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

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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 $1.2, \mathrm{p} 479]$.

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？



\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},

\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,

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 $\xi\left(T^{ }\right) \in L^{q}(P).磷r○○F.吨H和和q你一世v一种一世和nC和○F(一世)一种nd(一世一世)H一种sb和和ns和和n一种b○v和.F○r吨H和和q你一世v一种一世和nC和○F(一世一世)一种nd(一世一世一世)s和和s和C吨一世○n3.d.3\left(\xi(t)=E_{P}\left[d Q / d P \mid \mathcal{F}_{t}\right], \xi\left(T^{ }\right)=d Q / d P\right)$。