Questions tagged [stochastic-analysis]

For questions about stochastic analysis or stochastic calculus, for example the Itô integral.

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0answers
17 views

Malliavin derivative wrt time changed Brownian motion

The Malliavin derivative $D^W_\alpha$, $\alpha \in \mathbb{R}$, with respect to a standard Brownian motion $W_t$ is $$ D^W_\alpha W_t = 1_{[0,t]}(\alpha). $$ What would be the Malliavin derivative ...
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1answer
13 views

What is the stochastic analogue of convergence to the global minimizer when iterates are stochastic?

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a differentiable and strongly convex function with $m>0$ as follows: $$ f(y) \geq f(x) + \nabla f(x)^T(y-x) + \frac{m}{2}||x-y||^2 \quad \forall x,y \in \...
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1answer
16 views

Sampling stationary distribution of SDE by solving numerical scheme

currently I am learning the stochastic differential equation and their numerical approximations and something is bothering me. Some processes has stationary distributions if stationary Fokker-Planck ...
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0answers
27 views

Value of the $\limsup$ of a brownian motion

I have tried to solve the following exercise: let $B=(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in T},\{X_t\}_{t\in T},P)$ and $W=(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in T},\{W_t\}_{t\in T},P)$ two ...
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2answers
68 views

Killing the nonlinearity in SPDE

Consider the SPDE $$dX(t,x) = \left( \frac12 \Delta X(t,x)-\frac{|\nabla X(t,x)|^2}{2}+a(x)\right)\,dt - \nabla X(t,x)\cdot \,dW(t)$$ where $(x,t) \in \mathbb{R}^n \times (0,T)$, $a$ is a ...
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1answer
63 views

Quadratic variation is the right variation for continuous martingales of unbounded variation

Notation: $M_2^c$ are continuous square integrables martingales, $\langle X\rangle_t$ is the quadratic variation and $V_t^{(2)}(\Pi)$ the sum of square of increments of $X$ over the partition $\Pi$. ...
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1answer
47 views

Show that $X_t:=\mathbb{E}[Y|\mathcal{F_t}] $ is a martingale

I have this exercise about martingales: "Let $Y$ be a random variable with $\mathbb{E}(|Y|)<\infty$ and let $\mathbb{F}$ be a filtration as well as $X_t:=\mathbb{E}[Y|\mathcal{F_t}] $ for all $...
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1answer
50 views

Decomposition of a stochastic process into a continuous and pure jump part

We work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,T]},P)$. Let $X$ be an adapted làdlàg stochastic process (i.e. the left and right limits exist). For $0<t<T$ ...
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24 views

Why can we use limit inferior to calculate the expected value of a stopped process?

Consider ($\tau_n$) a diverging sequence of stopping times (e.g. $\inf\{t: X_t>n\}$). We can write the stopped local martingale $X_t^{\tau_n}$ = $X_{t\wedge \tau_n}$, which yields $\lim_{n\...
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19 views

Time marginal of Brownian motion and Heat Equation

I am studying the Propagation of chaos from Sznitman. We start with an $\mathbb{R}^d$ valued process that satisfies the following SDE: $$dX_t=dB_t+\left(\int_{\mathbb{R}^d}b(X_t, y)u_t(dy)\right)dt,\;\...
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1answer
91 views

Trouble with extending Doob's Optional Stopping Theorem

Let $\tau\geq0$ be a stopping time, $\mathbb{E}\tau<\infty$. Show $\{\tau\geq k\}\in\mathcal{F}_{k-1}$. Based on the identity $$ |X_{T\land n}|= \bigg|\sum_{k=1}^n(X_k-X_{k-1})\cdot\mathbf{1}\{\...
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1answer
22 views

Show that $\limsup_{s,\:t\:\to\:\tau}\left\|x(s)-x(t)\right\|_E\ge r$ implies $\left\|\Delta x(\tau)\right\|_E\ge r$

Let $E$ be a normed $\mathbb R$-vector space, and $x:[0,\infty)\to\mathbb R$ be right-continuous. Assume $x$ $$x(t-):=\lim_{s\to t-}x(s)$$ exists for all $t\in O$ and let $$\Delta x(t):=x(t)-x(t-)\;\;\...
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1answer
47 views

Is a right-continuous function on a compact space even "uniformly right-continuous"?

