a \geq V_{T^{*}}(\theta)$. Thus $\chi$ is an arbitrage in $\mathcal{V}$.

For a subspace $W \subseteq L^{p}(P), 1 \leq p \leq \infty$, let $W_{+}={h \in W \mid h \geq 0}$ and call a linear functional $\pi$ on $W$ strictly positive, if $h \in W_{+}, h \neq 0$ implies $\pi(h)>0$. Here $h=0$ means $h=0, P$-as. Let $q$ be the exponent conjugate to $p$, defined by $(1 / p)+(1 / q)=1$

If $Q$ is any measure on $\mathcal{F}$ which is absolutely continuous with respect to $P$ and satisfies $m=d Q / d P \in L^{q}$, then $\pi(h)=E_{Q}(h)=E_{P}(m h)$ defines a positive linear functional on $L^{p}$. This functional is strictly positive if and only if $m>0, P$-as., that is, iff $Q$ is equivalent to $P$.

微积分作业代写calclulus代考|Theorem

3.e.1 Theorem. Assume that $B=\left(B_{0}, \ldots, B_{n}\right)$ is a finite market containing the riskless bond $B_{0}=1,1 \leq p<\infty$ and $q$ is the exponent conjugate to $p$. Then the following are equivalent:
(i) $B$ is $S A F(p)$.
(ii) There exists a local martingale measure $Q$ for $B$ with $d Q / d P \in L^{q}(P)$.
(iii) There exists a local deflator $\xi$ for $B$ with $\xi\left(T^{}\right) \in L^{q}(P)$. Proof. The equivalence of (i) and (ii) has been seen above. For the equivalence of (ii) and (iii) see section 3.d.3 $\left(\xi(t)=E_{P}\left[d Q / d P \mid \mathcal{F}_{t}\right], \xi\left(T^{}\right)=d Q / d P\right)$.

Remark. We really do not care whether the Radon-Nikodym derivative of a local martingale measure $Q$ for $B$ is in $L^{q}(P)$ but this is a consequence of the condition $S A F(p)$. This suggests that $S A F(p)$ is too strong if all we want is a local martingale measure for $B$. Indeed, this condition was formulated so as to be able to apply Clark’s Extension Theorem and thus to stay within the bounds of reasonably simple functional analysis.

The conditions $S A F(p)$ become increasingly weaker as $p \uparrow \infty$ and one wonders what happens for $p=\infty$. This case is inaccessible to our means, as $L^{\infty}(P)$ is not a separable space. If $B$ is a finite market with locally bounded asset prices and time horizon $T^{*}=1$ the existence of an equivalent local martingale measure is equivalent to the condition NFLVR (no free lunch with vanishing risk) defined as follows:
There does not exist a sequence $\left(\theta_{n}\right)$ of self-financing trading strategies in $B$ satisfying $V_{0}\left(\theta_{n}\right)=0, V_{1}\left(\theta_{n}\right)^{-} \rightarrow 0$ uniformly and $V_{1}\left(\theta_{n}\right) \rightarrow f, P$-as., as $n \uparrow \infty$, for some function $f \geq 0$ with $P(f>0)>0$.
The reader is referred to [DS, Corollary $1.2, \mathrm{p} 479]$.

随机微积分作业代写stochastic calculus代考| THE GENERAL MARKET MODEL

微积分作业代写calclulus代考|Absence of arbitrage and existence of a local spot martingale measure

3.e 不存在套利且存在局部即期鞅测度。让乙=(乙0,乙1,…,乙n)是一个有限的市场。从 3.d.5 我们知道存在一个一种-局部鞅测度问为了乙排除空间中的套利策略 $ S \left(B^{A}\right).一世和吨你sn○在一世nv和s吨一世G一种吨和吨H和C○nv和rs和:D○和s吨H和一种bs和nC和○F一种rb一世吨r一种G和一世nS \left(B^{A}\right)$ 暗示存在 A 局部鞅测度?

