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4
votes
1answer
39 views

Martingale with respect to a decreasing filtration

I am trying to solve problem 2.16 from the book "Continuous Martingales and Brownian Motion" by Revuz and Yor. There are two things that confuse me from the exercise so hopefully someone can shed some ...
2
votes
1answer
40 views

How to combine two conditional exponential CDF's?

Suppose one has two machines (machine A and machine B) in sequence with time to machine break down exponentially distributed with rate parameters $\lambda_A$ and $\lambda_B$. Machine A and B have a ...
1
vote
0answers
20 views
+50

Conditional expectation with disjoint $\sigma$-algebras

Let $(B^1,B^2)$ be independent Brownian motions with corresponding filtration $\mathcal{F}_t$. Let $\mathcal{F}^2_t$ be the filtration generated by $B^2$. How does one prove that for any $s<t$ and ...
1
vote
2answers
52 views

Tossing two dice with sum equal to 4?

Exercise: Throw two dice. Suppose that eye sum are 4. Calculate the resulting conditional probability that a) the first dice gave a 3 . b ) the second dice gave two or fewer eyes. c ) ...
0
votes
0answers
10 views

Levy Processes - triplet for compound Poisson process

I'm stuck on 2 problems with Levy processes. People says that they are simple, but I can't solve it. Can anyone provide step by step solution? 1. Show that gamma distribution is infinitely divisible. ...
0
votes
0answers
16 views

Find pdf for solution of Stochastic DE

I have some troubles learning with Stochastic DE. There is a problem. Find the probability density function f(x,t), of $X_t$ where {$X_t$} is a solution of SDE: $dX_t = mdt + \sigma dW_t, X_0 = 0$ I ...
0
votes
0answers
22 views

Deciding if a measure is dominated by the Lebesgue measure

We define $X := \{0,1\}, \mu := \frac{1}{2} (\delta_0 + \delta_1)$ and $(\Omega, \mathcal{F},\mathbb{P}) : = \bigotimes_{n=1}^{\infty} \left( X, 2^X,\mu \right)$. For $\omega \in \Omega$ we denote the ...
0
votes
1answer
42 views

Deriving the Doob Meyer decomposition of a Sub Martingale using Ito's

Given the standard brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$ and defining the sub-m.g.: $$X_t =B^6_t+2t$$ I would like to derive its Doob-Meyer decomposition: [Sub-m.g.]= [increasing ...
0
votes
1answer
41 views

Show martingale space is a Banach space

Let $\mathcal{H}^1 = \{M \in \mathcal{M}, E[sup_{t\geq 0} |M_t|] < \infty\}$, where $\mathcal{M}$ is the space of right continuous with left limits martingales. Show that $\mathcal{H}^1$ is ...
1
vote
1answer
18 views

Definition of s-lim? (context: Trotter product formula)

I am searching for a definition of "s-lim", a notation I am seeing used sometimes in the statement of the Trotter product formula (for instance in Barry Simon's book Functional Integration and Quantum ...
1
vote
0answers
35 views

What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let $H$ be an ...
-1
votes
0answers
75 views

Find the relation between 2 stochastic integral

$g(s,t)(\omega)$ is an adapted stochastic process on $\mathbb R^2$ define: $$X=\int_0^1\int_0^1g(s,t) \,dW_s\,dt$$ $$Y=\int_0^1\int_0^1g(s,t) \,dt\,dW_s$$ Could we conclude that "$X=Y$ a.s"? I ...
2
votes
1answer
21 views

Conditional Ito's isometry

I am looking for a formal proof of the following (if true): $\mathbb E \left[ \int_0^1 g_1(s)\,dW_s \int_0^1 g_2(s) K_s\,dW_s \big|\mathscr F^K \right]=\int_0^1 g_1(s)g_2(s)K_s\,ds $, where ...
3
votes
0answers
41 views

Interchangeability of the malliavin derivative with a lebesgue integral

I was curious to know the most general conditions under which a malliavin derivative $\mathscr{D}_t \int^T_t F_v d\mu(v) = \int^T_t \mathscr{D}_t F_v d\mu(v)$ commutes with a lebesgue integral? I was ...
3
votes
1answer
40 views

Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

Suppose we have an sde of the form: \begin{eqnarray} dX_t=b(X_t)dX_t + \sigma (X_t)dB_t \end{eqnarray} where $b$ and $\sigma$ are Lipschitz. Then we have existence and uniqueness of the solution ...
0
votes
1answer
25 views

Ito integral's zero mean

My Sto Cal prof gave a long proof for the fact that $E[\int_{0}^{t} f_s dW_s] = 0$ where W is Brownian and f is Borel x $\mathscr{F}$-measurable, adapted and satisfies some integrability condition. ...
0
votes
0answers
19 views

markov property in Durrett's textbook

Assume $B_t(\omega)=\omega(t),\omega\in (C,\mathcal{C},\mathbb{P}^x)$ is a B.M.(C is the continuous function space ,$\mathcal{C}$ is generated by the coordinate maps) In Durrett's textbook,he proved ...
1
vote
1answer
33 views

