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# 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|MAST90059 Brownian motion

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Brownian motion

The stochastic process that we shall focus our attention on in this course is Brownian motion (BM) $W=\left(W_t\right){t \geq 0}$ (which is the building block for all stochastic calculus involving processes with continuous sample paths). Summarising it very briefly, BM is a continuous process with stationary, independent, normally distributed (or Gaussian) increments of mean zero and variance equal to the time elapsed during the increment. That is, for $0 \leq s \leq t<\infty$, the increment $W_t-W_s$ is distributed according to $W_t-W_s \sim \mathrm{N}(0, t-s)$ and is independent of $W_s$ (and, indeed, is independent of the entire history $\left(W_u\right){0 \leq u \leq s}$ ). Here, and elsewhere, we shall use the notation whereby $X \sim \mathrm{N}\left(m, s^2\right)$ denotes that a random variable $X$ is normally distributed with mean $m$ and variance $s^2$.

We shall give some rigorous definitions of BM shortly. BM has all the quintessential properties one could hope for: it is a martingale and a Markov process (as well as having continuous paths), and can also be thought of as the limit of a scaled and speeded-up random walk. However, the paths of BM are highly irregular, as we shall see; they are not differentiable, and are of infinite length over any finite time interval.

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Remarks on existence of BM

Remarks on existence of $\mathbf{B M}$. The existence of $\mathrm{BM}$ as a well-defined mathematical object is a non-trivial issue, but one that will not concern us – we shall assume the existence of BM, though we now make some brief remarks with references to where constructions of BM can be found.

One approach to showing the existence of $\mathrm{BM}$ is to write down what the finite-dimensional distributions of the process (based on stationarity, independence and normality of its increments) must be, and then construct a probability measure and a process on an appropriate measurable space in such a way that we obtain the prescribed finite-dimensional distributions.

This technique is a standard approach to constructing a Markov process (of which BM is an example), and can be lengthy and technical, see Karatzas and Shreve [13, Section 2.2]. That this procedure works is guaranteed by deep theorems initiated by Kolmogorov (the “consistency” theorem, and the “continuity” theorem). These say, respectively, that given a set of finite-dimensional distributions (FDDs), one can indeed construct a well-defined stochastic process with these FDDs, and under additional conditions on the moments of the increments (so, on $\mathbb{E}\left[\left|X_t-X_s\right|^\alpha\right], s \leq t$, for some positive $\alpha$ ), the process can be assumed to almost surely (so, with probability one) have continuous paths.

There are also a number of more direct constructions of BM. The first rigorous construction was by Norbert Wiener $[26,27]$ using Fourier series methods (his construction is outlined in Bass [1, Section 6.1]). Einstein, in 1905 [8], derived the transition density (the probability density $p(t, x)$ for the BM moving from 0 to $x$ in time $t$ ) for BM by considering the molecular-kinetic theory of heat (as being composed of many random collisions of molecules). An approach similar in spirit to Wiener’s (so also based on Hilbert space theory, namely sets of orthonormal functions), which uses the so-called Haar functions, was carried out by Lévy [18] and later simplified by Ciesielski [5]. Such a construction can be found in Karatzas and Shreve [13, Section 2.3].

Yet another another proof for the existence of BM is based on the idea of a weak limit (so a convergence in distribution) of a sequence of random walks. This construction can be found in Karatzas and Shreve [13, Section 2.4]. It involves constructing a sequence of probability spaces such that the corresponding sequence of probability measures satisfies a property known as tightness. One has a sequence of processes $\left(X^n\right){n \in \mathbb{N}}$ which induce a sequence of tight probability measures on $(C([0, \infty)), \mathcal{B}(C([0, \infty))))$ (the space of continuous functions on $[0, \infty)$ equipped with the Borel $\sigma$-field on this space), such that there is a limiting measure $\mathbb{P}*$ (called Wiener measure) under which the coordinate mapping process $W_t(\omega):=\omega(t), t \geq 0$ on $C([0, \infty))$ is a standard, one-dimensional Brownian motion. This is a lengthy and technical construction, to say the least. It is also possible to construct a probability space on which all the random walks are defined and converge to BM almost surely, rather than merely in distribution, as shown by Knight .

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Remarks on existence of BM

BM 也有许多更直接的结构。第一个严格的构造是由 Norbert Wiener $[26,27]$ 使用傅立叶级数方法（他的构造在 Bass [1，第 $6.1$ 节] 中进行了概述) 。咾因斯坦在1905年 [8]推导了跃迁密度 (概率密度 $p(t, x)$ 对于 BM 从 0 移动到 $x$ 及时 $t$ ) 对于 BM，通过考虑 热的分子动力学理论 (由许多分子的随机碰噇组成)。Lévy [18] 实施了一种与 Wiener 的方法 (因此也甚于 Hilbert 空间理论， 即正交函数集) 在本质上类似的方法，该方法使用所调的 Haar 函数，由 Lévy [18] 进行，后来由 Ciesielski [5] 进行了简化。这 种结构可以在 Karatzas 和 Shreve [13，第 $2.3$ 节] 中找到。

BM 存在的另一个证明是基于随机游走序列的弱极限（因此分布收敛）的想法。这种结构可以在 Karatzas 和 Shreve [13，第 $2.4$ 节] 中找到。它涉及构建概率空间序列，使得相应的概率测度序列满足称为苭度的属性。一个有一系列的过程 $\left(X^n\right) n \in \mathbb{N}$ 这引起 了一手列严格的概率测量 $(C([0, \infty)), \mathcal{B}(C([0, \infty)))$ ) (连紏函数的空间 $[0, \infty)$ 配备了 Borel $\sigma$-这个空间上的场)，这样就有一 个限制措施 $\mathbb{P} *$ (称为维妠测度) 下的坐标映射过程 $W_t(\omega):=\omega(t), t \geq 0$ 上 $C([0, \infty))$ 是标准的一维布朗运动。至少可以说， 这是一个几长的技术结构。还可以构建一个概率空间，在该空间上定义了所有随机游走并几夹肯定地收敛到 BM，而不是仅仅在分 布中，如 Knight 所示。

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