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198 views

stochastic exponential

I was able to show the following: $X$ a semimartingale, $X_0=0$ then the SDE $$ dZ_t=Z_tdX_t$$ with $ Z_0=1$ has the unique solution $Z_t:=\exp{(X_t-\frac{1}{2}\langle X\rangle_t)}$. I was able ...
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1answer
157 views

Is the expectation $E[\xi U'(\xi)]$ finite?

I encounter the following problem today. It seems a simple question. Let $U$ be a real function from $R^+\rightarrow \bar{R}$ satisfying the following conditions: (1) $U$ is concave, continuous, ...
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1answer
174 views

a question on Stochastic Calculus

I encounterred a question on Stochastic Calculus as following, but I don't understand the meaning of $\mathcal{N}$ here, can any expert explain me a little bit? Thank you very much in advance! ...
2
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1answer
160 views

The variance of bilateral filtered random variables

I am glad to have found this great site. There is a problem I am trying to solve for a while. I want to analyze the noise attenuation behavior of the bilateral filter. So given the unnormalized ...
3
votes
1answer
251 views

Futures pricing and futures price process under the real world measure

This is something that keeps bothering me about the Benchmark approach of Platen, which (very) shortly is as follows: Compare the development of an economic value with a growth optimal portfolio. ...
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1answer
111 views

Why is $ \operatorname{sign} B_t $ a predictable process?

Does anybody know why $ \operatorname{sign} {B_t} $ is a predictable process if $ B_t $ is a Brownian motion and sign denotes the signum function with the convention that $ \operatorname{sign} (0) := ...
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0answers
473 views

Variance of a Wiener process

Problem statement: a continuous wiener process $w(t)$ with unit incremental variance and $w(0)=0$ is given, and then we check the wiener process at every $h$ seconds, $h>0$ is a positive number. If ...
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3answers
1k views

On hitting time of Brownian motion and Ito's lemma

I have two possibly related questions. Let $\tau:=\min\{t\geq0:B_t=1\}$, where $B_t$ is a standard Brownian motion. I am supposed to derive the fact that $\mathbf{E}\tau=\infty$ by applying some ...
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1answer
308 views

Convergence of quadratic variation of Ito processes

I need to find an example of an Ito process $X=\{X_t:t\in[0,T]\}$ with non-zero Ito integral part and a sequence of Ito processes $\{X_n\}$ such that $X_n$ converges uniformly to $X$, as ...
4
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2answers
193 views

Stratonovich SDE coefficient selection

Is it possible to find a strictly positive function $\sigma:\mathbb{R}\to\mathbb{R}$, such that a solution $X_t$ to an SDE $$dX_t=-X_tdt+\sigma(X_t)\circ dB_t,$$ with $X_0$ being arbitrary, is a ...
3
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1answer
123 views

stochastic analysis problem

Suppose $X$ and $Y$ are Ito processes, $X_t=x+\int^t_0Y_sdB_s$ and $Y_t=y-\int^t_0X_sdB_s,\ t\geq 0$, here $B$ is a standard Brownian motion. I need to prove that ...
3
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1answer
143 views

Solving SDE's on subsets of $R^n$.

It is well-known (see for instance Oskendal's text) that if $T>0$ and $$b(\cdot,\cdot): [0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^n~~~~~~\sigma(\cdot,\cdot):[0,T] \times \mathbb{R}^n ...
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0answers
74 views

Langevin equation with uniform noise

Given the Langevin equation written in the form: $$\ddot{x}(t)+\lambda \dot{x}(t)=\mu(t)$$ if $\mu(t)$ is noise with gaussian $pdf$, the solution is well known in therms of the spectrum of the ...
2
votes
1answer
208 views

Reference request for Optimal Stopping (Stochastic Analysis)

I would like to start and get into the habit of reading some publications in different areas of mathematics, to get used to the writing style / mathematical sophistication etc. that is expected. In ...
1
vote
2answers
189 views

Random diffusion coefficient in the Fourier equation

I'm stuck on the following simple problem: It's given the Fourier equation: $$\partial_t{u(x,t)}=\partial_x[k(t)\partial_xu(x,t)]$$where the diffusion coefficient $k(t)$ is a random variable with a ...
3
votes
2answers
534 views

calculate all the equivalent martingale measure

Under the assumption of no arbitrage without vanish risk, in an incomplete market $(\Omega,{\cal F}, P)$, the set of equivalent martingale measure is NOT empty, i.e. ${\cal P} = \{Q: Q \sim P\}\neq ...
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1answer
197 views

basic application of Strong Law of Large Numbers

In $$ \sum_{j=0}^q {q\choose j}{1\over n}\sum_{i=1}^n X_i^j(-\bar X)^{q-j} \quad \overrightarrow{a.s.} \quad \sum_{j=0}^q {q\choose j} \mathbb{E}(X^j) (-\mathbb{E}(X))^{q-j} $$ using the Strong ...