# Tagged Questions

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### The relationship of $\sigma(f(X))$ and $X$

If X is a random variable and f is a measurable function, 1) Is f(X) measurable with $\sigma(X)$ ? 2) Is X measurable with $\sigma(f(X))$ ? Please give proof & example or counter example. Ok ...
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### Please explain $E[S_{min(n,T)} ]= E [S_{0}]=0$

If $S_{n}$ is a simple random walk i.e $X_{k}= +/- 1$ with prob = 0.5 T = inf {n > = 0 |$S_{n}$ = 1} is a stopping time. T is finite almost surely. .Explain $E[S_{min(n,T)} ]= E [S_{0}]=0$ I know ...
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Show that $$\text{ess-sup}_\Omega g (x+ B_T) = \sup_{y \in \mathbb R ^d }g(y)$$ where $B$ is a d-dimentional brownian motion , $x \in \mathbb R ^d$ and g a Lipschitz bounded function of $\mathbb R ... 0answers 37 views +50 ### Approximation of stochastic processes in Protter I'm reading Stochast integration and stochastic differential equation by Protter. In particular I have a question about Theorem 10 in chapter 2.4. Here Protter defines a simple predictable processes ... 1answer 24 views ### Markov processes and semimartingales Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. ... 0answers 28 views ### Stochastic Differential Equation- When martingale? Suppose I'd like to check the martingale property for some SDE. What do I have to require for it to be martingale? I know that no drift is one requirement, but what are the others? 0answers 34 views ### If two stochastic integrands are equal on some measurable set, will the stochastic integrals be equal on that set? Given a$X$semi-martingale on a filtered probability space$(\Omega,\{\mathcal F_t\}_{t\le\infty},P)$I am trying to prove: For any$B\in\mathcal F_\infty$and processes$a_1,a_2$such that ... 3answers 39 views ### The uniqueness of solution for stochastic differential equation involved with sign function. When I read a paper about Levy distribution thoerem (http://www.maphysto.dk/publications/MPS-RR/1998/22.pdf). In the first page, the author mentioned the following: There is a unique strong solution ... 1answer 46 views ### Diffusion processes I am trying to work out a problem to which I have not found similar solutions on the website. Perhaps you can help me out. Let$X = (X_t)_{t\geq0}$be a non-negative diffusion process which solves ... 0answers 21 views ### Autocovariance of an Ito process [closed] How do you compute the unconditional autocovariance$Cov[dX_t,dX_u]$for an Ito process of the form$dX_t=\mu_{t}dt+\sigma_{t}dW_t$? Thanks in advance. 1answer 26 views ### Distribution of Stopped Brownian motion at hitting time of another Brownian motion. Suppose$B_t$and$W_t$are two independent Brownian motions and$\tau$is the first hitting time of$B_t$to some$a >0$. Compute the distribution of$W_{\tau}$. We can try the characteristic ... 1answer 25 views ### A few questions about Stochastic Processes and Numerical Methods I am having a few problems understanding the Ornstein Uhlenbeck solutions, on wikipedia under solution (http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process) it described using variation of ... 0answers 57 views ### Determine if this is a Martingale I am trying to check if the process$S_t$is a martingale, where$\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$,$S_0 = 1$. We know that$S_t$is a local martingale because if we stop it ... 1answer 42 views ### Sum of two Markov processes another Markov process? Let$dX_{t} = m_1(l_1-X_{t})dt+\sigma_1 dW_{t}$and$dY_{t} = m_2(l_2-Y_{t})dt+\sigma_2(\rho dW_{t}+\sqrt{1-\rho^2}dW_{t}^{1})$where the$m_i$'s,$l_i$'s and$\sigma_i$'s are constants,$\rho \in ...
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We know that any Semimartingale has Quadratic variation. I am interested to know if the converse is also true i.e. if a process has quadratic variation then it is semimartingale. Can some one ...
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### Correlation between two stochastic processes [closed]

