w*****e
发帖数: 197 | 1 The classic problem is following:
B(t) is a brownian motion and determine the probability
F( X, T ) = P{ B(t) <= X for 0 <= t <= T }
where X > 0
This one has a very nice solution using reflection
principle. So my question is how to generalize this?
How about Y(t) = mu * dt + sigma * dw, where mu and sigma
can be state/Y dependent, again calculate:
F( X, T ) = P{ Y(t) <= X for 0 <= t <= T }
I vaguely recall we can use some kind of PDE and then
go back to integration to solve it. But I can't work | p*****k
发帖数: 318 | 2 wushine, this seems str8forward by change of measure
(Girsanov) and taking into account the R-N derivative.
not sure which book is the best reference, but Joshi
definitely talked about this exact topic when he
discussed the barrier options | w*****e
发帖数: 197 | 3 Joshi's book only handles constant mu and sigma case.
I am very interested in the general case, where mu and
sigma might be state dependent.
I think I saw someone posted something here or on Wilmott
about a general PDE based approach. But I just can't find
it anymore. | l******i
发帖数: 1404 | 4 As long as
dXt=miu(t,Xt)dt+sigma(t,Xt)dWt
where miu and sigma are deterministic functions of (t,Xt),
(note that only adapted process property is not enough.)
Xt is a markov process.
You can convert the stochastic problem into a PDE framework:
Solve the classic Kolmogorov backward equation (KBE)
or Kolmogorov forward equation (KFE)
to get the transition density function p(x,t) explicitly.
(I usually use KFE, known as Fokker–Planck equation.)
Here is some introduction:
http://en.wikipedia.org/wiki/Kolmogorov_backward_equation
Hope it helps.
Lichen | J**********g
发帖数: 213 | 5 Can I know when we can convert the stochastic problem into a PDE framework?
being a Markov process is one? any other condition? I am still trying to
learn Stochastic calculus...
【在 l******i 的大作中提到】
: As long as
: dXt=miu(t,Xt)dt+sigma(t,Xt)dWt
: where miu and sigma are deterministic functions of (t,Xt),
: (note that only adapted process property is not enough.)
: Xt is a markov process.
: You can convert the stochastic problem into a PDE framework:
: Solve the classic Kolmogorov backward equation (KBE)
: or Kolmogorov forward equation (KFE)
: to get the transition density function p(x,t) explicitly.
: (I usually use KFE, known as Fokker–Planck equation.)
| l******i
发帖数: 1404 | 6 As long as
dXt=miu(t,Xt)dt+sigma(t,Xt)dWt
where miu and sigma are KBE deterministic functions of (t,Xt),
Mostly you can use PDE framework.
The deep reason is that you can use Feynman–Kac formula
to convert a stochastic problem into a PDE framework.
After such implementation, we can have
Blask-Scholes PDE, Term structure equation, KBE/KFE and etc.
?
【在 J**********g 的大作中提到】
: Can I know when we can convert the stochastic problem into a PDE framework?
: being a Markov process is one? any other condition? I am still trying to
: learn Stochastic calculus...
| J**********g
发帖数: 213 | 7 You mean convert martingale-related stochastic problem into PDEs, right?
Thanks.
【在 l******i 的大作中提到】
: As long as
: dXt=miu(t,Xt)dt+sigma(t,Xt)dWt
: where miu and sigma are KBE deterministic functions of (t,Xt),
: Mostly you can use PDE framework.
: The deep reason is that you can use Feynman–Kac formula
: to convert a stochastic problem into a PDE framework.
: After such implementation, we can have
: Blask-Scholes PDE, Term structure equation, KBE/KFE and etc.
:
: ?
| l******i
发帖数: 1404 | 8 For short, I think it is not confined to dynamics under martingale measure.
It has nothing to do with martingale.
Generally speaking, we usually require a martingale measure to make the
market arbitrage free and derive the risk-neutral valuation formula. However
in the other way, we can use PDE framework as I introduced above to derive
analytic price formula instead of risk-neutral valuation formula to avoid
Monte Carlo simulation. That's why Markov property is so important.
【在 J**********g 的大作中提到】
: You mean convert martingale-related stochastic problem into PDEs, right?
: Thanks.
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