# Questions tagged [brownian-motion]

In mathematics, Brownian motion is described by the Wiener process; a continuous-time stochastic process named in honor of Norbert Wiener.

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### Integral of Brownian motion w.r.t. time

Let $$X_t = \int_0^t W_s \,\mathrm d s$$ where $W_s$ is our usual Brownian motion. My questions are the following: Expectation? Variance? Is it a martingale? Is it an Ito process or a Riemann ...
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### Geometric Brownian motion - Volatility Interpretation (in the drift term)

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
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### Why should we expect geometric Brownian motion to model asset prices?

Disclaimer: I am a complete ignoramus about finance, so this may be an inappropriate forum for me to ask a question in. I am a mathematician who knows nothing about finance. I heard from a popular ...
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### Estimation of Geometric Brownian Motion drift

One can find many papers about estimators of the historical volatility of a geometric Brownian motion (GBM). I'm interested in the estimation of the drift of such a process. Any link on this topic ...
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### Why is Brownian motion useful in finance?

The following is an interview question from Mark Joshi et al. Quant Job Interview. Question: Why is Brownian motion useful in finance? I am from a Pure Maths PhD background (functional analysis, ...
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### Why is Brownian motion merely 'almost surely' continuous?

Why is Brownian motion required to be merely almost surely continuous instead of continuous? For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener ...
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### GBM 3d plot with R

I want to plot the density of the GBM in a 3d plot. So I have on one axis the stock price, on the other the time and on the z axis the density. At the end I want to produce this graph. The formula I ...
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### How to simulate correlated Geometric brownian motion for n assets?

So I'm trying to simulate currency movements for several currencies with a given correlation matrix. I have the initial price, drift and volatility for each of the separate currencies, and I want to ...
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### Derivation of Ito's Lemma

My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ...
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### How to compute the Radon-Nikodym derivative?

Suppose $B(t)$ is a standard Brownian motion, and $B_{1}(t)$ is given by $dB_{1}(t)=\mu dt+dB(t)$. Suppose $P$ is the Wiener measure induced by $B(t)$ on the $C[0,\infty)$, and $P_{1}$ is the Law ...
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### Determining Hurst exponent of a Brownian motion

I am trying to determine the Hurst exponent of a simple Brownian motion, however, I seem to get a result that differs from 0.5. I am following the instructions given on the Wikipedia-page, and here is ...
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### More questions about integral of Brownian Motion w.r.t time

A similar question have been posted earlier but one part has remained unanswered. Let us define: $$X_t = \int_0^t W_s ds,$$ where $W_t$ is a standard Brownian Motion. Is $X_t$ an Itô process or a ...
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### Stochastic process with non-independent increments

All stochastic process I see always have independent increments. It is true for: standard brownian motion geometric brownian motion (?) Ornstein Uhlenbeck (?) in general, Levy process etc. What are ...
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Say I have an asset following arithmetic Brownian motion $$dX(t) = \sigma dW^\bot (t)$$ with $\sigma$ constant, and I have prices of vanilla options on $X$. I introduce a Brownian bridge $$dY(t) = ... 1answer 177 views ### Basic question on Ito integrals Let \space X(t) =\begin{cases} 2, \qquad\text{if} \space 0\le t \le 1 \\ 3, \qquad\text{if} \space 1 < t \le 3 \\ -5, \qquad\text{if}\space 3 < t \le 4 \end{cases}  or in one forumala ... 2answers 927 views ### Risk neutrality correction for Monte Carlo Bootstrapping according to PRIIP regulation for products of category III The PRIIP (packaged products) regulation prescribes Monte Carlo bootstrapping simulation for calculation of VaR for products of category III (non-linearly leveraged products). The idea is based on ... 1answer 567 views ### Modelling EUR/USD with Ornstein-Uhlenbeck + jumps? I'm trying to simulate a process as close as possible to EUR/USD of the ten past years. I've used a Ornstein-Uhlenbeck process:$$d X_t = -\theta (X_t - \mu) d t + \sigma d B_t$$with the parameters \... 0answers 167 views ### On a time integral of Brownian motion up to the hitting time Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space (\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\... 3answers 723 views ### Expectation of exponential of 3 correlated Brownian Motion Consider, are correlated Brownian motions with a given I want to calculate the, , I can't think of a way to solve this although I have solved an expectation question with only a single exponential ... 4answers 654 views ### Find a formula for the price of a derivative paying \max(S_T(S_T-K),0) Develop a formula for the price of a derivative paying$$\max(S_T(S_T-K))$$in the Black Scholes model. Apparently the trick to this question is to compute the expectation under the stock measure. So,... 2answers 449 views ### Variance of a time integral with respect to a Brownian Motion function Let process$$I_t = \int_0^t f(s) W_s \,\mathrm d s $$where W_s is standard Brownian motion. My question are the following: We know that \mathbb{E} (I_{t})=0 for all t and f a integrable ... 1answer 4k views ### Can I always use quadratic variation to calculate variance? Suppose we have a Brownian Motion BM(\mu,\sigma) defined as X_t=X_0 + \mu ds + \sigma dW_t The quadratic variation of X_t can be calculated as dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt ... 2answers 195 views ### What's the name of this nearly-brownian stochastic process? 1) Does the following algorithm (my question is math, not programming-related): ... 5answers 768 views ### Math background required to understand geometric brownian motion What mathematical concepts are required before I can understand what exactly is a Geometric Brownian motion as applicable to stock prices? I mean which branches of probability, calculus, statistics ... 2answers 1k views ### Does Black Scholes need to assume no arbitrage? Since Girsanov's theorem guarantees a risk neutral measure for Geometric Brownian motion, by the fundamental theorem of asset pricing there can be no arbitrage. So, why does the model assume no ... 2answers 2k views ### Why do we usually use normal distribution and not Laplace distribution to generate stochastic process? When working with a stochastic process based on brownian motion, the increments have normal (gaussian) distribution. However, it seems that a Laplace distribution, with density:$$f(t) = \frac{\...
Let $(\Omega,\mathcal{F},P)$ be a probability space, equipped with a filtration $(\mathcal{F})_{0 \leq t \leq T}$ that is the natural filtration of a standard Brownian motion \$(W_{t})_{0 \leq t \leq T}...