Let $E$ be a normed $\mathbb R$-vector space, $I\subseteq\mathbb R$ be nonempty, $O\subseteq I$ be open and $x:I\to\mathbb R$. Assume $x$ is right-continuous on $O$ and $$x(t-):=\lim_{s\to t-}x(s)$$ ...
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48 views

Some Grownwall type inequality for stochastic integral equation

Suppose $B_t$ is a one dimensional Brownian motion. If I know that $X_t$ is positive and $$ X_t \geq x + \int^t_0 b X_s ds + \int^t_0 \sigma X_s d B_s,$$ can I obtain that $$ X_t \geq x \exp\left( (b-\...
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1answer
83 views

If $x$ is a càdlàg function and $f$ has compact support, how can we approximate $\sum_{s\in(a,\:b]}f(\Delta x(t))$?

Let $E_i$ be a normed $\mathbb R$-vector space and $x:[0,\infty)\to E_1$ be right-continuous. Assume $$x(t-):=\lim_{s\to t-}x(s)$$ exists for all $t\ge0$ and let $\Delta x(t):=x(t)-x(t-)$ for $t\ge0$. ...
5
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1answer
57 views

How can we adapt the Ito's formula if $g \in C^{2}(\mathbb R\setminus \{ x_{1},...,x_{n}\})$ and $g^{''} \leq M$

Let $B$ be Brownian motion. Consider $g:\mathbb R \to \mathbb R$ that is $C^{2}$ except for some exceptional set $\{ x_{1},...,x_{n}\}\subseteq \mathbb R$. How can we adapt the Ito's formula if $g^{''}...
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1answer
46 views

How to show that $dX_{t} = \sqrt{2c\lambda} dB_{t}-\lambda X_{t}dt$ has the following solution

Let $B$ be the brownian motion. How can I show that $dX_{t} = \sqrt{2c\lambda} dB_{t}-\lambda X_{t}dt$ is solved by $$ X_{t}=X_{0}e^{-\lambda t}+\sqrt{2c\lambda}\int\limits_{0}^{t}e^{-\lambda(t-s)}dB_{...
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1answer
22 views

How to calculate the jump of $e^{{\rm i}N_ty-\operatorname E[N_t](e^{{\rm i}y}-1)}$ for a Poisson process $N$?

Let $(N_t)_{t\ge0}$ be a (nonhomogeneous) càdlàg Poisson process on a probability space $(\Omega,\mathcal A,\operatorname P)$. Assume $\alpha(t):=\operatorname E\left[N_t\right]$ for $t\ge0$ is ...
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1answer
58 views

Stochastic integration of $s^2$ with respect to the Brownian motion

When applying integration by part formula for Itô's integrals, I get $$\int_0^T s^2dB_s=T^2B_T-\int_0^T2sB_sds$$ but how do we deal with the term $\int 2sB_sds$?
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1answer
32 views

Interchanging limit and expectation regarding integrated brownian motion

I am currently working with integrated brownian motion and have to find its mean and variance. I know they are $0$ and $t^3/3$, but I am having trouble showing this. Using the Îto formula we can ...
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69 views

Equivalent characterization of Poisson processes

Let $\alpha>0$. We usually say that an $\mathbb N_0$-valued process $(N_t)_{t\ge0}$ on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge0},\operatorname P)$ is Poisson with ...
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1answer
83 views

Prove that $N(t,A):=\left|\left\{s\in(0,t]:\Delta X(s)\in A\right\}\right|$ has independent increments

Let $E$ be a normed $\mathbb R$-vector space, $(X(t))_{t\ge0}$ be an $E$-valued càdlàg Lévy process on a probability space $(\Omega,\mathcal A,\operatorname P)$ and$^1$ $$N(t,A):=\left|\left\{s\in(0,t]...
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1answer
48 views

Is a càdlàg martingale on a finite time horizon square-integrable?

