Questions on the calculus of stochastic processes, or processes that have a random component.

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58 views

moving window supremum of a Wiener process

Let $W$ be a Wiener process. For each fixed $t>\frac12$, is it true that,$$\sup_{s\in[0,\frac12]}|W(t-s)|$$ has the same law as $$\sup_{s\in[0,\frac12]}|W(t)-W(s)| ?$$
5
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1answer
296 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
1
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1answer
104 views

Variance of a stochastic integral?

Does there exist a variance formula for stochastic integrals? Suppose we have $dX = \sigma (X) dW + \mu(X) dt$ Do we have a formula for $Var(X_t)$ or an intergral of $X$ against $B$ More ...
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1answer
334 views

Mean and variance of a brownian bridge

I am trying to compute mean and variance of the stochastic process $X_t$, which is a Brownian bridge from x to y, in the time-interval $[t,T]$. $$X_t = y + ...
1
vote
1answer
47 views

Not using stochasstic integral how to prove $E\int_0^T W^2(t)dt<+\infty$?

Can anyone help me to prove this? Suppose $W_t$ ~ $N(0,t)$, then not using stochasstic integral (or anything related with Ito) how to prove $E\int_0^T W^2(t)dt<+\infty$? Thanks.
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1answer
137 views

Ito vs Stratonovich SDE with irregular time-dependence in coefficients

Suppose I am interested in the Stratonovich SDE $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) \circ dB_t $$ If the coefficients are smooth enough in time and space, I can show this is equivalent to the Ito ...
2
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1answer
87 views

Joint distribution of $W(t)$ Brownian motion and $B(s)=\int_0^t \operatorname{sgn} ( W(s) ) dW(s)$

Let $(W(t))$ be a brownian motion and $B(s)=\int_0^t \operatorname{sgn} ( W(s) ) dW(s)$. Does one know the joint distribution $(W(s),B(s))$ for a given $s$? I know some related theory like Tanaka's ...
2
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0answers
72 views

an exetension of Doob's inequality

Doob's inequality gives an estimation of $$\mathbb{P}(\sup_{0\leq t\leq 1}|X_t|\geq\varepsilon)$$ where $X$ is a martingale. Now I wonder how to estimate $$\mathbb{P}(\sup_{0\leq t,s\leq 1, ...
2
votes
1answer
111 views

Characteristics of stochastic integral?

I need to describe a couple of integrals which are supposed to be evaluated in terms of Ito calculus. $$ I_1 = \int_0^t e^{-2\tau}dW(\tau); \\ I_2 = \int_0^t e^{-3 W(\tau)} dW(\tau); $$ Here ...
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0answers
72 views

question about the sequential continuity of the set of probability measures

I have a question about the sequential continuity of the set of probability measures. Let $\Omega$ be the space of continuous functions defined in $[0,1]$ taking values in $\mathbb{R}$. Let ...
1
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1answer
82 views

Covariance combined with normal distribution

We have $N_1$ and $N_2$, normal distributed random variables with averages $µ_i=E[N_i]$ and variances $σ_i^2=Var[N_i]$ and $c = Cov(N_1, N_2)$. We want to compute $E[e^{N_1} I(N_2>0)]$, where I is ...
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0answers
50 views

Contradiction on equality with stochastic integrals

I want to compute $E[∫_0^tB_u \, du ∫_0^sB_u \, du]$ and I know from another source that should be equal to $ts^2/2$. But when I try to compute it like: $$\begin{align} & E\left[(tB_t- \int_0^tu ...
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1answer
95 views

Computing expectation of a stochastic integral

I need to compute the expectation $$E\left[\int_0^tu \, dB_u \int_0^s u \, dB_u \right].$$ Being that is my first question, how can I initialize MathJax if I have it on my hard drive.
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1answer
151 views

If quadratic variation of a local martingale is zero then it is itself zero

Let $M$ be a local martingale, if we need it, we can assume that $M$ is continuous. We know that $\langle M\rangle =0$. This implies that $M$ and $M^2$ are local martingale. Can we conclude that ...
4
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1answer
125 views

How to compute $E\left[W_t \int_0^t s \, dW_s\right]$?

I want to compute $E\left[W_t \int_0^t s \, dW_s\right]$ where $W_t$ is a Brownian motion. My attempt below is based on some very shaky mathematics; in particular I have no justification of the 4th ...
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1answer
42 views

Stochastic Infinitesimal Generator Definition Confusion

I have seen an operator $A$ called the Infinitesimal Generator. Given $b: \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $\sigma: \mathbb{R}^n \rightarrow \mathbb{R}^{n m}$ and $f:\mathbb{R} ...
0
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0answers
86 views

A stochastic programming with a chance constraint

Let $X$ be a bounded positive variable with an unknown probability density function (PDF) and $f(X)$ be a differentiable positive function. $$\begin{align*} &\min/\max ...
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0answers
44 views

Space of stochastic process $\mathcal M (\mathcal C [0, T], E)$

A simple notation question, what is the precise definition of the space $\mathcal M (\mathcal C [0, T], E)$ ($\mathcal M^p (\mathcal C [0, T], E)$) in the context of stochastic processes where $E$ is ...
2
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1answer
173 views

Girsanov transformation and preservation of independence

If we create a weak solution of an SDE using the Girsanov transformation, are the initial condition and parameters independent of the transformed Wiener process if they are independent of the original ...
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1answer
973 views

How to integrate a Wiener process that freezes at a determined time?

