The stochastic-integrals tag has no wiki summary.
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stochastic differential equation
Xt is a weak solution to the SDE with dXt = ( −αXt + γ )dt + β dBt , ∀t ≥ 0
X0 = x0. α, β , and γ constants, and Bt is a brownina motion.
need to find the PDE for the transition density of X at ...
-1
votes
1answer
246 views
Scalar product of Gaussian process
Assume that $n(t)$ is a White Gaussian Noise (WGN) process with $E[n(t)]=0$, $E[n(t)^2]=\sigma^2$ and $x(t)$ a deterministic function defined in $[0,T]$. How can I compute from first principles the ...
4
votes
1answer
183 views
Solution to the stochastic differential equation
Let $X_o=x$, $dX_t=\frac{1}{X_t}dt+X_tdW_t$, $W_t$ is a brownian motion i am thinking of trying $Y_t=\frac{X_t^2}{2}$ and apply ito's lemma on $Y_t$
6
votes
1answer
245 views
Does Itō isometry have different versions?
Itō isometry from Wikipedia:
Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical
real-valued Wiener process defined up to time $T > 0$, and let $X :
[0, T] \times \Omega \to ...
3
votes
1answer
196 views
what's the difference between RDE and SDE?
what's the difference between random differential equation and stochastic differential equation?
does stochastic differential equations include random differential equation?
1
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0answers
186 views
Show that this semimartingale is a local martingale
Let $B_t$ be a standard Wiener motion, $I_t=\int_0^t|B_s|^2\!\text{ds}\ $and $S_t=\max_{0\leq s\leq t}B_s$. Let also $F:\mathbb{R}^2_+\times\mathbb{R}\times\mathbb{R}_+\rightarrow\mathbb{R}$ a ...
7
votes
2answers
628 views
Is this local martingale a true martingale?
Using the Ito's formula I have shown that $X_t$ is a local martingale, because $dX_t=\dots dB_t$, where
$$X_t = (B_t+t)\exp\left(-B_t-\frac{t}{2}\right),$$
$B_t$ - is a standard Brownian motion
I ...
4
votes
1answer
361 views
Expectation value of a product of an Ito integral and a function of a Brownian motion
this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated.
Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
2
votes
2answers
97 views
Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$
Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?
