A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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Learning stochastic calculus, want to know what the notation of this function means

$f(x) = \sum_if_i 1_{[ai;bi)}(x)$ This is a function that is piecewise constant equal to f_i on finite set of intervals [ai; bi) in a set F: I am a little confused about what this 1 in the summation ...
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3answers
265 views

Ito formulae for stochastic processes with finite cubic, quartic … n-tic variation

Many stochastic processes that you encounter are kind of well-behaved, i.e. have infinite variation, yet finite quadratic variation. My question revolves around stochastic processes that have ...
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522 views

asymmetric random walk, hitting time probability

Let's consider an asymmetric Random Walk on $Z$, with transition probabilities $p_{i, i+1}=p$, $~~p_{i, i+1}=q$, $\forall i \in \mathcal{Z}$, $p+q=1$ and $p>q$. I am interested in the probability ...
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1answer
96 views

Nonstandard definition for the generator of a standard Ito diffusion

For a standard Brownian motion, the generator of the diffusion is $$ L = \frac12 \frac{d^2}{dx^2}. $$ Is there a nonstandard definition of this generator?
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134 views

Constructing black noise with non-standard analysis

With noise in the sense of i.i.d. random sequence, a noise is black if it is not isomorphic to standard Gaussian white noise. Tsirelson showed the existence of black noise through the scaling limit ...
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184 views

Reflected Brownian Motion

Let $Y(t)$ be a reflected Brownian motion, and $G(t)$ is the process which keeps count of number of times that $Y(t)$ has hit the X axis. How do I approach to find distribution of $G(t)$, or almost ...
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0answers
94 views

Trying to get an idea of the maths I could use for this optimization problem

Firstly, apologies if some of the notation or terminology is odd, or if I am defining functions that have standard notation associated with them already - I am not familiar with the concepts in this ...
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2answers
152 views

If $\mathcal{F}_t$ is separable why is $\mathcal{F}_\infty$ generated by a random variable?

I am reading this introduction to enlargement of filtration and at the beginning of section 2.4 there is a claim that I cannot justify but seems like it should be well known. The author claims that ...
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90 views

A terminal coalgebra of a certain functor on Mes

Let $\mathfrak C = \mathsf{Mes}$ be the category of meausurable spaces and measurable maps. For any object $X\in \mathfrak C_0$ we assign a measurable space $\mathcal P(X)$ whose elements $\mu$ are ...
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1answer
131 views

Upper bound concerning Snell envelope

Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the usuual ...
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1answer
219 views

On martingale representation theorem

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(\mathcal{F_{t}})_{0\le t\le T}$ a filtration generated by standard Brownian motion $W_t$. Let $f(x)$ be $C^1$ function such that $|f'(x)| ...
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128 views

Do there exist (almost surely) $C^{\infty}$-smooth Gaussian random fields?

Let $d \ge 1$. Do there exist Gaussian random fields on $\mathbb R^d$ which are (almost surely) $C^{\infty}$-smooth, but which are not analytic? If so, what are necessary and sufficient conditions ...
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345 views

Converse to Girsanov's theorem?

Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to ...
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78 views

Markov renewal process with failure?

I hope this question is not too elementary for this site, and that it contains a sufficient degree of detail. I have a problem where I want to model sequences of variable length $\boldsymbol{e}_i = ...
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1answer
166 views

Maximal inequalities for certain functions of a martingale difference sequence

Suppose $\xi_1,\ldots \xi_T$ is a martingale difference sequence. Then, 1) For any $a\in \mathbb{R}^{+}$, can we say something about the sequence $\xi_1^2\mathbb{1}(\xi_1\geq a),\ldots, ...
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221 views

Exit probability of a Brownian particle.