Brownian motion on the circle and Itô processes

Consider the differential system \begin{cases} dX_t &=& -\frac{1}{2}X_t dt - Y_tdB_t, \\ dY_t &=& -\frac{1}{2}Y_tdt + X_tdB_t, \end{cases} $X_0 = 1$, $Y_0 = 0$. Let $X_t$ and $Y_t$ ...
3
votes
1answer
106 views

perfectly correlated processes

I am really stuck in this question: Let $\{S_t\}$ and $\{S'_t\}$ be two stochastic processes, satisfying \begin{equation} dS_t = S_t ( \sigma_t \,dB_t + r_t \,dt), \quad dS'_t = S'_t (\sigma'_t ...
0
votes
0answers
33 views

Equality of two spaces of stochastic processes

Let $(\Omega, \mathcal F, P, \mathcal F_t)$ be a filtered probability space. Consider two spaces $M$ and $S$ defines as follows: $M$ is a collection of all continuous $\mathcal F_t$-adapted processes ...
2
votes
2answers
42 views

Limit of $P(X_n > a_n)$ where $X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$ and $a_n\xrightarrow[n \to \infty]{} \infty$

I've been working on following problem and could need some help. Let $X_n$ be a sequence of RV with $$X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$$ for some $\mu \in \mathbb{R}$ and ...
1
vote
1answer
24 views

Sufficient condition for u.c.p. convergence of processes

Assume that all processes are cadlag. I am trying to prove the following claim: Let a sequence of processes $X_n$ be given. Assume that for all $s$ in a dense subset of $\mathbb R^+$ $ X_n(s) ...
1
vote
1answer
52 views

Uniform integrable proof

Lets be $E[\sup_{t\in[0,T]}e^{-rt}\Psi(S_{t})]<\infty$. I want to show that $J_{t}$ defined by \begin{align} J_{t}:=\mathrm{ess sup}_{\tau \in \mathcal F_{t,T}}E[e^{-r\tau}\Psi(S_{\tau})|\mathcal ...
0
votes
1answer
32 views

Approximation of stochastic integrals by Riemann sums

I know that for $f:[0,1]\to \mathbb{R}$, the Riemann Integral converges in the sense that $$\sum_{k=1}^Kf(t_k)(t_{k} - t_{k-1})\longrightarrow \int_0^1f(t)dt$$ as the grid becomes smaller and smaller. ...
1
vote
0answers
13 views

Is stochastic modelling a subset of Frequentist and Bayesian points of view?

From what I know of stochastic modelling it seems to me that this technique takes a Frequentist approach. For example, and please correct me if I am wrong, but isn't a Monte Carlo Simulation a ...
3
votes
0answers
98 views

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let ...
6
votes
0answers
66 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
1
vote
1answer
19 views

A question on the extension of of integrants from simple processes t0 $L^2$?

I have a question. While defining the Stochastic integral w.r.t to the Brownian Motion we begin with simple processes which are adapted and left continuous and then extend it to the square integrable ...
2
votes
0answers
35 views

How to calculate probability of an event in a stochastic setting?

Let $\left(\, B_{t}\,\right)_{t\ \geq\ 0}$ be a Brownian motion. Calculate the probability of the event: $$ E\equiv\left\{\,\exists\ \epsilon > 0 : \forall\ 0 < h < \epsilon, \max_{t\ \in\ ...
0
votes
1answer
128 views

A Liar's Paradox Mathematically

The premise (you can skip to the mathematical part below): You are driving back to the town where you were born. You haven't been home for a very long time and you are unsure if you are even on the ...
0
votes
0answers
7 views

Derive bound for a function satisfying stochastic differential equations/inequalities

I have a function $U(t)=\frac{1}{2}\|{x}(t)\|^2$ satisfying $ d{U} \le -\frac{\lambda_0}{2}dt+\frac{\sqrt{2T}}{N} \sum_i {dB_i(t)} \cdot {x}, \ \ if\ \ \|{x}\|\ge d_0,$ $d{U} \le \lambda_1 ...
1
vote
0answers
33 views

Stochastic integral density of simple functions no1

I am trying to understand proposition 2.6 page p.134 from Karatza's book Brownian motion and stochastic calculus. If $M$ is continuous square integrable martingale on $(\Omega, \mathcal{F}, P)$ and ...
1
vote
1answer
31 views

First-order stochastic dominance and truncation

Suppose we have two distributions $F$ and $G$ over $\left[0,1\right]$. Suppose $F(x) \leq G(x)$ for all $x$, i.e. $F$ first-order stochastically dominates $G$. Is it true that $F(x|x\leq k) \leq ...
0
votes
1answer
39 views