Let $$dX_t = k_1 X_t \, dt + \sigma_1 \, dW_t$$ and $$dY_t = k_2 Y_t \, dt + \sigma_2 \left( \rho \, dW_t + \sqrt{1-\rho^{2}} \, dW_t^1\right)$$ where $W_t$ and $W_t^1$ are independent. What is ...
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### Differential of the integral of a stochastic process

In the HJM model one considers the forward rates to be on the form $$\mathrm df(t,T) = \alpha(t,T)\,\mathrm dt + \sigma(t,T)\,\mathrm dW(t)$$ In the proof of showing the drift condition on $\alpha$ ...
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### First hitting time in the one-dimensional case by solving a boundary value problem

If have a question about section 3.1 in the paper Kramers' law: Validity, derivations and generalisations by Nils Berglund. (See http://arxiv.org/abs/1106.5799 page 7 - 9) On page 8 it says, that ...
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### Is any FV-Process a special Semimartingale?

Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special ...
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### lower bounds for a stochastic integral

for all $t \in [0,T]$, consider a stochastic integral as follows: $\int_0^{min \{t^*,T \}} f(t,\omega) dt$ where $f \geq 0$ is a nonnegative stochastic process and $t^*$ is a random stopping time. I ...
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### Proof that the predictable sigma algebra is also generated by continuous and adapted processes

I'm reading George Lowther's blog and have a question about the proof of lemma 2. We want to verify that the predictable sigma algebra is also generated by the continuous and adapted processes. One ...
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### Estimation of a Ito's semi-martingale linear functional

Could someone check my solution for the following problem please? Or maybe propose a smarter/shorter solution. Consider a stochastic process $X=(X_t)_{t \in [0,1]}$ defined in a filtred ...
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### Is it sensible to always assume that the “usual conditions” always hold?

I've read in several places that it is reasonable to assume that the usual conditions (that the filtered space is complete, and that the filtration is right-continuous) hold since one can always ...
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### Oksendal SDE book mistake?

I am reading through Oksendals SDEs. I think there may be a mistake in question 5.18b and I can not find an errata so I was looking for some confirmation. The problem concerns the following SDE ...
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### Integration of Gaussian process

Let $\textbf{G}(t)$ be a zero-mean tight Gaussian process and $f(t)$ be a deterministic function. What theorem can be used to prove that $\int_0^\tau \textbf{G}(t)df(t)$ is a zero-mean Gaussian ...
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### Prove that integral is a Gaussian random variable, compute its mean and variance

I have to prove that $X_t=\int_0^t W_s ds$ is a Gaussian random variable. I need also to compute it's mean and variance. My attempt: Let $W_t$ be a simple adapted process ...
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### Laws and Moments of two dimensional brownian motions

I am a bit rusty on this. So let us consider the following two dimensional standard Brownian motion issued from zero defined on the probability space $(\Omega, \mathcal{F},\mathbb{P})$ (note that, in ...
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### 2-D exponential functional brownian motion

I'm looking for the distribution of $X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$ and $Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$ (where $W_t$ is a standard brownian motion) On most ...
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### When does almost sure convergence of stochastic integral imply $L^2$ convergence?

Consider a probability space $(\Omega, \mathcal{F}, P)$ equipped with a Brownian motion $W$. Let $(\xi_n)_{n=1}^\infty$ be a sequence of adapted $\mathcal{F}(t)$-progressively measurable processes. ...
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### Proving the martingale property of stochastic exponentials of pure jump processes

I am playing with different versions of compound-Poisson like processes with regime-switching features. Then I take stochastic exponentials of these to define a change of measure process. However, how ...
### Stochastic differential equation for $Y(t)=\sqrt{X(t)}$
Assume that $X(t)$ solves the stochastic differential equation $$dX(t)=\sigma(t)dW(t)+\mu(t)dt$$ with $\mu(x)=bx+c$ and $\sigma^2(x)=4x.$ Assume that $X(t)\ge 0$. Find the stochastic differential ...