Geometric Brownian Motion SDEs
Tags: #Financial #EconomicsEquation
$$\mathrm{d}Y(t) = \mu Y(t)dt + \sigma Y(t) \mathrm{d}Z(t) \\ \mathrm{d}[\ln Y(t)] = (\mu - \frac{\sigma^2}{2}) \mathrm{d}t + \sigma \mathrm{d}Z(t) \\ Y(t) = T(0) e^{(\mu - \frac{\sigma^2}{2})t + \sigma Z(t)}$$Latex Code
\mathrm{d}Y(t) = \mu Y(t)dt + \sigma Y(t) \mathrm{d}Z(t) \\
\mathrm{d}[\ln Y(t)] = (\mu - \frac{\sigma^2}{2}) \mathrm{d}t + \sigma \mathrm{d}Z(t) \\
Y(t) = T(0) e^{(\mu - \frac{\sigma^2}{2})t + \sigma Z(t)}
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Introduction
Equation
Latex Code
\mathrm{d}Y(t) = \mu Y(t)dt + \sigma Y(t) \mathrm{d}Z(t) \\
\mathrm{d}[\ln Y(t)] = (\mu - \frac{\sigma^2}{2}) \mathrm{d}t + \sigma \mathrm{d}Z(t) \\
Y(t) = T(0) e^{(\mu - \frac{\sigma^2}{2})t + \sigma Z(t)}
Explanation
Latex code for the Geometric Brownian Motion.
: Observed value Y(t) at time stamp t
: Any normal random variable
: Drift coefficient
: Volatility
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