# Tagged Questions

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### $E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t )ds$

I was trying to compute $E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t) ds$, $\mathcal{F}$ is associated to $W$. I tried the following. 1) Splitting the integral ...
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### Determine the distribution of $\int_0^t (W_s-\frac{s}{t}W_t) ds$, where $(W_s)_{s\geq 0}$ is a brownian motion

I have to find the distribution of $X_t:=\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is ...
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### Expectation of product of stochastic integral and brownian motion

Find the covariance: $$COV((\int_t^T(T-s)dW_s), W_t)$$ I used the covariance formula: COV(X,Y) = E(XY) - E(X)E(Y) = E(XY) as E(X)=E(Y)=0 But I am stuck on figuring out the expectation of the ...
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### Stochastic Integral Help

Let W(t) be a Brownian Motion. Show that the integral: $$\int_t^T W(s)ds$$ can be written in terms of the stochastic integral: $$\int_t^T (T-s)dW(S)$$ Is there an error with this question? I ...
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### Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
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### Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$dV_t = - \beta V_t dt + \sigma dW_t$$ with $V_0 = v$, where $W_t$ is a Wiener ...
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### Stopping times, open sets and Brownian Motion

Let $B_t$ be a brownian motion started at 0. I am trying to prove that $\tau$, defined as: $$\tau = \inf\{t > 0 \mbox{ }|\mbox{ } \left|B_t\right| \geq \frac{1}{1+t} \}$$ is a stopping time with ...
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### Extended (or augmented) stopping times

I am trying to prove that $\tau$, defined as: $$\tau = inf\{t > 0 \mbox{ }|\mbox{ } B_t < t-1 \}$$ is a stopping time with respect to the filtration $(\mathscr{F}_{t+}^B)_{t\geq 0}$ where ...
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### Mean and Variance of Gaussian Process

Let $B = (B_t : t \geq 0)$ be a standard Brownian Motion. Fix $0 \leq s \leq t$. How can I prove that, conditionally on $\{B_s = x, B_t = z\}$, the intermediate value $$B_{\frac{t+s}{2}}$$ has ...
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### Brownian motion transition density question

Let $Y_t = M_t - W_t$ where $M_t$ is the running maximum of brownian motion and $W_t$ is brownian motion. I want to show that $P^0[Y_{t+s} \in dy| Y_t = x] = p(s,x,y)+p(s,x,-y)$ where $p$ is the ...
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### Ornstein-Uhlenbeck process and Markov property

There isn't a similar question in the forum, so here it goes. Firstly, let the O-U velocity process be defined as $$dV_t = -\beta V_t dt + \sigma dB_t$$ with $V_0 = v$, and $B = (B_t), t \geq 0$ a ...
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### Running average of Brownian motion

Question : Let us define the cumulative sum (Brownian motion): $$x_k = \sum_{i=1}^k y_i$$ and the running average : $$\overline{x_k} =\frac{1}{W}\sum_{i=k-W+1}^k x_i$$ for $k>W$, $W$ ...
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### $\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
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### $dX_t=-\mu X_tdt + \sigma dW_t$. Prove that $X_t = e^{-\mu t}X_0 + \sigma \int_0^t e^{-\mu(t-u)}dW_u$

So the solution says use Ito-s formula, taking $Y_t:= e^{\mu t}X_t$ to obtain $dY_t = [\mu e^{\mu t}X_t - e^{\mu t}\mu X_t + e^\mu t \sigma dW_t]$. As far as I can see though, Ito's formula says ...
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### Integrating brownian motion times exponential function

I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that ...
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### An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
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### Ito's Lemma and Geometric Brownian Motion With Jumps

I have a price process: $$dF_t = d\Pi_t - \mu_\pi \sigma_t F_t \gamma \, dt + \sigma_t F_t \, dz$$ And wish to simulate the process $x_t = \ln(F_t)$ by Euler method, ...
I have to do the following exercise: Let $(W_t)$ be a Brownian motion. (a) Does X given by $X_t:=W_t^3$ have constant expectation? (b) Is it a martingale? (c) Does it have independent increments? ...