I'm tying to solve a stochastic differential equation of stock price. The equation is(adsbygoogle = window.adsbygoogle || []).push({});

[itex]dX = X(\mu dt + \sigma dW)[/itex]

where [itex]\mu, \sigma[/itex] are constants and greater than zero.

It is easy to show analytically that the expectation value to the solution is

[itex]E[X(t)] = E[X(0)] e^{\mu t}[/itex]

Then I solved this equation numerically by standard Euler method to check my analysis.

I found that if [itex]\sigma[/itex] is less than some particular value, my numerical solution is

consistent with the analytical solution which is increasing with time. The problem is

when [itex]\sigma[/itex] is greater some number, the numerical solution tends to go to zero

instead of increasing with time. Note that [itex]\mu[/itex] is the same.

What happens?

I read another book. They told me that if [itex]\mu > \sigma^2/2[/itex] then

[itex]Prob\{X(t \rightarrow \infty )=\infty \} = 1[/itex],

and if [itex]\mu < \sigma^2/2[/itex]

[itex]Prob\{X(t \rightarrow \infty ) = 0 \} = 1[/itex],

where [itex]X(t)[/itex] is the Ito solution.

For the second case, the probabilistic value is not consistent the expectation value, isn't it?

Can you help me about these two different answers?

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# Geometric Brownian motion/stock price

Can you offer guidance or do you also need help?

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