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* [http://www-stat.wharton.upenn.edu/%7Esteele/Publications/PDF/AoMtSC.pdf http://www-stat.wharton.upenn.edu/~steele/Publications/PDF/AoMtSC.pdf] | * [http://www-stat.wharton.upenn.edu/%7Esteele/Publications/PDF/AoMtSC.pdf http://www-stat.wharton.upenn.edu/~steele/Publications/PDF/AoMtSC.pdf] | ||
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| − | + | ==Ito SDE== | |
| − | + | ;def | |
| + | A stochastic process $X(t)$ is said to satisfy an Ito SDE, written as, | ||
| + | $$ | ||
| + | dX(t)= \alpha(X(t),t)dt+\beta(X(t),t)dW(t)\label{ito} | ||
| + | $$ | ||
| + | if for $t\ge 0$ it satisfies the integral equation, | ||
| + | $$ | ||
| + | X(t)= X(0) +\int_{0}^{t}\alpha(X(\tau),\tau)\,d\tau+\int_{0}^{t}\beta(X(\tau),\tau)dW(\tau) | ||
| + | $$ | ||
| + | ===Kolmogorov equation=== | ||
| + | * let $p(x,t)$ be the p.d.f. of the stochastic process $X(t)$ satisfying \ref{ito}. Then | ||
| + | $$ | ||
| + | \frac{\partial p}{\partial t} = -\frac{\partial(\alpha(x,t)p)}{\partial x}+\frac{1}{2}\frac{\partial^2 (\beta^2(x,t)p)}{\partial x^2} | ||
| + | $$ | ||
==example== | ==example== | ||
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* [[Brownian motion]] | * [[Brownian motion]] | ||
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| + | ==computational resource== | ||
| + | * http://mathematica.stackexchange.com/questions/30558/solving-a-stochastic-differential-equation?rq=1 | ||
| + | * http://mathematica.stackexchange.com/questions/83645/martingale-pricing-simulation-random-walk-stock-price | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
2016년 5월 23일 (월) 01:50 판
introduction
- start with Brownian motion
- http://www.mathematica-journal.com/issue/v9i4/contents/StochasticIntegrals/StochasticIntegrals_1.html
- http://www-stat.wharton.upenn.edu/~steele/Publications/PDF/AoMtSC.pdf
Ito SDE
- def
A stochastic process $X(t)$ is said to satisfy an Ito SDE, written as, $$ dX(t)= \alpha(X(t),t)dt+\beta(X(t),t)dW(t)\label{ito} $$ if for $t\ge 0$ it satisfies the integral equation, $$ X(t)= X(0) +\int_{0}^{t}\alpha(X(\tau),\tau)\,d\tau+\int_{0}^{t}\beta(X(\tau),\tau)dW(\tau) $$
Kolmogorov equation
- let $p(x,t)$ be the p.d.f. of the stochastic process $X(t)$ satisfying \ref{ito}. Then
$$ \frac{\partial p}{\partial t} = -\frac{\partial(\alpha(x,t)p)}{\partial x}+\frac{1}{2}\frac{\partial^2 (\beta^2(x,t)p)}{\partial x^2} $$
example
- Loewner equantion