We work on a filtered probability space with finite time horizon $T$. Let $X$ be a càdlàg martingale. Question: Is $X$ square-integrable, i.e. does $\sup_{t\in[0,T]}E[X_t^2] <\infty$ hold? We know ...
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200 views

Forward vs backward formulation in Feynman-Kac

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a nice filtered probability space with an $m$-dimensional standard Brownian motion $W$. Fix a time horizon $T>0$. Let $\mu \colon [0,T] \times \mathbb{R}^d \...
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0answers
46 views

Linear SDE with commuting matrices.

Consider the linear $d$-dimensional SDE $$dX_t=-AX_tdt+\sqrt{2}BdW_t$$ for $X_t\in\mathbb{R}^d,A\in \mathbb{R}^{d\times d}$, and an invertible matrix $B\in \mathbb{R}^{d\times d}$, here $W_t$ is a $d$-...
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0answers
59 views

A continuous $\overline{\mathbb{F}^W}$ - adapted process X is $\sigma(W)$ - measureable.

Let $(\Omega,\mathcal{F},\mathbb{F}=(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space satisfying the usual conditions, $W$ an $\mathbb{F}$-Brownian motion and $X$ a continuous $\...
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0answers
30 views

Is a piecewise semi-martingale again a semi-martingale?

We work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P})$, where $T$ is finite. We consider stopping times $0=\tau_0<\tau_1<\dots<\tau_n\leq T$, ...
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0answers
23 views

Understanding the definitions of weak and strong solutions in SDEs [duplicate]

I am studying a course on stochastic differential equations and I have the definitions of weak and strong solutions of the SDE \begin{align} dX_t = f(t,X_t)dB_t + f_0(t,X_t)dt \ \ \ \ \ \ \ \ \ \ \ (...
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0answers
24 views

Evaluating probability of absolute value of stochastic integral being more than some number

in a book published in Japan, I came across an inequality, which I am stuck at. Let a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\geq 0}, P)$ given. $X=\{X_t\}_{t\geq 0}$ is ...
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0answers
34 views

Bounds (or moment estimates) on the stochastic exponential of a continuous Brownian integral martingale?

I'm trying to find results that estimate the stochastic exponential $\mathcal{E}(M)$ of a continuous local martingale $M = (M_t)_{0 \leq t \leq T}$ starting at zero, where $T \in (0,\infty)$. By ...
5
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0answers
128 views

If a diffusion is Gaussian, what does it imply to its drift and volatility?

Let $(Y_t)$ be a stochastic process solution to the SDE $$dY_t= \lambda(Y_t,t) dt + \sigma(Y_t,t) dB_t. $$ If we know that $(Y_t)$ is a Gaussian process, what does it inform us on the drift $\lambda$ ...
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1answer
38 views

Filtration enlargement and Brownian bridge SDE

I am looking at section 2.2 of this pdf: https://www.minet.uni-jena.de/Marie-Curie-ITN/EoF/talks/jeanblanc_introduction.pdf Here is what it says: Let us start with a Brownian Motion ($\operatorname{BM}...
2
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1answer
37 views

Showing an identity of two uniform distributed random variables by using characteristic functions and the inversion formula

I have to show that for $a,b > 0$ $$\int_\mathbb{R} \dfrac{\sin(at)\sin(bt)}{t^2}dt = \pi\min(a,b)$$ by using characteristic functions and the inversion formula. We do have the hint that we should ...
2
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0answers
30 views

Are all semimartingale Gaussian processes weak solutions to a diffusion SDE?

Let $(X_t)$ be a Gaussian semimartingale process with mean $\mu(t)$ and covariance function $U(s,t)$. Is it true that we can always find functions $\lambda$ and $\sigma$ such that $(X_t)$ is solution ...
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0answers
34 views

Dubins-Schwarz and independence property

If $M$ is a continuous local martingale such that $M_0=0$ and $\langle M \rangle_{\infty}=+\infty$, then $M$ can be seen as changed time brownian motion. More precisely, there exists a Brownian motion ...
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0answers
27 views

Find number of solution tuples to a given inequality with combinatorics/stochastics involved?

