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# 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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I ran through an equality in a paper I was reading but couldn't check if it is correct. Let $W^1_t$, $W^2_t$ and $W^3_t$ be three brownian motions, not necessarily independent, is it true that the ...
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### Differential of integral of Wiener process over time

I am trying to compute this quantity: $\frac{d}{dt}\int_{0}^{t} W_s ds$ Where $W_t$ is a Wiener process. Is there a theorem which tells how this can be computed? I have tried https://en.wikipedia....
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### Measure of a Brownian motion = normal distribution?

Consider some model where the process increments are normally distributed, e.g. Vasicek: $$dr(t) = \left(\theta - ar(t)\right)dt + \sigma dW(t).$$ We usually say that $W(t)$ is a Brownian motion ...
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### Calculate drift of Brownian Motion using Euler method

I am working on a project to approximate numerically the solution $X_t$ of a stochastic diﬀerential equation (SDE) using the Euler method. I have do to this for the Brownian motion with drift. I am ...
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### Fractional Brownian motion references

Does anyone know any good references to understand the fractional Brownian motion and its numerical simulation, preferably applied to finance.Thank you.
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### Fourth moment of a itos integral

$I(t)=\int_0^t \sqrt sdW_s$ What is $E(I(t)^4)$
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### Theoretical distribution of (geometric) Brownian motion (with drift)

I am working on a simulation study which focuses on both the Brownian motion with drift (1) and the geometric Brownian motion (2). I denote them by $X_t$. What are the theoretical distributions of ...
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### Quadratic variation of an integral of a function of a Brownian motion

I'm asked to find the quadratic variation of the integral $\int_{0}^{t} W_s^2 ds$.
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### Brownian bridge with time varying volatility

I have a question to ask about the Brownian bridge for a process with deterministic volatility varying over time. In other words, we have this dynamic: $dS_t = \sigma_{t} * dW_t$. We want to know the ...
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### Asset price simulation under Monte Carlo for option pricing using market data

I am trying to use Monte Carlo to price some exotic options. I have in mind to simulate asset prices under GBM (say S&P prices) using Monte Carlo and price the option accordingly from the payoffs ...
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### Conditional Probability - Geometric Brownian Motion

Background I am trying to find a way to price a variant of a gap option by using closed-end expressions. What makes this option a bit tricky is that it can be exercised at four predetermined dates (t=...
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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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### 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 ...