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8

I can clarify 100% that $(dw)^2$= $dt$ and recommend you to accept it as a fact. Like any other differential, this differential is defined in terms of its integral: $$\int_{t_{0}}^{t_{1}}(dW)^{2}\equiv\lim_{n\rightarrow\infty}\sum_{k=0}^{n-1}[W(t_{k+1})-W(t_{k})]^{2}$$ Where $t_{k}=t_{0}+k(t_{1}-t_{0})/n$. Since $$W(t_{k+1})-W(t_{k})=\sqrt{t_{k+1}-t_{k}}\... 8 We know that (\tilde{W}_t) := (-W_t) is also a Wiener process so$$ E[W_pW_qW_r] = E[\tilde{W}_p\tilde{W}_q\tilde{W}_r] = (-1)^3E[W_pW_qW_r] and that implies that E[W_pW_qW_r] = 0. 8 @Ivan's comment regarding the covariances is the key. Consider an equally spaced partition \Pi_n = \left\{ t_0 = 0, t_1 = \Delta_n, \ldots, t_n = t \right\} of the interval [0, t], where t_i = i \Delta_n and \Delta_n = t / n so that X_t = \lim_{n \rightarrow \infty} X_n, \qquad X_n = \sum_{i = 1}^n W_{t_i} \left( t_i - t_{i - 1} \... 7 Firstly, m_T=\min\limits_{t\in[0,T]} B_t = -\max\limits_{t\in[0,T]} -B_t \overset{Law}{=} -\max\limits_{t\in[0,T]} B_t = -M_T. So, you can either consider the running maximum or minimum. Let \tau be a stopping time and (B_t) a Brownian motion. Then, \begin{align*} W_t =\begin{cases} B_t & t\leq \tau, \\ 2B_\tau - B_t & t\geq \tau, \end{cases} ... 7 Let (W_t) be a standard Brownian motion and a>0. We define X_t=e^{aW_t-\frac{1}{2}a^2t}. Then, the process (X_t) is adapted and integrable which are the first two conditions of being a martingale. Finally, for any s<t, \begin{align*} \mathbb{E}\left[ X_t\mid\mathcal{F}_s\right] &= \mathbb{E}\left[ e^{aW_t-\frac{1}{2}a^2t}\mid\mathcal{F}... 7 The trick is to start with the highest power, rewrite it as something you know (a third order moment) and then work backwards on the remaining terms. By that I mean you can complete the cube as follows:E[W_t^3 - 3tW_t|\mathcal{F}_s] = E[(W_t-W_s)^3 - C -3tW_t|\mathcal{F}_s]$$where you'll need to find C such that the equality holds (i.e. C=W_s^3 + ... 6 yes, by definition. The integral is by definition the limit$$ \int_0^T f_s dW_s = lim_{ n \to \infty } \sum_{ s_i \in \mathcal{P}_n } f_{s_i} ( W_{ s_{i+1} } - W_{s_i} ), $$where \mathcal{P}_n is a partition of n points of the interval [0,T]. In your case, for any partition, the sum above is telescopic and always evaluates to W_T. so the limit ... 5 Set Z_t := f(W_t,t) where f(x,t) = e^{ax -\frac{1}{2}a^2t}. Then applying Ito's lemma we get$$dZ_t = aZ_tdW_t.This means that Z_t is a local martingale. For Z_t to be a martingale you have to prove that E \int_0^t|a e^{aW_t -\frac{1}{2}a^2u}|^2 du <\infty. For that you can use Fubini's theorem. \begin{align*} E \int_0^t|a e^{a W_u - \frac{1}{... 4 If you examine a standard definition of a Wiener process, W_t - W_0 follows the normal distribution with mean zero and variance t. If you want the variance to be something else, you have to scale it. 4 In this particular case, the simplest way to compute the expected value is to write \cos(x) = \Re(e^{ix}) and use the formula for the characteristic function of a Gaussian variable: if Z \sim \mathcal{N}(\mu,\sigma^2), E[e^{iuZ}] = e^{iu\mu - \frac{1}{2}u^2 \sigma^2 } (simply write the expected value as an integral \int_{\mathbb{R}} e^{iuz} \frac{1}{\... 4 This is wrong! Notice that dX_t=\mu(t,X_t)dt + \sigma(t,X_t)dW is a shorthand for\int_0^tdX_s = \int_0^t \mu(s,X_s)ds + \int_0^t\sigma(s,X_s)dW_s$$Integrating:$$X_t-X_0 = \int_0^t \mu(s,X_s)ds + \int_0^t\sigma(s,X_s)dW_s \text{ (eq.1)} $$If we take expectations, remembering that \mathbb{E}[\int_0^t\sigma(s,X_s)dW_s]=0, we have$$\mathbb{E}[...

3

Using only words and no equations: Knowing the Variance (or standard deviation) of a Brownian Motion we can calculate the uncertainty in the future position of a particle. Knowing $\sigma^2$ and assuming the particle starts at $S_0$ we can say that $S_T$ will be in $[S_0-1.96 \sigma, S_0+1.96 \sigma]$ 95% of the time. In other words 95% of the trajectories ...

3

Note that, for $t>s>0$, \begin{align*} X_t-X_s &= \frac{1}{t}\int_0^t udW_u - \frac{1}{s}\int_0^s udW_u\\ &=\frac{1}{t}\bigg(\int_s^t u dW_u + \int_0^s udW_u \bigg)- \frac{1}{s}\int_0^s udW_u\\ &=\frac{1}{t} \int_s^t u dW_u + \Big(\frac{1}{t} -\frac{1}{s}\Big)\int_0^s udW_u\\ &=\frac{1}{t} \int_s^t u dW_u - \frac{t-s}{t} X_s. \end{align*...