Perhaps the answer is common folklore among probabilits and stochasticians(!)? But I would like a good lower estimate for the probability that a particle undergoing brownian motion in 1 dimensions ...
2
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1answer
169 views

Limit of a Wiener integral

How to show that $$ \lim_{\alpha \rightarrow \infty} \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.}$$ where $\left (B_s ...
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1answer
85 views

Concerning Jump process (Lévy process)

Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is $$ \nu \left( dx\right) = A ...
2
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1answer
118 views

Completion time of a process on a tree

Given is a constant degree rooted tree of depth $D$. It is also known that the total number of nodes in the tree is at most $D^2$. There is a probabilistic process with discrete time steps on the ...
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65 views

Lévy processes in the space of tempered distributions

We know that the brownian motion is dominated by some polynomial function (consequence of the law of the iterated logarithm for instance). Then, one can say that almost surely, a brownian motion is in ...
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1answer
168 views

Coupling of vectors

Let $X = (X_1,X_2)$ and $\hat X = (\hat X_1,\hat X_2)$ be two random variables where $X_i,\hat X_i$ are taking values over the Polish space $E_i$ endowed with their Borel $\sigma$-algebras, where ...
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214 views

Convergence rate for product of stochastic matrices

Hi, I have a system of the form $$x(t+1) = A(t + 1) x(t),$$ for $t \geq 1$, and some fixed initial condition $x(1)$. Here $A (t)$ is a time-varying $m \times m$ matrix that is stochastic at all ...
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1answer
355 views

Topological conditions of Kolmogorov Extension Theorem

KET is often used to construct stochastic processes in continuous time when the state space is $\Bbb R^d$. As far as I am familiar with its proof, it uses standard monotonic class-like arguments ...
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2answers
123 views

Is there a general process for conditioning a stochastic process above a boundary?

$(X_t, Y_t)$ is a two-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. Given its transition function $a(x, y | x', y')$, I would like to condition the process on $\inf_{s ...
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119 views

Can we express a one-dimensional raised Bessel Bridge as a function of a single Brownian Motion?

A Bessel Bridge is a Brownian Motion, conditioned such that $B(0) = B(1) = 0$ and $B([0, 1]) \ge 0$. A raised Bessel Bridge is a generalization of this: it's a Brownian Motion conditioned such that ...
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76 views

Multinomial — how many trials in order to see all the values with prob 1-\alpha

Let suppose that I have a box with $k$ different balls, each one with a different color. At each time I have to extract a ball and observe the color. Then I put the ball back in the box. How many ...
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2answers
422 views

minimum of different independent Poisson random variables

Let $X_1,\ldots,X_N$ be independent Poisson distributed random variables with unequal parameters $\lambda_1,\ldots,\lambda_N$. Is there any closed form expression or at least a good approximation for ...
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1answer
156 views

Area under Gaussian sample path curve

Dear All, We know that the coastline paradox is related with fractal dimension of a curve. Now I want to know how to estimate the area under a sample path curve of a Gaussian process: Given a ...
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133 views

A question on Wiener Process

Suppose we have a Wiener process $W$, and $U_x$ is the amount of time spent below the level of $x$ during the time interval $(0,1)$. How can I calculate the probability density function of $U_x$. Does ...
4
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1answer
295 views

Measurability issues in the proof of Fujisaki, Kallianpur and Kunita for stochastic filtering

I'm currently looking over the proof(s) of the theorem of Fujisaki, Kallianpur and Kunita regarding the MRT-like characterisation of square integrable random variables measurable with respect to the ...
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136 views

log-likelihood of ito diffusion

Consider a diffusion process: $ \text{d}X_t = f(X_t)\text{d}t + \text{d}W_t$ I've seen it given that the log-likelihood of the path is proportional to the Onsager-Machlup functional $ \int_0^T ...
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127 views

sum of stochastically continuous processes

Hallo, is the sum of two stochastically continuous processes again a stochastically continuous process? why? Thank you very much, Paolo.
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133 views

First moment of a function of a normally distributed random variable

I'm trying to find the first moment of the following function: $f(x) = \frac{(-ax+\sqrt{1-a^2})(-bx+\sqrt{1-b^2})}{\sqrt{x^2+1}}H(-ax+\sqrt{1-a^2})H(-bx+\sqrt{1-b^2})$ where $H(x)$ denotes the ...
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312 views

total variation distance between two solutions of SDE

Suppose we have two stochastic differential equations with the same initial conditions: $$d X_t^1= b_1(t,X_t^1)dt + dW_t$$ $$d X_t^2= b_2(t,X_t^2)dt + dW_t,$$ $X_0^1=X_0^2=x_0$; $W_\cdot$ is a ...
4
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1answer
270 views

