# 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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### Distribution of the random variable $X (t) = ∫ s*B_sds$ [duplicate]

What is the distribution of the random variable $$X (t) = ∫ s*B_sds ?$$ The integral is taken over [0,t]. $$B_s$$ is a brownian motion.
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### Two- (multi) dimensional geometric Brownian Motion

I am trying to calculate the value of a Basket Option with two stocks and the following information: S1 = 100, S2 = 120, r = 0.06 L = Volatilitymatrix = ((0.3, 0.1), (0.0, 0.2)), weight of Stock 1 = 1/...
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### Risk-neutral Simple Return Moment Log-return Moment

I am trying to find a way to link Risk-neutral moment of simple return to risk-neutral moment of log-returns. Specifically, by making the same standard assumptions of the Black-Scholes model with the ...
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### Does it make sense to simulate from the multidimensional GBM?

Suppose I have times series data on 3 assets and I do $N$ simulations (GBM) first for each of assets individually and then from a multidimensional GBM since their log-returns are correlated (I use ...
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### Interpretation of drift parameter $\mu$ in GBM

Currently studying Ito's calculus. Looking on the GBM model: $\frac{d S_t}{S_t} = μ dt + \sigma d B_t$ we end up on the expected stock price at time t: $E[S_t]=s_0 e^{\mu t}$.What does actually $\mu$ ...
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### Proof that integral of Brownian motion wrt time is not a martingale

Let $X_t=\int_0^t W_s ds$ where $W_s$ is Brownian motion, so $E[W_s]=0$. Then $E[X_t]=\int_0^t E[W_s] ds=\int_0^t 0 ds=0$. So $E[X_t|{\cal F}_s]=0\neq X_s$, almost everywhere. So by previous ...
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### Expectation of the product of two Brownian motions [closed]

Could you please let me know the steps to follow to get to the solution?
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### Probability distribution and Stock Price Movement [closed]

How can we use normal distribution for finding the probability of a stock price offer where current price offer depends upon the last price offer. The price offer on some day can go 10% above (at the ...
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### Differential product Correlated processes

I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now: We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$...