3

I am curious about the target of your question. It is rarely put as $$\mathbb{E}\left[\int_0^{\tau}X_t{\rm d}t\Bigg|X_{\tau}=a\right],$$ because, as per your statement, $\tau$ is defined as $$\tau=\inf\left\{t>0:X_t=a\right\}.$$ Following this definition, it is a must that $X_{\tau}=a$. Hence it is unnecessary to be conditioned. As far as I know, ...

3

\begin{align*} E\Big(W_t^3-3tW_t \mid \mathcal{F}_s\Big) &= E\Big((W_t-W_s+W_s)^3-3t(W_t-W_s+W_s) \mid \mathcal{F}_s\Big) \\ &=E\Big((W_t-W_s)^3+W_s^3+3(W_t-W_s)^2W_s + 3 (W_t-W_s)W_s^2\\ &\qquad \qquad -3t(W_t-W_s)-3tW_s \mid \mathcal{F}_s\Big) \\ &=E\Big((W_t-W_s)^3\Big) + W_s^3+3W_sE\Big((W_t-W_s)^2\Big)\\ &\qquad \qquad + 3W_s^2 E(...

3

 My "answer" below is not a really an answer for I have completely misinterpreted your original question. I thought you asked about the covariance of 2 processes over a given time horizon (i.e. for a fixed $\omega$) and not the covariance of two random variables (fixed $t$). Also note that $\text{cov}(x,y)=0$ does not mean that $x$ and $y$ are ...

3

Your solution is correct. Generally speaking, for any $\alpha,\beta\in\mathbb{R}$, the drift $\mu_{X^{\alpha\beta}}$ of the process: $$X_t^{\alpha\beta}:=e^{\alpha t+\beta W_t}$$ will be equal to: $$\mu_{X^{\alpha\beta}}=\left(\alpha+\frac{\beta^2}{2}\right)X_t^{\alpha\beta}$$

3

The proof uses the martingale property of the Ito integral. For an adapted stochastic process $X_t$ such that $$\mathbb{E}\int_0^{t}|X_s|^2ds <\infty$$ we have $$\mathbb{E}\int_0^{t}X_sdW_s =0$$ Now your result follows by setting $$X_t=\frac{1}{W_t^2+1}.$$ To see that the square integrability condition is satisfied note $$\mathbb{E}\int_0^{t}\frac{1}{(W_s^... 2 I believe that the process you postulate has a Beta conditional distribution. If my memory serves me well, I have encountered it in the book by Liptser and Shiryayev "Statistics of Random Processes" as the evolution of the conditional probability in a HMM. This was 10+ years ago, therefore I might be well off. In that case you should be sampling from Beta ... 2 The following is not a proof but some reasoning: If you consider the L2-limits then you see something along the lines:$$ dW^2 = \lim_{n \rightarrow \infty}\sum_{j=1}^n (W_{t_{j+1}} - W_{t_{j}})^2 \rightarrow t $$For W_1, W_2 with correlation \rho this transaltes to$$ dW^1 dW^2 = \lim_{n \rightarrow \infty}\sum_{j=1}^n (W_{t_{j+1}}^1 - W_{t_{j}}^1)(W_{...

2

$\sigma S$ is in units of dollars per square root of a unit of time. $\sigma$ is usually quoted as an annual or daily percentage. $dX ^2$ is in units of time, as $E[(dX)^2] = dt$. Here is an online tutorial which you may find helpful. EDIT by kotozna: $\sigma$ has dimensions 1/(square root of time) and $dX$ has dimensions square root of time. ...

1

You state $X(t_j) - X(t_{j-1}) \backsim \mathcal{N}(0, \frac{t}{n})$. Thus: $$X(t_j) - X(t_{j-1}) = \sqrt{\frac{t}{n}} Z ,$$ where $Z \backsim \mathcal{N}(0, 1)$. Note that: \begin{align} & \mathbb{E} \sqrt{\frac{t}{n}} Z = 0 \\ & \mathbb{E} \left( \sqrt{\frac{t}{n}} Z \right)^2 = \frac{t}{n} \mathbb{E} Z^2 = \frac{t}{n} \\...

1

The Bachelier model which assumes that price follows a normal distribution is a correct approximation for the Black-Scholes one for short times t. When time is short it's fine to ignore drift because the movements will be more driven by volatility For longer time intervals you cannot ignore drift any more. But the actual distribution of the stock price at ...

1

Based on the form of your equation, we can consider the SDE \begin{align*} dX_t = \sigma dW_t, \end{align*} where $W$ is a standard Brownian motion. Since, for $0 \leq t \leq T$, \begin{align*} X_T = X_t + \sigma (W_T-W_t), \end{align*} based on Feynman–Kac formula, the solution is given by \begin{align*} F(t, x) &= E\left(X_T^2 \mid X_t = x\right)\\ &...

1

The first part has already been answer by @Uditg_ucla, so I am only providing answer of your 2nd part. Rewriting your SDE in more sophisticated way: $$dS=k(b-S)dt+\sigma S dz$$ You want SDE for $S^2$. Using Taylor series, it can be written as: $$df(S)=f'(S)dS + \frac{1}{2!}f''(S)(dS)^2+\cdots$$ $$df(S)=2SdS+(dS)^2$$ df(S)=2S[k(b-S)dt+\sigma S dz]+\...

1

Not sure I fully understand your question. However, I'd suggest using the Ito's lemma (second equation on wikipedia page https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma) to solve for dY. In both cases, dY will have both a drift term and a stochastic term. The coefficient of the stochastic term will indicate what sort of probability process Y follows. e.g., ...

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