# Tagged Questions

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### Optimal stopping problem

Consider the OU process: $dX_t = -X_tdt + dW_t$, $X_0 = 0$. Compute the optimal stopping time for the following problem: $$v = \sup_{\tau} E[|X_{\tau}| - \tau]$$ So far I have set $L\phi = 0$, ...
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### Filtration right continuity completion

I have a question about filtration. Now fix a measurable space $(\Omega,\mathcal{M})$. Let $(\mathcal{M}_{t})_{t\in[0,\infty)}$ be a filtration on $(\Omega,\mathcal{M})$. We set \begin{eqnarray*} ...
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### Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions. However, I have the following statement ...
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### Integral of Squared Brownian Motion given the integral of a regular Brownian motion?

So I'm looking to calculate the variance of the following integral: $$\operatorname{Var}\left(\int_0^b W(s)^2\,ds \mid \int_0^b B^y(s)\,ds\right)$$ where $W(s)$ is a standard Brownian motion, and ...
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### On “for all” in if and only if statements in probability theory and stochastic calculus

1 In my friend's Probability Theory long test there was this question: Let $(\Omega, \mathfrak{F}, P)$ be a probability space on which is defined all sub-$\sigma$-algebras, events and random ...
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Let $X,Y$ be square integrable Right continuous martingales. If $Z$ is the total variation of $\langle X,Y\rangle$, how can I show that $$Z \leq \frac{1}{2}[\langle X\rangle + \langle Y\rangle].$$ I ...
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### Solve $dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t) \, dt + \sqrt{1+X_t^2} \, dW_t$ explicitly

Solve explicitly the 1-dimensional equation: $dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t)dt + \sqrt{1+X_t^2}dW_t$ I have hopelessly been guessing solutions to this. Does anyone know how to solve this ...
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### Invariant mesure of a reflected random walk

Let $(X_n), n \geq 0$ be a Reflected Random Walk defined by: $X_0 = 0$ and: $X_{n+1}=\max( 0 , X_n + \xi )$ $\xi$ is a random variable such that $P(\xi=a)=\theta$ and $P(\xi=-b)=1-\theta$ for a ...
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### Reflected random walk

Suppose that $X_n$ is a reflected (in 0) random walk with parameter $\theta$. So $X_{n+1}-X_n = 1$ with probability $\theta$ , and -1 with probability $1-\theta$ when $X_n \geq 1$, if $X_n=0$ then ...
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### Question on generators in the proof of Kolmogorov's Backward Equation

Here is a part of the proof of the Kolmogorov's Backward Equation. I cannot see why $Y_t$ has been picked as it has. In particular, I cannot see why you would want to subtract t in the first bit of ...
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### Questions about expectation of stochastic integrals

I am considering the following SDEs: $$dX_1=-\theta(X_1-a_1)dt+\sqrt{X_1}(1-X_1)dW_1-X_1\sqrt{X_2}dW_2$$ $$dX_2=-\theta(X_2-a_2)dt-X_2\sqrt{X_1}dW_1+\sqrt{X_2}(1-X_2)dW_2$$ Here $W_1$ and $W_2$are ...
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### The completed natural filtration of brownian motion is right-continuous, proof?

I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique. Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real ...
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### More on the Existence and Uniqueness of the solutions of an SDE Proof

An extract from the proof of the existence and uniqueness of the solution of a SDE from Oksendal. I cannot see how holders inequality and the ito isometry are applied.
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### Part of Proof of the Uniqueness of the Solution of SDE's

This is an extract from Oksendal's SDE of the proof of the uniqueness of the solution of a SDE. I cannot see how the $P[|X_t-\hat{X_t}|=0 \ \ \ \text{for all t} \in \mathbb{Q} \cap [0,T]]=1$ is ...
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### Finding the unique Martingale Convergence Representation of a given r.v.

According to the martingale representation there exists a unique $g(t,\omega) \in \mathcal{V}(0,T)$ such that $M_t = E[M_0]+\int^{t}_{0} g(s,\omega) dB(s); \ \ \ t \in [0,T]$ Find g in the case ...
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### More preliminaries of the Martingale Convergence Theorem

Really struggling with this lemma. Not sure about the general structure of the proof. Why have we chosen g to be orthogonal to all functions of the form 4.3.1? Why should $G(\lambda)=0$, does it ...
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### A Property of the Ito Integral

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then $E[\int^{T}_{S}f dB_t]=0$ Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ...
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### Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
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### generator of a function (stochastic) [closed]

How do I find a generator of $$g(Y_t)=Y_t^2-10Y_t+25 \, ,$$ where $Y_t$ is a geometric BM: $$dY_t=-1Y_tdt+2Y_tdW_t \, ,$$ and $W_t$ is white noise
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### Solve Itô integral with power

$$\int_0^t e^{Ws} W_s^r dW_s$$ where $W_s$ is Wiener process and r> in $\mathbb{Z}$ My first approach would be to use Ito's lemma, however, coming up with the function $g(t,x)$ is difficult The ...
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### What is a.e. a.s

I am reading a paper which uses almost everywhere almost surely (a.e.,a.s.) simultaneously, I am not quite sure what it means then. To be specific, they consider a stochastic process $\{X_t\}$ such ...
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### Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...
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### Poisson/ jump process distribution for process $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$

For the process: $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$, where $X(t)$ is a poisson process with paramater $\lambda$, and: $J_k$ are i.i.d . random variables (jumps). $B(t)$=brownian motion. I want to ...
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### Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
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### Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$dV_t = - \beta V_t dt + \sigma dW_t$$ with $V_0 = v$, where $W_t$ is a Wiener ...
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### Mean and Variance of Gaussian Process

Let $B = (B_t : t \geq 0)$ be a standard Brownian Motion. Fix $0 \leq s \leq t$. How can I prove that, conditionally on $\{B_s = x, B_t = z\}$, the intermediate value $$B_{\frac{t+s}{2}}$$ has ...
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### Sufficient condition for time-changed quadratic covariation to vanish in probability

Let $(M_t^n)_{t \geq 0}$ be a sequence of continuous martingales of the form $M^n_t = \int_0^t X^n_s \, dB_s$ where $B_s$ is a Brownian motion. Let $\tau^n(t)$ be the time change associated to $M_t^n$ ...
There isn't a similar question in the forum, so here it goes. Firstly, let the O-U velocity process be defined as $$dV_t = -\beta V_t dt + \sigma dB_t$$ with $V_0 = v$, and $B = (B_t), t \geq 0$ a ...