Questions tagged [stochastic-calculus]

Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.

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2
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1answer
138 views

Calculate Radon-Nikodym derivative

For the laws of two pure-jump Markov processes $\mu_1$ and $\mu_2$ on $\mathbb R^n$, which generators are $H_1f(x)=\int h(x,dy) (f(y)-f(x))$ and $H_2f(x)=\int e^{-g(x,y)} h(x,dy) (f(y)-f(x))$ (...
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0answers
22 views

Can one change the dimension of a Bessel process by a Girsanov change of measure?

Recall that a (squared) Bessel process $X_t$ with the dimension $\delta_0>0$ is the solution of the SDE $$d X_t = 2\,\sqrt{X_t}\,d W_t+\delta_0\,d t.$$ A naive application of the Girsanov Theorem ...
5
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1answer
84 views

Scalar product of random unit vectors

Let $X,X'$ be two random vectors on the sphere $S^{d-1}$. What is the distribution of their dot product $X\cdot X'$ in the following cases: $X,X'$ independent with uniform distribution on the sphere $...
3
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1answer
154 views

Do Lyapunov functions imply exponential integrability of hitting times?

I have a question of some integrability of hitting times. Let $X=(\{X_t\}_{t \ge0},\{P_x\}_{x \in E})$ be a diffusion process on a locally compact separable metric space $E$. We assume that there ...
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39 views

Fourier transform of a general normal distribution [closed]

How can I calculate the Fourier transform of this equation? Here is a known equation that may be helpful.
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0answers
25 views

Dominance convergence theorem to compute expectation of a sequence of random variables defined by their time derivatives

Let $ (X_t) $ be a stochastic process, and define a new stochastic process by $ Y_t = \int_0^t f(X_s) ds $. Then consider a sequence $X_t^0,X_t^1,\ldots, X_t^n$ for which we get $Y_t^0,Y_t^1,\ldots, ...
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2answers
168 views

Exponential or sub-exponential ergodicity?

Consider the one-dimensional stochastic differential equation $$d X(t) = -sgn(X(t))dt + dW(t),$$ where $W$ is a standard Brownian motion, and $sgn(x) = 1$ if $x > 0$ and $-1$ if $x\le 0$. It can be ...
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1answer
27 views

Can the joint law $P \circ (X,Y)^{-1}$ of two random variables $X$ and $Y$ be written as $P \circ (X,\phi(X,U))^{-1}$ for $U$ uniform in $[0,1]$?

I want to know whether there is some general assumpitons we can make on two measurable spaces $E$ and $F$ (e.g. polish, complete, separable,...) such that we can ensure that the following "Theorem" ...
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0answers
30 views

Stopping times about Brownian motion with draft

Assumet $M(t) = B(t) + \mu t$ where $B(t)$ is a standard Brownian Motion. Denote: $$T_a := \inf \{ t \geq 0, \, M(t) = a\}, \quad T_b := \inf \{ t \geq 0, \, M(t) = b\}$$ The question asks to ...
1
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1answer
62 views

The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$

In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X_{n}(t)=nt1_{[0,1/n]}(t)+(2-nt)1_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the ...
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1answer
58 views

Find a conditional expectation of a difference of two independent Poisson process

Consider two independent Poisson processes $N,M$ with rate $\lambda$, and define $$X(t):=x+\dfrac{1}{\sqrt{n}}[N(t)-M(t)].$$ From this formula we know that $X(0)=x$. Now, I want to compute the ...
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21 views

Stochastic differential equations with correlated Brownian Motions

let's consider an sde of this kind: \begin{equation} \label{eq:system} \begin{cases} dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t \\ X_0=x_0 \\ dY_t=B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^2 \\ Y_0=y_0 \end{...
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0answers
16 views

Bifractional Brownian motion admit a representation in the form of a stochastic integral?

good morning. You know the fractional Brownian motion, multifractional Brownian motion and sub-fractional Brownian motion, can be represented as a wiener integral ( moving average representation ). ...
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2answers
60 views

Show that if $A_{0}(t)+A_{1}(t)W(t)=0$ for all $t$ with $A_{0}$ and $A_{1}$ differentiable in $t$ and $W(t)$ a Wiener process, then $A_{0}=A_{1}=0$

I am learning the quadratic variation of stochastic process, and I am working on an exercise stating that If for all $t$, we have $$0=A_{0}(t)+A_{1}(t)W(t),$$ where $(A_{0}(t),\mathcal{F}_{t})$ ...
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0answers
22 views

Sufficient condition for weak existence of solution of a SDE

Please be adviced that I'm cross-posting this question from MSE since it's very likely it will remain unsolved, and I haven't been able to obtain an answer from my colleges/professors. It's a well ...
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0answers
207 views

On the level of measure theory, what does it mean for a drift to be deterministic?

