c**********e
发帖数: 2007 | 1 X_t is a stochastic process which satifies
d X_t = X_t^{1.5} dB_t?
What is the distribution of X_t? | p*****k
发帖数: 318 | 2 did you ask this before about the analytical solution? for
that one i would think no.
this is a different question though - more later. | c**********e
发帖数: 2007 | 3 pcasnik, thanks for working on it. Yes, I asked about analytical solution
before, but I think you are right, there probably no analytical solution for
it.
The general problem is to ask the distribution of X_t if we have SDE d X_t =
X_t^alpha dB_t.
When alpha = 0, X_t is BM.
When alpha = 1.5, X_t is GBM.
(The original question is about alpha = 1.5.)
So there should not be a general analytical solution for all alpha. But I
think there should be some research papers on this. | w*****e
发帖数: 197 | 4 Maybe I am missing something:
if dx = x^(3/2) dB
d(logx) does not seem to be BM, so how come
x is GBM as you point out below?
for
=
【在 c**********e 的大作中提到】
: pcasnik, thanks for working on it. Yes, I asked about analytical solution
: before, but I think you are right, there probably no analytical solution for
: it.
: The general problem is to ask the distribution of X_t if we have SDE d X_t =
: X_t^alpha dB_t.
: When alpha = 0, X_t is BM.
: When alpha = 1.5, X_t is GBM.
: (The original question is about alpha = 1.5.)
: So there should not be a general analytical solution for all alpha. But I
: think there should be some research papers on this.
| c**********e
发帖数: 2007 | 5 Sorry. It is a typo. Corrected.
GBM if alpha=1.
【在 w*****e 的大作中提到】
: Maybe I am missing something:
: if dx = x^(3/2) dB
: d(logx) does not seem to be BM, so how come
: x is GBM as you point out below?
:
: for
: =
| p*****k
发帖数: 318 | 6 careerchange, in general to get the p.d.f., one needs the
Kolmogorov forward equation:
http://en.wikipedia.org/wiki/Kolmogorov_backward_equation
dp(X,t)/dt = 1/2 * d^2[X^(2*alpha)*p(X,t)]/dX^2
with p(X,0)=delta(X-X_0)
but for alpha=1.5 here, there is a way to exploit known results;
set Y_t=1/X_t, one can get: dY_t = dt - sqrt(Y_t)*dW_t
this is a limiting case for the well-known CIR process
(e.g., in Heston model)
http://en.wikipedia.org/wiki/CIR_process
so take theta->0 with theta*mu=1, sigma=-1 | c**********e
发帖数: 2007 | 7 老大的水平不是一般的高。多谢。
【在 p*****k 的大作中提到】
: careerchange, in general to get the p.d.f., one needs the
: Kolmogorov forward equation:
: http://en.wikipedia.org/wiki/Kolmogorov_backward_equation
: dp(X,t)/dt = 1/2 * d^2[X^(2*alpha)*p(X,t)]/dX^2
: with p(X,0)=delta(X-X_0)
: but for alpha=1.5 here, there is a way to exploit known results;
: set Y_t=1/X_t, one can get: dY_t = dt - sqrt(Y_t)*dW_t
: this is a limiting case for the well-known CIR process
: (e.g., in Heston model)
: http://en.wikipedia.org/wiki/CIR_process
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