For questions about stochastic analysis or stochastic calculus, for example the Itô integral.
Questions tagged [stochastic-analysis]
2247 questions
3
votes
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Canonical probability space for Brownian motion
Let $\Omega$ be the space of continuous functions $\omega: [0,T]\to \mathbb{R}^{d}$, $\mathcal{F}=\mathcal{B}(C[0,T))$ and $\mathbb{P}$ be the Wiener measure. Therefore the coordinate processes $W$ defined by $W_t(\omega)=\omega_t$ is a Brownian…
Jackie
- 39
2
votes
0 answers
under what conditions stochastic exponential is a usual exponential?
Stochastic exponential is
$\mathcal{E}_t(X)=e^{X_t-\frac{1}{2}_t}\prod_{s\leq t}(1+\Delta X_t)e^{-\Delta X_s}$
and usual exponential is $e^{X_t}$
I guess, if process is continuous and of finite variation then it they should be the same, right?…
c-walk
- 447
2
votes
0 answers
Definition improper Itô's integral.
Let $\{B_t:t\geq 0\}$ be a standard Brownian Motion and let $\{\mathcal{F}_t\}_{t\geq 0}$ be the natural filtration associated to Brownian Motion (that is, $\mathcal{F}_t=\sigma(B_s:0\leq s\leq t)$). Fix $T>0$. Let…
user311684
- 21
2
votes
1 answer
Help with proof of Girsanov Theorem
I'm studying this proof of Girsanov Theorem and trying the figure out the details however I need some help with this. I noticed there are, also here on stackexchange, a lot of different versions of the Theorem so I start by stating Girsanov the way…
Dylan
- 21
2
votes
0 answers
Smoothness requirement for Stratonovich Integral
Every place I've seen defines the Ito formula for the Stratonovich integral as $df(X_t) = f'(X_t) \circ dX_t$ for $f \in C^3(\mathbb{R})$ and $X_t$ brownian motion, while the Ito integral only requires $f \in C^2$. Why is this so? More explicitly,…
Rikimaru
- 834
1
vote
0 answers
Bound on the conditional expectation of a stochastic integral given W_T (multidimensional case)
This is a follow-up of a previous question I asked.
Suppose that $f \colon \mathbb R^n \to \mathbb R^n$ is a bounded vector field (not necessarily the gradient of another function), that $(W_t)_{0\leq t \leq T}$ is a standard Brownian motion in…
Roberto Rastapopoulos
- 2,122
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0 answers
$ \mathbb{E}|f(X)|^2_{\infty, \eta t \wedge \tau} \leq C_1 \int_0^t \mathbb{E}|f(X)|^2_{\infty, \eta s \wedge \tau} ds. $ implies $f(X)=0$ or $X\in M$
I am self-studying the following material, whose source is https://sayanmuk.github.io/StochasticAnalysisManifolds.pdf
I am stuck at the final step of the following propsotion proof.
Proposition 1.2.8. Let $X$ be the solution of the extended…
Pedro Gomes
- 3,891
1
vote
0 answers
is Debut of an optional set a stopping time? (under no usual hypothesis)
Do not assume usual hypothesis. Let $A$ be a set measurable with respect to optional sigma-algebra.
Then is debut of $A$, $D_A$, a stopping time with respect to $F$ or $F_+$ filtration?
I know that if filtration is complete then the debut of a…
c-walk
- 447
1
vote
0 answers
Martingales and non-exploding solutions of a specific nonlinear SDE with no drift.
Suppose I have the following nonlinear ODE for a function $X:\mathbb{R}\rightarrow\mathbb{R}$ such that
\begin{equation}
\frac{dX(t)}{dt}=k^{1/2}((X(t))^{2}(X(t)-1)^{1/2})
\end{equation}
with initial data $X(0)=X_{o}=1$ and $k>0$. The formal…
DR M
- 11
1
vote
0 answers
Itô's Lemma in proof of Feynman-Kac's Formula
Let $h$ and $V$ be bounded continuous functions on $\mathbb{R}^d$. Suppose $u$ is continuous on $\mathbb{R}_+ \times \mathbb{R}^d$, bounded on $[0,T] \times \mathbb{R}^d$ for eacht $T < \infty$ and $u \in C^{1,2}((0,\infty) \times \mathbb{R}^d)$.…
iJup
- 1,999
1
vote
1 answer
Proving a probability mass function to be $\le 1$
Assume $\Omega = \mathbb{N}_0$ and $k > 0$. Prove that $f(\omega) = e^{-\lambda} \cdot \frac{\lambda^{\omega}}{\omega !}$ is a mass probability function.
Showing $f \geq 0$ is trivial as well as $\sum \limits_{\omega \in \Omega} f(\omega) = 1$. I…
user234723
- 11
1
vote
0 answers
Langevin equation with uniform noise
Given the Langevin equation written in the form:
$$\ddot{x}(t)+\lambda \dot{x}(t)=\mu(t)$$
if $\mu(t)$ is noise with gaussian $pdf$, the solution is well known in therms of the spectrum of the $x(t)$
Now, my question is: if $\mu(t)$ is a random…
Riccardo.Alestra
- 10,546
0
votes
1 answer
Different statements of the Feynman-Kac formula
In many books on finance, the PDE solved by Feynman-Kac is often formulated by the following:
$$\begin{aligned}
\frac{\partial}{\partial t}u(t,x)+\mathcal{L}u(t,x)&=V(t,x)u(t,x),\\
u(0,x)&=\varphi(x)
\end{aligned}$$
where the generator is defined as…
Mochizuki_Mk19
- 80
- 8
0
votes
0 answers
Derivative of stochastic integral "$\int_0^t \exp(X_s)\sin(s+2t) dW_s$"?
If $W$ be the Wiener process and $X_t$ be an stochastic process,how can compute the derivative of stochastic integral,
$V(t)=\int_0^t \exp(X_s)\sin(s+2t) dW_s$,
$dV/dt=?$
0
votes
1 answer
Quadratic variation of Square of a continuous martingale
Problem:
Show that if $(M_t)$ is a continuous local martingale, then $M^2$ has quadratic variation
$$\langle M^2\rangle_t=4\int_0^tM_s^2\mathrm{d}\langle M\rangle_s$$
It seems to me that $M^2$ is a semi-martingale with decomposition $M_t^2 =…
I.D.D
- 75