Given a drift $F\in W^{1,2}([0,T])$ adapted to the filtration of a Brownian motion $B(t)$ on Wiener space $(C[0,T],\mathcal B(\|\cdot \|_\infty)$ with Wiener measure $\mu_0$, there is another measure $...
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0answers
19 views

SDE conditional expectation

Let's suppose I have a bidimensional SDE of the form: \begin{equation} \label{eq:system} \begin{cases} dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\ X_0=x_0 \\ dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
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1answer
66 views

Question about the exit time of a time-homogeneous Itô diffusion

Consider a one-dimensional Itô diffusion: $$\mathrm{d} X_{t}=b\left(X_{t}\right) \mathrm{d} t+\sigma\left(X_{t}\right) \mathrm{d} B_{t}$$ where $X_0 = 0$ and $B_t$ is the standard Brownian Motion. ...
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0answers
44 views

conditional expected value and in Stochastic differential equations

Let's suppose I have a bidimensional SDE of the form: \begin{equation} \label{eq:system} \begin{cases} dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\ X_0=x_0 \\ dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
0
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1answer
58 views

Stochastic invariant subset

Let us consider a stochastic differential equation (SDE), $$ dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}% $$ and a compact set $C\subset\mathbb{R}^{n}$. Given a stochastic ...
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1answer
59 views

Does having the derivative in the limit suffice to solve the function at the limit? [closed]

Suppose that I have a function $f(x, \epsilon)$ and I know that $$ \lim_{\epsilon \to 0} f'(x, \epsilon) = g'(x). $$ Now let $g(x)$ be the function whose derivative appears above. How can I ...
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2answers
110 views

Transience of 3-dimensional Brownian motion

I'm attempting Exercise 5.33 of Le Gall's Brownian motion, Martingales and Stochastic Calculus. Let $B_t$ be a 3-dimensional Brownian motion starting from $x$. Part 6 asks me to show that $$|B_t| = |...
2
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2answers
94 views

Convergence of fraction of expectation values

Let $X_1,...,X_n$ be iid normal random variables. I am looking for a strategy to establish the following limit for fraction of expectation values $$\lim_{N \rightarrow \infty} \frac{E(\prod_{1\le i ...
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0answers
43 views

Exponential functional of an Ito processes

Let $\sigma_t \in L^2 (\mathbb{R})$ an adapted square integrable process and $W_t$ a brownian motion. Does the closed form of law of the following process $I_t$ existe? $$I_t(\sigma_t) = \int_0^t e^...
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0answers
29 views

Associating “weak” solutions of stochastic differential equation on manifolds with real valued weak solutions

Let's say we are working on a (real) differentiable manifold $M$. For smooth vector fields $A_0,A_1,...,A_r$ on $M$ we define stochastic differential equations as $$dX_t = A_{\alpha}(X_t) \circ dB^{\...
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2answers
159 views

Continuity of Brownian motion constructed from Kolmogorov extension theorem?

I'm trying to construct Brownian motion using the Kolmogorov extension theorem. I am happy with the construction of a process with the required FDDs as (the canonical process associated with) a ...
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1answer
50 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 ...
0
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1answer
79 views

Is the integral of an adapted, measurable process adapted?

Let $X_s(\omega)$ be measurable and adapted. Under what conditions will the process $F_s(\omega) = \int_0^t X_s(\omega) \, ds$ also be adapted? To me it seems that adaptedness and measurability ...
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0answers
25 views

Gap between optimal policies for optimal control problems

Suppose that one is given a stochastic optimal control problem $$ J(u)\triangleq \mathbb{E}\left[ \int_0^T L(t,u_t,X_t^u) dt + g(X_T) \right], $$ where $L$ and $G$ are convex, non-negative valued, and ...
3
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0answers
59 views

Martingale polynomial functions

If $B_t$ is a Brownian motion then using Hermite polynomials one can find that $$1, B_t, B_t^2-t, B_t^3 - 3tB_t,...$$ are martingales. If $X_t$ is a diffusion $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)...
8
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2answers
254 views

Show that this process is not a martingale

I am cross-posting this question from MSE since I did not received any answer, furthermore I tried asking some professors in my university but still we could not find an answer. The most surprising ...
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0answers
26 views

Reference for an infinite system of SDEs

Consider system of the following form, \begin{align*} \mathrm{d} X_k(t) = \big(AX(t)\big)_k\mathrm{d}t + B_k(X_k(t))\mathrm{d}t+\mathrm{d}W_k(t),\quad k\in\mathbb{Z}, \end{align*} where $A$ is matrix, ...
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0answers
22 views

Step verification in derivation of Ito formula

At the page 125 (see 4.90) (see images below, this proof is from "Statistics of Stochastic processes" by Lipster and Shiryaev) we consider function $u(t,W_t)=f(t,at+bW_t)$ and $a(\omega), b(\omega)$ ...
2
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0answers
45 views

Representation of optimal controls as diffusions

In reading this post I couldn't help but wonder the following question: Let $\sigma>0$ and suppose, as in the motivational post, we are given a stochastic optimal control problem: $$ \begin{...
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0answers
34 views

Reference request: Ito formula for function $G(t, x)$ when $G$ depend on $\omega$

There is proved Lemma in book : Let the function $G(t,x)$ is defined when $t\in [0,T], x\in(-\infty,\infty)$, $G$ has continuous derivative with respect to $t$ and twice continuously diferentiable ...
2
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1answer
74 views

What's the role of commutation relations in stochastic mechanics?