I would like to calculate the expected variance of the average of a Wiener process from time $0$ to time $1$. The equation I believe I am trying to solve is: $$ \mathbb{E} \left[ \left( \int_0^1 W_t ...
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0answers
123 views

Example of a regular strong solution of an SDE, which doesn't satisfy a Lyapunov condition?

Let $$dX_t = a(t,X_t) \, dt + b(t, X_t) \, dW_t, \quad t \in [0,T]$$ be a stochastic differential equation, where $W$ is an $m$-dimensional Brownian motion, $X_0 = x \in \mathbb{R}^d$, and the ...
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votes
2answers
146 views

Calculating covariance, with multiplication by stochastic variable.

As an exercise I'm supposed to calculate; $\text{cov}(X \cdot Y,X)$, where $X$ and $Y$ are independent discrete stochastic variables, with probability functions given by; $$ p\left(var\right) = ...
0
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1answer
43 views

Proving that a discrete stochastic variable is binomial distributed.

Given a discrete stochastic variables, with the probability function; $$ p_{X}\left(x\right)=\left\{ \begin{array}{cc} \frac{1}{4} & \text{if } x = -1 \\ \frac{1}{4} & \text{if } x = 0 \\ ...
0
votes
1answer
48 views

Error, calculating covariance between two stochastic variables

In my exercise, I'm given two independent discrete stochastic variables, with the probability function; $$ p_{X}\left(x\right)=\left\{ \begin{array}{cc} \frac{1}{4} & \text{if } x = -1 \\ ...
2
votes
1answer
160 views

Square root of a stochastic process

i need help with the following problem. how can i derive d√v using Ito's lemma for the following process: d√v=(α−β√v)dt+δdX The parameters α, β, δ are constant. Using Itô's lemma show that dv = ...
1
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1answer
123 views

Need to Prove Result in Stochastic Calculus using Ito's Lemma

I can't figure out where : \begin{align} \delta^2\,dt\\ \end{align} comes from. Consider the process $$ d\sqrt{v} = = (\alpha - \beta\sqrt{v})\,dt + \delta \,dW $$ Here $\alpha, \beta,$ and $\delta$ ...
0
votes
1answer
59 views

limit of the first moment of solution of stochastic differential equation

Suppose $X^x$ is solution of $$d X_t = X^3_t dW_t, \quad X_0 = x>0.$$ In the above, $W$ is a Brownian motion in a given filtered probability space. Such an equation has unique strong solution, ...
4
votes
1answer
100 views

Reversing a diffusion bridge.

Suppose I have an $n$-dimensional Itô SDE $$dX_t = \sigma(X_t) dW_t + \lambda(X_t)dt$$ and I'm interested in diffusion bridges from $X_0=a\in\mathbb R^n$ to $X_T=b\in\mathbb R^n$. Now let $Y_t$ be a ...
2
votes
1answer
101 views

Limit of a stochastic integral

Let $W_t$ be a one-dimensional Brownian motion and I would like to prove $$\lim_{\beta\rightarrow+\infty}\sup_{0\leq t\leq T}\left|e^{-\beta t} \int_0^te^{\beta s}\mathrm dW_s\right|=0$$ This is an ...
2
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0answers
812 views

Integrating deterministic function with respect to Brownian motion

I have looked everywhere for a satisfactory answer to this, including Shreve's textbooks, but I can't find one. If I want to integrate a some deterministic function f(t) with respect to brownian ...
5
votes
2answers
452 views

Almost sure convergence of stochastic process

Suppose that we have a (almost surely) continuous stochastic process $\{ X_{t} \}_{t \geq 0}$ on $[0,1]$ with non-stochastic initial value $X_{0} = x_{0} \in [0,1]$ and exponentially decreasing ...
3
votes
1answer
66 views

Convergence in distribution and normality of the limit

Let $Z=(Z_1,Z_2)$ be a bivariate standard normal vector and $Y_{1,n},Y_{2,n}$ two sequences of real valued random variables with finite variance such that $Y_{1,n}\xrightarrow{d}Z_1$ and ...
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0answers
66 views

Uniform convergence of random distribution functions

Let $(X_i)_{i\in\mathbb{N}}$ be a strictly stationary sequences of real valued random variables with finite variance. We have the empirical distribution functions $F_{n}(u):=\frac{1}{n} \sum_{i=1}^n ...
3
votes
2answers
490 views