我们将假设 numeraire一种是市场上的证券乙. 那么市场乙一种包含无风险债券一种一种(吨)=1. 等效地,替换市场乙和乙一种,我们可以假设
乙0(吨)=一种(吨)=1
并寻求一个等价的概率问在F=F吨. 这样所有的资产乙一世在乙是问-本地鞅。让我们这样称呼问局部鞅测度乙(以前术语中的局部 1-鞅测度)。

事实证明,在比 $ S (B)小得多的交易策略空间中没有套利就足够了.一世nd和和d,一世和吨\数学{V}b和吨H和sp一种C和○F一种一世一世吨r一种d一世nGs吨r一种吨和G一世和s\theta \in S(B)s你CH吨H一种吨吨H和pr○C和ssV_{t}(\theta)一世s你n一世F○r米一世和b○你nd和d,吨H一种吨一世s,\left|V_{t}(\theta)\right| \leq CF○rs○米和C○ns吨一种n吨C一种nd一种一世一世t \in\left[0, T^{ }\right]一种nd一世和吨$ V=\left{V_{T^{ }}(\theta) \mid \theta \in \mathcal{V}\right} \subseteq L^{\infty}(\Omega, \mathcal{F}, P )
$$
是所有终端收益的空间H=五吨∗(θ)的交易策略θ∈五. 写一世∞=一世∞(Ω,F,磷),一世p=一世p(Ω,F,磷)和一世0=一世0(Ω,F,磷). 回想起那个一世0简直就是所有人的空间F- 可测量的随机变量。

空间五包含足够多的交易策略来暗示资产流程的属性乙一世他们自己。考虑一项资产乙一世在乙. 由于过程乙一世不一定有界,五一般不会包含买入并持有策略的投资乙一世. 然而,固定到>|乙一世(0)|和设置
\tau_{k}=\inf \left{t>0:\left|B_{i}(t)\right|>k\right},\tau_{k}=\inf \left{t>0:\left|B_{i}(t)\right|>k\right},

我们有|乙一世(吨∧τ到)|≤到,吨∈[0,吨],通过连续性乙一世. 如果τ是任何可选时间,其值为[0,吨], 让ρ到=τ∧τ到并考虑策略θ=(θ0,…,θn)只投资于乙0=1和乙一世, 买入一股乙一世在时间为零,在时间出售这个ρ到, 用所得购买股份乙0并持有这些直到时间吨. 更正式的θj=0,j≠0,一世,θ一世=1[0,ρ到],θ0=1乙ρ到,吨\rr括号乙一世(ρ到).
我们声称θ∈五. 清楚地五吨(θ)=乙一世(吨∧ρ到)是一致有界的。现在让我们验证一下θ是自筹资金。确实,自从d乙0(s)=0, 我们有
\begin{aligned} V_{t}(\theta) &=B_{i}\left(t \wedge \rho_{k}\right)=B_{i}(0)+\int_{0}^{t \wedge \rho_{k}} 1 d B_{i}(s) \ &=V_{0}(\theta)+\int_{0}^{t}\left{1_{\left[0, \rho_ {k}\right]} d B_{i}(s)+1_{] \rho_{k}, T^{}\right]} B_{i}\left(\rho_{k}\right) d B_ {0}(s)\right} \ &=V_{0}(\theta)+\int_{0}^{t} \theta(s) \cdot d B(s), \quad \text { 根据需要. } \end{对齐}\begin{aligned} V_{t}(\theta) &=B_{i}\left(t \wedge \rho_{k}\right)=B_{i}(0)+\int_{0}^{t \wedge \rho_{k}} 1 d B_{i}(s) \ &=V_{0}(\theta)+\int_{0}^{t}\left{1_{\left[0, \rho_ {k}\right]} d B_{i}(s)+1_{] \rho_{k}, T^{}\right]} B_{i}\left(\rho_{k}\right) d B_ {0}(s)\right} \ &=V_{0}(\theta)+\int_{0}^{t} \theta(s) \cdot d B(s), \quad \text { 根据需要. } \end{对齐}价格功能。假设没有套利五. 如果θ∈五, 元素H=五吨(θ)∈五可以被认为是一次随机的回报(索赔)吨可以在零时间以成本实施五0(θ)根据策略进行交易θ. 现在很想定义价格圆周率0(H)在零时H作为圆周率0(H)=五0(θ). 为了看到这个价格是明确定义的,我们必须证明θ,φ∈五和五吨(θ)=H=五吨(φ)暗示五0(θ)=五0(φ). 为此,足以表明θ∈五和五吨(θ)≥0暗示五0(θ)≥0. 这个的证明和下面的证明很相似θ∈五,五吨(θ)≥0,五吨(θ)≠0⇒五0(θ)>0,
这是下面需要的,并且很容易从没有套利的情况下得出五. 首先注意五包含策略ρ购买一股资产乙0=1在时间为零并保持到时间吨. 现在假设(与(2)相反)θ∈五,五吨(θ)≥0,五吨(θ)≠0(因此磷[五吨(θ)>0]>0) 但五0(θ)≤0. 放一种=−五0(θ)≥0并考虑策略χ=θ+一种ρ∈五走多远θ在时间为零并使用盈余一种购买并持有乙0. 然后χ在时间为零和收益为零时成本为零五吨(χ)=五吨(θ)+一种≥五吨∗(θ). 因此χ是一种套利五.