In a stochastic context, we can understand a term like $$ \int_0^T \frac{d q(t)}{dt} dq $$ either as the (Ito) limit $$ \lim_{N\to\infty} \sum_{i}^N dq(t_i) \frac{d q(t_i)}{dt} $$ or the (Anti-...
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0answers
47 views

Scaling Property Ito diffusion processes

It is well-known that a Brownian motion $W$ has the following scaling property $$ c^{-1/2}W_{ct} \qquad \mbox{ for any $c,t>0$}. $$ In particular this means that the increments of the processes $W$ ...
15
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2answers
503 views

Why do stochastic integrals depend on the choice of partitioning points?

When we integrate a function, we must make some choice about how we approximate it before we take the limit. In principle, we can choose $\tau_i$ to be any value between $t_{i-1}$ and $t_i$. But for ...
3
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1answer
311 views

White noise vs. black noise

In this excellent lecture ("2d Percolation Revisited") Stanislav Smirnov mentioned the connection of the theory of percolation with the notion of the so called black noise—see at 29:42 (the notion ...
1
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1answer
82 views

Explicit densities for Brownian motion hitting times

I'm looking for functions $g: \mathbb{R}_+ \to \mathbb{R}$ such that the hitting time $$\tau := \inf \{t \geq 0 : B_t \nleq g(t) \} $$ has an explicit density with respect to the Lebesgue measure, ...
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0answers
35 views

differential of posterior probability distribution over the mean drift of brownian motion

Let $W_t$ be the Weiner process, and let $X_t = W_t + \mu t$, where $\mu$ is either 0 or 1. We wish to get information about $\mu$ by looking at $X_t$. Let $q_t$ be the probability we assign to $\mu=1$...
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0answers
28 views

Applications of Kazamaki Conditions

I'm interested in applications of this theorem by Sekiguchi Kazamaki: "Continuous Exponential Martingales and BMO" - Theorem 1.12: Let $M$ be a continuous local martingale and $Z(M):= \exp(M-\frac{1}{...
1
vote
1answer
75 views

A generalized Mercer's Theorem?

If $X_t$ is a mean zero, square integrable process with covariance kernel $k(s,t),$ Mercer's theorem states that there exists an orthogonal basis $\{\phi_i\}$ in $L^2$ and eigenvalues satisfying $$\...
2
votes
1answer
150 views

Pathwise stochastic integral as a linear operator on continuous functions

Let $B$ be a Brownian motion. Definining a pathwise stochastic integral $I(f):=\int f~dB$ for certain classes of deterministic functions is straightforward: For instance if $f=\sum_ic_i1\{[t_i,t_{i+1})...
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0answers
150 views

Proving that $dX_t=a_tX_td\tilde{B}_t$ is a martingale

Fix $T>0. $Consider the probability space $(\Omega,F,Q)$ and a Brownian motion $\{B_t\}_{t\leq T}$ and filtration $\{F\}_t$ generated by the Brownian paths. Suppose $a_t,\gamma_t$ are random ...
2
votes
0answers
84 views

Malliavin derivative of Ito process

Let $X_t= X_0 + \int_0^t \mu(s,X_s)ds + \int_0^t \sigma(s,X_s)dW_s$ where $\mu$ and $\sigma$ are $C^1$ functions satisfying the usual growth restriction and $W_t$ is a $d$-dimensional Brownian motion. ...
2
votes
2answers
208 views

Weak convergence in Skorohod topology

Let $D([0,T];R^d)$ be the space of càdlàg functions endowed with the usual Skorohod topology. $X_t(\omega):=\omega(t)$ denotes the usual canonical process. Assume that a family of probability ...
0
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0answers
34 views

Derivation of a differential equation from a SDE

Suppose there is a non-homogeneous Markov process with state space $\mathbb{R}_{+}$ driven by this McKean-Vlasov-tipe SDE: $$ dY_t = a \mathbb{E}[Y_t]\ dt - b\ Y_t\ dt - Y_t\ dN_{aY_t}$$ where $...
0
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0answers
66 views

Eigenfunction expansion and ultracontractivity

I have a question on heat kernels and eigenfunctions of symmetric Markov processes. Let $X=(\{X_t\}_{t \ge0 },\{P_x\}_{x \in M})$ be a $\mu$-symmetric Markov process on a locally compact separable ...
0
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0answers
90 views

About martingales induced by iterative processes

Suppose I have a discrete stochastic process $\{ X_i \}_{i=1,\ldots..}$ defined as, $X_{i+1} = X_i - \eta \nabla f(X_i) + \sqrt{\eta} \xi_i$ where $f : \mathbb{R}^d \rightarrow \mathbb{R}$ and $\xi_i \...

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