Probability brainteaser

Normal 52 card deck. Cards are dealt one-by-one. You get to say when to stop. After you say stop you win a dollar if the next card is red, lose a dollar if the next is black. Assuming you use the ...
2
votes
1answer
127 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
2
votes
1answer
349 views

Ito's formula for multivariable Ito integral

I'm having trouble finding something that I think should exist, which is an integral formula of the multivariable Ito lemma. Simply put, suppose I have a function $f$ of two stochastic processes, ...
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0answers
46 views

question about time change for filtration

I have a question: Let $T$ be a bounded stopping time and let $(\mathcal{F}_t)_{t\geq 0}$ be a filtration satisfying the usual conditions. Define $\mathcal{G}_t:=\mathcal{F}_{T+t}$, $t\geq 0$. Then ...
3
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1answer
104 views

Question on step in the proof of Itō's formula (along the book of Revuz and Yor)

I am working through the proof of Itō's formula contained in the book "Continuous Martingales and Brownian Motion" by Revuz and Yor and am stuck at a point in the proof. Theorem (Itō's formula). ...
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1answer
124 views

Integration by part formula in Malliavin Calculus

The set $S$ of smooth random variables is the set of random variables $F : \Omega \rightarrow \mathbb R$ such that there exist a function $f$ in $ \mathcal C_p^{\infty}(\mathbb R^n)$ (for some $n ...
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0answers
50 views

right continuity of martingale constructed by $X_t=E[X|\mathcal{F}_t]$

$X\in L_1$ is a random variable, and $(\mathcal{F}_t)_{t\geq 0}$ is a filtration satisfying the usual conditions, so could we find a version of martingale defined by $X_t=E[X|\mathcal{F}_t]$. I think ...
2
votes
1answer
134 views

Lp integrable local martingale is a true martingale.

Is the following argument correct: Let $(M_t)_{t\geq 0}$ be a local martingale s.t. for some $p>1$ $E[\sup_{s \leq t} |M_s|^p]<\infty$ for all $t \geq 0$. Then $E[\sup_{s \leq t} |M_s|]\leq 1+ ...
7
votes
1answer
152 views

Prove the density of this SDE is not smooth in a parameter

Consider the following, 1-dimensional, equation $$X_t^x = x + \int_0^t \mathbb{E} |X_s^x| \, ds + B_t , $$ where $B$ is a Brownian motion. This a McKean-Vlasov equation, sometimes called a nonlinear ...
2
votes
0answers
100 views

Absolute Continuity and simple discontinuity

I am reading a book called Stochastic Process, Estimation, and Control, in P.32 it states that a function with finite simple discontinuities can still be absolutely continuous, which confused me, I ...
4
votes
3answers
302 views

Proof of Levy's zero-one law

Let $(\Omega, \mathcal{F},\mathbb P)$ be a probability space and let $X$ be a random variable in $L^1$. Let $(\mathcal{F}_k)_k$ be any filtration, and define $\mathcal{F}_{\infty}$ to be the minimal ...
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1answer
259 views

Uniform integrability of a backward submartingale

Let $\{\mathcal{F}_n\}_n$ be a decreasing sequence of sub-$\sigma$-fields of $\mathcal{F}$($\mathcal{F}_{n+1}\subset\mathcal{F}_n$) and let $\{X_n\}_n$ be a backward ...
4
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1answer
340 views

Continuous Square integrable martingale Quadratic Variation

We know that given a continuous square integrable martingale there exists unique (up to indistinguishability) continuous, natural and increasing process which is quadratic variation process of the ...
2
votes
1answer
37 views

Argument that vector space is closed with respect to bounded monotone convergence

Let $(\Omega, \mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a probability space. Let $C$ be the stochastic processes which can be written on the form $\sum _{i=1}^n K_i 1_{(a_i,b_i]}$ for ...
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0answers
39 views

Unit Root Test interpretation

I am analyzing data collected hourly over $9$ years (around $80000$ hours) and I have following result for Unit Root Test : How can I interpret the data? Can I use the Autoregressive model?
5
votes
2answers
699 views

Why isn't the Ito integral just the Riemann-Stieltjes integral?

Why isn't the Ito integral just the Riemann-Stieltjes integral? What I mean is, given a continuous function $f$, some path of standard brownian motion $B$, and the integral: $$\int_0^Tf(t)\;dB(t).$$ ...
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0answers
52 views

Total set of functions in $L^2(\Omega)$

Are the sets of functions $\{e^{\int_0^T h(s)dB_s -\frac{1}{2}\int_0^T h(s)^2 ds}\}$ and $\{e^{\int_0^T h(s)dB_s}\}$ total in $L^2(\Omega)$? What is the difference? What should one use to prove weak ...