My problem has some practical background but on essence it circles around the following mathematical problem: let $f(a,b,c)$ be a given function of a,b and c. a,b and c can, totally random and ...
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0answers
41 views

$\lim_{r \to +\infty}\frac{1}{\sqrt{2r \ln(\ln(r))}}(B_r-B_{\left \lfloor{\sqrt{2r \ln(\ln(r))}}\right \rfloor})= 0$ a.s.?

Consider a Brownian motion $B$ and let $f(r)=\sqrt{2r \ln(\ln(r))}.$ Is it true that $\lim_{r \to +\infty}\frac{1}{f(r)}(B_r-B_{\left \lfloor{f(r)}\right \rfloor})= 0$ a.s. ? If so, how to prove it? ...
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0answers
18 views

Stochastic approximation algorithm for finding smallest root of nondecreasing function

We know that the e.g. Polyak-Ruppert averaging algorithm (from the stochastic approximations field) allows us to find the root of an increasing function, when we can only observe noisy evaluations of ...
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2answers
64 views

Box A has m balls, box B has n balls. Draw one ball randomly from one box. What is the expectation of remaining balls when one box is empty?

There are $m$ balls in box A and $n$ balls in box B. For every time, you draw a ball from either box with equal probability. (i.e. 1/2) You stop drawing when there is an empty box (i.e. you stop as ...
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0answers
14 views

Convergence results on projected stochastic approximation?

For general projection-free stochastic approximation(SA) $x_{n+1}=x_n+\alpha_n (f(x_n)+M_n)$, there has been many convergence results rely on $sup\|x_n\|<\infty, a.s.$. This condition is however ...
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0answers
30 views

Regarding the construction of stochastic integral: are the left-end points enough?

My question about the stochastic integral is elementary and somehow descriptive. In Karatzas and Shreve's Book (pp. 132-134), there are three steps to construct the general Ito integral (with ...
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0answers
19 views

Test function for stochastics differential equations

Hi … I study the stochastic differential equations and I like to know how we can define and apply the “test functions” for this group of equations? Moreover, can we define the weak solution using the ...
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0answers
17 views

Definition of closure of space with respect to a norm?

I am failing to understand the definition of the following space. Let $E$ be a Banach space and $\mu$ a (Gaussian) measure on it. Then one defines the Cameron-Martin space as follows: The Cameron-...
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0answers
30 views

Covariance operator of Gaussian

I am reading through M.Hairer's "An Introduction to Stochastic PDEs" and am slightly confused with the term covariance operator. Let $\mu$ be Gaussian measure on a Banach space $E$. Then we ...
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0answers
29 views

Does a Random Variable Bounded in L^P imply a Bounded Expectation

I have an elementary question regarding a random variable bounded in $L^p$. By definition, it satisfies $$\sup_{t\ge0}\mathbb{E}(|X_t|^p)<\infty.$$ However, can we say equivalently there exists ...
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0answers
51 views

Independence of discrete random variable from itself (please check and comment on my attempt)?

Could you please check and comment my following attempt of a proof: Let $(\Omega,\mathcal F,\Bbb P)$ be a probality space and $(E,\mathcal E)$ a measurable space. $E$ countable and $\mathcal E = 2^E$. ...
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1answer
31 views

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)&=\...
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2answers
41 views

A continuous Markov process that is not Gaussian?

Given a probability space, we say that $(X_t)_{t \geq 0}$ is Markov w.r.t its own filtration $(\mathcal F_t)$ if for all $s<t$, $$ P(X_t \in \cdot | \mathcal F_s) = P(X_t \in \cdot | X_s).$$ ...
0
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1answer
29 views

OU process generator

Consider the centered OU process given by $X(t)=\frac{\beta}{\sqrt{2\alpha}}e^{-\alpha t}B(e^{2\alpha t})$ where $(B(t))_t$ is a standard Brownian motion. It is obvious why this is a 0 mean process ...
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0answers
42 views

Cylindrical Brownian motion?

There's two different notions of cylindrical Brownian motions on a Hilbert space and I can't quite link them together: The first definition (for example used in Liu/Röckner SPDE'S:An introduction) of ...

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