All Questions
18 questions
1
vote
0
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31
views
$\alpha$ stable processes without jumps
Levy processes with jumps can be formulated following the Levy-kinchkine representation, which provide a decomposition of the characteristic function into three factors corresponding to the diffusion (...
- 305
-1
votes
1
answer
169
views
joint density of two relevant random variables
It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
2
votes
0
answers
201
views
Continuity of density of SDE
Consider a stochastic differential equation in $\mathbb R^m$ with a parameter $\theta\in\mathbb R$:
\begin{equation}
dX_t^{\theta,x} = v(\theta,X_t^{\theta,x})dt+\sigma(X_t^{\theta,x})\circ dW_t,~...
- 21
5
votes
1
answer
336
views
Joint distribution of drawdown time and value of geometric Brownian motion
Let $X$ be a geometric Brownian motion, satisfying the SDE
$$dX_t = \sigma X_t \, dW_t, X_0 = 1.$$
for $W$ a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Define the ...
- 6,195
0
votes
1
answer
493
views
Fokker-Planck: uniqueness and convergence to stationary distribution
Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
- 305
5
votes
1
answer
392
views
Uniqueness of the solution to some SDE
Consider the stochastic differential equation as follows:
$$X_t=X_0+t+\int_0^t\frac{dW_s}{1+m(s)},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable and $m(t)=\mathbb P[...
- 1,334
5
votes
2
answers
311
views
A comparison of diffusions
Consider two diffusions given by
$$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$
for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
- 128k
-2
votes
1
answer
138
views
Problem arising from martingale solutions to SPDE: $Law(u)=Law(v)$ on $C([0,T]; X)$, can $Law(u)=Law(v)$ on $C([0,t]; X)$ for $t<T$?
I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find ...
1
vote
1
answer
305
views
Existence of a Lyapunov function for a log-concave measure
Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1$$ and $\mu$ denote the measure with density $e^{-f}$ with respect to the Lebesgue measure on $\...
- 167
1
vote
1
answer
435
views
How to calculate the probability of 2 events happening in time series under only cdf information?
In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...
- 29
4
votes
1
answer
322
views
Asymptotic form of pdf of Escape Time of arithmetic fBm
I am trying to apply the Girsanov formula and Doobs optional sampling theorem to obtain an asymptotic form of first passage density of an fbm process with drift, but the answer i am getting seems ...
- 281
2
votes
1
answer
534
views
Time interval of existence of an SDE solution with locally Lipschitz drift
Consider the stochastic ODE $$
dX = F(X) \, dt + dB
$$
where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "...
- 51
2
votes
0
answers
107
views
Markov chain approximates a fractional diffusion
Let assume that
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R}
$$
Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...
- 323
0
votes
0
answers
57
views
Parametric distribution where the parameter follows a diffusion process
I'm looking for a distribution $P_{\theta}$ with pdf $f (t,\theta)$ over $\mathbb{R}^{+}$ such that there exists functions $\mu(\theta)$ and $\sigma(\theta)$ such that for all $t>0$:
$$\mu(\theta)\...
- 1,902
2
votes
1
answer
139
views
Stochastic inverse
Let $X_t$ be a semi-martingale and $H_t$ be a predictable process and $g$ be a measurable bijective function with measurable inverse. Does there exist a function $f(h,x)$ satisfying
$$
\int_0^Tf(H_t,...
- 5,405
4
votes
1
answer
463
views
Variance and expectation of timed-change squared Bessel process
Let $X_t$ be a squared Bessel process satisfying the SDE:
$$
dX_t=\left(1-\frac{\beta}{(1-\beta)(1-\rho^2)} \right) dt +2\sqrt{X_t}dW^{(1)}_t
$$
and $v_t=v_0e^{-\alpha^2 t/2+\alpha W^{(2)}_t}$ be a ...
- 323
3
votes
0
answers
276
views
Processes with the same finite dimensional distributions as the solutions to SDEs
Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
- 131
0
votes
1
answer
379
views
What is the derivative of this integral?
I have asked this question here
https://math.stackexchange.com/questions/1536018/how-to-find-derivative-of-this-intergral
but still has no response.
Might I ask it here ?
Let $\alpha(t)\in\{0,1\}: ...
- 131