Trajectorial version of Doob's $L^2$ inequality

In the paper http://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0154.pdf you can find a trajectorial version of Doob's inequality. It is given by: ...
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1answer
257 views

Probability that a “closable” self-avoiding random walk forms a polygon

Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...
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134 views

Existence of multidimensional Levy process with dependent structure

Levy process is frequently cited recently. When we come to multidimensional Levy process, the components are usually assumed to be independent. Are there any examples on how to construct a Levy ...
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171 views

Limit of the stochastic process at time 0

This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S_t$ is a continuous stochastic ...
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2answers
167 views

Properties of the Euler Discretization of a diffusion

Let $X$ be a continuous 1-d diffusion: $$ dX_t = a(X_t)dt + b(X_t)dW_t, X_0 = x. $$ W is a standard Brownian Motion and $a(\cdot)$ and $b(\cdot)$ can have nice regularity properties. Let ...
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75 views

Parameter Sensitivity of Stochastic Process

How do I compute the derivative \frac{\partial X_t}{\partial \sigma}? Where dX_t=\theta (\mu-X_t)dt+\sigma \sqrt{X_t}dZ_t
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Wiener-Hopf Integral/Lindley's Equation

Lindley's equation is well known within queueing theory and is as follows $F(y) = - \int_0^\infty F(x)dH(y-x)$ However, many textbooks only consider the case where 0 $\le$ y $\le \infty$ (which ...
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119 views

Computing a density function for the integral of a stochastic process, given its transition function

$P$ is a one-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. I know its transition function: $P(0) = x_0$ and for any $0 \le t_a < t_b \le t_f$, the function $f(x_b | ...
4
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Integrating a Bessel Bridge

Preliminaries An order-3 Bessel Process is the one-dimensional stochastic process $X$ described by $X(t) = \sqrt{W_1(t)^2 + W_2(t)^2 + W_3(t)^2}$, where each $W_k$ is an independent Brownian Motion. ...
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131 views

Norm bound of the entrywise logarithm of a stochastic matrix stationary matrix

Hello, Denote $\log_\star$ as the entrywise logarithm operation, and let $A$ be some row-stochastic matrix such that $\lim_{p\rightarrow\infty}A^p$ exists and all its entries are non-zero. As a part ...
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186 views

Differentiability of integral w.r.t. hitting time of Brownian Motion

I have been trying to prove the following conjecture for a while, but so far to no avail. Would be very grateful for some tips! (I edited the entire thing to make it clearer) The conjecture is the ...
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1answer
55 views

What is the distribution of the distance between a specific word in a Text which is generated by a markov process?

What is the distribution of the distance between a specific word in a Text which is generated by a markov process? For example for a text which is generated by a multinomial distribution over words, ...
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139 views

Stochastic optimal control with no diffusion

Classical stochastic optimal control problem is to minimize functional $$ J(u) = \mathbf E \int_0^T f(t,x_t,u_t)dt, \tag{1} $$ subject to SDE $$ dx_t = b(t,x_t,u_t)dt + \sigma(t,x_t,u_t)dW_t, \quad ...
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1answer
686 views

Bochner integral of stochastic process = path by path Lebesgue integral?

After some helpful comments, I realized that I had to repost this question in a more systematic way. On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...
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199 views

modification of Doob inequality

Hi everyone, please consider the following problem: Let $(M_t)_{t\geq 0}$ be a continuous and positive submartingale and $S_t=\sup_{0\leq s\leq t}M_s$. Please prove that for any $\lambda>0$ we ...
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1answer
136 views

contraction property for conditioned SDEs

Consider a strictly convex potential $U: \mathbb{R}^d \to \mathbb{R}$ and the Langevin diffusion $$dX = -\nabla U(X) dt + dW \qquad (*)$$ where $W$ is a standard Brownian motion. If $(X_t)_{t \geq 0}$ ...