对于子空间在⊆一世p(磷),1≤p≤∞, 让在+=H∈在∣H≥0并调用线性泛函圆周率在在严格正数,如果H∈在+,H≠0暗示圆周率(H)>0. 这里H=0方法H=0,磷-作为。让q是指数共轭p, 被定义为(1/p)+(1/q)=1

如果问是任何措施F这是绝对连续的磷并满足米=d问/d磷∈一世q, 然后圆周率(H)=和问(H)=和磷(米H)定义了一个正线性泛函一世p. 这个泛函是严格正的当且仅当米>0,磷-as.,即当且仅当问相当于磷.

微积分作业代写calclulus代考|Theorem

3.e.1 定理。假使,假设乙=(乙0,…,乙n)是一个包含无风险债券的有限市场乙0=1,1≤p<∞和q是指数共轭p. 那么以下是等价的:
(i)乙是小号一种F(p).
(ii) 存在局部鞅测度问为了乙和d问/d磷∈一世q(磷).
(iii) 存在局部通货紧缩指数X为了乙与 $\xi\left(T^{ }\right) \in L^{q}(P).磷r○○F.吨H和和q你一世v一种一世和nC和○F(一世)一种nd(一世一世)H一种sb和和ns和和n一种b○v和.F○r吨H和和q你一世v一种一世和nC和○F(一世一世)一种nd(一世一世一世)s和和s和C吨一世○n3.d.3\left(\xi(t)=E_{P}\left[d Q / d P \mid \mathcal{F}_{t}\right], \xi\left(T^{ }\right)=d Q / d P\right)$。

评论。我们真的不关心局部鞅测度的 Radon-Nikodym 导数是否问为了乙在一世q(磷)但这是条件的结果小号一种F(p). 这表明小号一种F(p)如果我们想要的只是一个局部鞅测度,那么它就太强了乙. 事实上,这个条件的制定是为了能够应用克拉克扩展定理,从而保持在相当简单的泛函分析的范围内。

条件小号一种F(p)变得越来越弱p↑∞有人想知道会发生什么p=∞. 这种情况是我们无法获得的,因为一世∞(磷)不是一个可分离的空间。如果乙是一个具有局部有界资产价格和时间范围的有限市场吨∗=1等效局部鞅测度的存在等效于定义如下的条件 NFLVR(没有消失风险的免费午餐):
不存在序列(θn)自筹资金交易策略乙令人满意的五0(θn)=0,五1(θn)−→0均匀地和五1(θn)→F,磷-as.,作为n↑∞, 对于某些函数F≥0和磷(F>0)>0.
读者参考[DS,推论1.2,p479].

随机微积分作业代写stochastic calculus代考| THE GENERAL MARKET MODEL
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