Help with SDE - Geometric brownian motion exercise

[omega_squared]

Hi guys,

It's been a while since high school, and now I'm faced with a problem I need to solve in a few days (attached). Would someone please help me through that? I would really appreciate support.

 

[chiro]

Hey omega_squared and welcome to the forums.

We can't give you the complete answers here at PF for the simple reason that the OP doesn't learn if this wasn't the case.

I can give you a few hints though:

For the 4th problem Jensens inequality is way of relating convexity in a statistical sense. The result of this convexity relationship is what you see in terms of E[f(t)] >= f(E[t]) which is used in many financial situations including insurance, and unsurprisingly is something you are working on.

Ito's lemma could be considered as more or less a 'substitution' where the brownian motion is modeled using a standard convention (i.e. B(t+h) - B(t) ~ N(B(t),h) if i recall correctly). If this is the case with your Weiner process then you can use the substitution.

The expectation is ok once you have some kind of explicit distribution (number 3).

The SDE you are quoting looks like a stochastic process with a drift parameter.

http://en.wikipedia.org/wiki/Wiener_process#Related_processes

If you have any more questions, I'll do my best to answer them.

 

[kai_sikorski]

Ito's lemma could be considered as more or less a 'substitution' where the brownian motion is modeled using a standard convention (i.e. B(t+h) - B(t) ~ N(B(t),h) if i recall correctly). If this is the case with your Weiner process then you can use the substitution.

Not really sure what you mean by this, but Ito's lemma is the stochastic calculus counterpart of the chain rule. Also B(t+h) - B(t) ~ N(B(t),h) doesn't really make sense. Think you meant N(0,h). There is no question about how the Weiner process is modeled though. Weiner process and Brownian motion are the same exact thing and increments over non overlapping time intervals being independent gaussian variables with variance given by the interval length is part of the definition.

 

[kai_sikorski]

Hi guys,

It's been a while since high school, and now I'm faced with a problem I need to solve in a few days (attached). Would someone please help me through that? I would really appreciate support.

In your class has the distribution for the P already been derived, and now its a question of using this to get the distribution of P^ρ? If not I must confess that I don't see a way to derive the distribution for P^ρ without solving the SDE or the Fokker-Planck equation for P, and that doesn't seem consistent with the question saying it can be done in a straightforward manner.

 

[chiro]

Not really sure what you mean by this, but Ito's lemma is the stochastic calculus counterpart of the chain rule. Also B(t+h) - B(t) ~ N(B(t),h) doesn't really make sense. Think you meant N(0,h). There is no question about how the Weiner process is modeled though. Weiner process and Brownian motion are the same exact thing and increments over non overlapping time intervals being independent gaussian variables with variance given by the interval length is part of the definition.

Yeah you're right in terms of the delta. I should have wrote B(t+h) ~ N(B(t),h) which means that B(t+h) - B(t) ~ N(0,h).

You have to remember that Ito's lemma is specific for Weiner processes and not for general distributions: you can't just use things like that for general distributions.

If the Brownian or motion or associated processes were not based on normal distributions, then you would other specific results.

 

[kai_sikorski]

You have to remember that Ito's lemma is specific for Weiner processes and not for general distributions: you can't just use things like that for general distributions.

Well Ito's lemma is often written out in it's simplest form which only applies to the Weiner process, but there is a more general formula for any X that is a semimartigale. The formula will include a quadratic covariation term [X,X]. For the Weiner process d[W,W]=dt.

 

[chiro]

Well Ito's lemma is often written out in it's simplest form which only applies to the Weiner process, but there is a more general formula for any X that is a semimartigale. The formula will include a quadratic covariation term [X,X]. For the Weiner process d[W,W]=dt.

Thankyou for that information: I wasn't aware that a general result existed: I'm doing a financial calculus course starting in a week so I'll have to check it out.

I do think however the OP realizes the limitations of the Ito lemma because it would not be a good thing for them to apply this knowledge without really understanding it.

 

[omega_squared]

only the standard solution is given by
X(t) = x₀ * exp((a-1/2*σ)t + σW(t)).. nothing else :/

In your class has the distribution for the P already been derived, and now its a question of using this to get the distribution of P^ρ? If not I must confess that I don't see a way to derive the distribution for P^ρ without solving the SDE or the Fokker-Planck equation for P, and that doesn't seem consistent with the question saying it can be done in a straightforward manner.

 

[kai_sikorski]

Okay if you have that, then basically you need to apply this a bunch of time. First to W(t) to get the probability law for P, then to P to get the probability law for P^ρ.