For questions about stochastic analysis or stochastic calculus, for example the Itô integral.
Questions tagged [stochastic-analysis]
2247 questions
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Solving a stochastic differential equation with multiplicative drift
What's the solution to the following SDE?
$$dX_t = X_t W_t dt + dW_t;\ X_t|_{t=0} = X_0,\ t \in \mathbb{R}_+$$
Joe Shmo
- 977
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proving the given random variable a stopping time
Let $T>0$ and $\{\mathcal{F_s}\}_{s\in[0,T]}$ be a filtration on the space $\Omega$. Define,$$T' = \sum_{n=0}^k\,t_n\,1_{X_n}\,,$$ where $0=t_0
user823777
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construction of stochastic integral
Let $(\Omega,\mathcal F,P)$ be a probability space and $B_t$ be a Brownian motion on it. If a progressively measurable space $X_t$ satisfies $E\int_0^T|X_t|^2dt<\infty$, then the integral $\int_0^TX_tdB_t$ is a martingale. If we have…
SHAN
- 487
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Product of orthogonal martingales
Let $(\Omega,\mathcal F,\mathcal F_t,P)$ be a probability space and $W(t)$ be a Brwonian Motion defined on it. Let $M(t)$ be a bounded martingale orthogonal to $W(t)$, i.e., there exists a constant $C>0$ such that $\sup_{t,\omega}|M(t)|\leq C$ and…
SHAN
- 487
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Does $X\sim B(Y1,\delta)$ first-order stochastically dominate(FSD) $W\sim B(Y0,\delta)$ when $Y1$ FSD $Y0$
It is very straightforward to prove $X\sim Binomial(n+1,\delta)$ first-order stochastically dominate $W\sim Binomial(n,\delta)$. However, is it also true that $X\sim B(Y1,\delta)$ first-order stochastically dominate(FSD) $W\sim B(Y0,\delta)$ when…
user3509199
- 175
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Interprete Expected Value $\frac{E(X)}{n \cdot \log n}$ and Variance $\frac{V(X)}{n}$
in my stochastics homework i have to plot the expected value and variance of a given random variable.
Expected value $\frac{E(X)}{n \cdot \log n}$ converges against 1.3 for $n \rightarrow \infty$ and variance $\frac{V(X)}{n}$ is constant at around…
MagikarpSama
- 125
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'||' symbol in multidimensional Brownian motion
Let $B_t \in \mathbb{R}^2$ and we define $X_t = \ln|B_t|$, we also define $\tau = \inf\{t>0:|B_t| \le \epsilon\}$, where $0 < \epsilon < \infty$.
I am confused about the $||$ symbol in these definition. I think it is absolutely value but $B_t$ is…
Fly_back
- 123
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2 answers
Exercise in stochastic analysis
The problem comes from Karatzas's book 'Brownian motion and stochastic analysis'. Exercise 5.20.
Suppose $X$ is in the space of square integrable martingales with stationary, independent increments. Then $\langle X \rangle_t = t(EX_1^2), T \ge 0$.
Fly_back
- 123
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Why $\mathbb{P}(\max\limits_{0 \le t \le 1} B(t) > \epsilon n) = 2\mathbb{P}(B(1) > \epsilon n)$, $B(t)$ is a Brownian motion
When I try to read the proof of Law of Large Numbers, there is a step as the tittle of the question, I would like to know why it works. Thanks beforehand.
Fly_back
- 123
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1 answer
How can I show that $\mathbb{E}(Y|\sigma(X))$ satisfies the abstract definition of conditional expectation?
More precisely let $(X,Y)$ is a pair of continuous random variables with joint density function $f(x,y)$ and we assume $\mathbb{E}(|Y|) < +\infty$. Define
$$H(X) = \int\limits_{-\infty}^\infty \frac{f(x,y)}{f_X(x)}y dy,$$
where $f_X(x) =…
Faragó Dávid
- 55