For a real-valued random variable it holds: $ E(|X|)<\infty\Leftrightarrow \sum_{n\in\mathbb{N}}P(|X|>n)<\infty$

Let $(\Omega,\mathcal{A},P)$ be a measurable space and $X$ be a real-valued random variable on $\Omega$. I want to show that it holds: $$E(|X|)<\infty\Leftrightarrow ...
2
votes
1answer
63 views

Ito Integral representation for bounded claims

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable and bounded. Does it ...
0
votes
1answer
61 views

The ito integral is gaussian [duplicate]

Let $\Omega, F, P)$ be the classic setting. I saw that if $f$ is a function which satisfies some assumptions then the integral with respect to the brownian motion is Gaussian. Ie $\int_{0}^{t} f_u ...
3
votes
1answer
111 views

Ito's Lemma for negative exponential

I'd been reading on Hull-White model, when I encountered the bond-pricing formula, that is if $$ dr(t) = (\alpha(t)-\beta(t)r(t))dt + \sigma(t)dW(t)$$ for some deterministic function $\alpha, \beta, ...
0
votes
0answers
30 views

Processes adapted to time changes

I have a question regarding a passage in Chapter X of "Calcul Stochastique et Problèmes de Martingales"J.Jacod(1979). In (10.13) they define an adapted process $X$ to the time change $\tau(t)$ as a ...
1
vote
1answer
11 views

How to interpret this variance

If I have a probability measure defined by $P( \Omega ) = \int_{\Omega} (1-a) \delta(x) + a \delta(x-a^2) dx,$ then I noticed that the variance is given by $a^5(1-a)$. This is somewhat strange, cause ...
2
votes
1answer
139 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
1
vote
1answer
71 views

Integral with respect to brownian motion

Let $f$ be a continuous function on $[0,\infty)$ and $B_t$ be a standard Brownian motion. Define $X_t=\int_0^t f(s) dB(s).$ a) Show that $X_t$ is Gaussian and computer its covariance $C(X_s, X_t)$ ...
0
votes
1answer
31 views

Equivalent Stopping Times for Brownian Motions

For standard Brownian motion $B$, define stopping time $T_1:=\inf\{t>0: B_t = 3\}$ and $T_2:=\inf\{t>0: B_t = -3\}$ and $T_3 := \min\{T_1, T_2\}$. Can I say that $T_3 = \inf\{t>0, B_t \in ...
0
votes
1answer
22 views

Brownian Motion and Progressive Process

Let $B_t$ be a Brownian motion. Define sign function as follows. $sign(0) = 0$ and $sign(x) = \frac{x}{|x|}, \forall x \neq 0$. I do not know how to show the following two questions, especially on the ...
0
votes
1answer
29 views

Stopping Time and Brownian Motion [closed]

Let $B_t$ be a Brownian motion. Let $a < 0 < b$. Consider $\tau: = \min\{T_a, T_b\}$ where $T_a := \inf\{s \geq 0: B_s \leq a\}$ and $T_b := \inf\{s \geq 0: B_s \geq b\}$, namely, the first ...
1
vote
1answer
43 views

Progressively Measurable for Rigth Continuous Adapted Processes

Any adapted and right continuous process $X_t$ is progressively measurable. For the above statement, I found proof in several books. They all have similar argument as follows. For a given $t > ...
1
vote
1answer
55 views

Variance process of stochastic integral and brownian motion

Let $(W_t)$ be a Brownian motion with respect to a filtration $(\mathcal{F}_t)$. For all $t \geq 0 $ set $$X_t = \int_0^t W_s^2 \mathrm{d} W_s,\qquad Y_t = W_t^7.$$ Find the covariance process ...
0
votes
2answers
37 views

Brownian and Brackets

A continuous martingale with deterministic bracket must be a Brownian motion. Is this statement ture or not, please? If true, how to show it? If not, what is a counter example?
4
votes
1answer
72 views

Why is the canonical filtration of a Brownian motion left-continuous?

Let $\{W_t, t\geq 0\}$ be a Brownian motion, and has a.s. continuous sample paths. Let $\{\mathcal{F}^W_t, t\geq 0\}$ be the canonical filtration, i.e. $\mathcal{F}^W_t=\sigma(W_s, 0\leq s\leq t)$. ...
1
vote
1answer
16 views

Remove drift from exponential Weiner process

I have the following problem: let $X_t$ solve $$ dX_t = b X_t \, dt + \sigma X_t \, dW_t$$ where $W_t$ is a Weiner process. Find $s(\cdot)$ such that $Y_t = s(X_t)$ is a martingale. We can see by ...
1
vote
0answers
42 views

Stochastic process using Markov chain (thief on the run!!)

I'm given an exercise where we are to simulate a thief escaping from an officer. The thief (let's call him/her T for simplicity) and an officer (O) have four cities to be in. Let's call the cities A, ...