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1answer
348 views

Haar basis on $L^2(0,1)$ - proof?

I have the following problem. We defined $\mathbb{H}=\{f_0,\quad f_{j,n} \quad j=1,...,2^{n-1} \quad n=1,2,...\}$ where for all $t\in[0,1]$ we put $f_0(t)=1$ and setting $K=2j-1$, $$f_{j,n}(t)=\left\{ ...
1
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1answer
38 views

Generating function of a random variable

I've got the following problem: Give the generating function of the random variable $X$ whose mass function is defined by: $$f(m) = P(X=m) = (m+1) p^2 (1-p)^m,$$ where $m$ is a positive integer ...
1
vote
1answer
107 views

Why is the following function not càdlàg?

I have constructed the following function but I can't see why it is not càdlàg on $[0,1]$: $$f(x)=\begin{cases} 1, & ...
1
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0answers
65 views

Stochastic Exponential: $dZ=-\lambda Z dM + dL$ to $dZ=-\lambda Z dM + Zd\tilde{L}$ while $\tilde{L}$ is still orthogonal to $M$

I have a question concerning the paper http://www.researchgate.net/publication/228648002_No_arbitrage_and_the_growth_optimal_portfolio, Lemma 6.3, which is based on ...
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1answer
45 views

Stochastic Processes Question

Give an example of a stochastic process $X_{n}$ that is not a Markov chain, such that $P_{y}(N(y)=\infty)=0$ but $E_{y}N(y)=\infty$
2
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2answers
1k views

Ito Isometry and quadratic variation

Here is a confusion regarding stochastic integrals. Let $Y_t=\int_0^tW_sds$ where $W_t$ is a Brownian Motion. Now $dY_t=W_tdt$. So from this expression one can conclude that $dY_t \cdot ...
-1
votes
1answer
239 views

How can I prove it is a martingale when there is a jump process

Let $N_t$ be a Possion process, $M_t=N_t - \lambda t$ we can easily show that $M_t$ is a martingale. Now $\int_0^t\Phi_udM_u=.....=\sum_{i=1}^{N(t)}\Phi_{\tau_i}-\lambda\int_0^t \Phi(u)du $ $\tau_i$ ...
0
votes
2answers
98 views

One stochastic integrability problem

On a lecture notes, there is a following arguement: To make $\int_0^T \pi_t dW_t$ well-defined, (maybe it means to make $\int_0^T \pi_t dW_t<\infty \ \ a.s.$) we only need $\int_0^T \pi_t^2 ...
3
votes
1answer
77 views

Given the SDE: $dX_t=dB_t+b(X_t) dt$ with $(x,b(x)) \leq 0, \forall x \in \mathbb{R}^n$, prove that $E[|X_t|^2] \leq nt+E[|X_0|^2]$

I'm working on this problem: Given a solution $X_t$ to the SDE $$dX_t=dB_t+b(X_t) dt$$ where $B_t$ is an $n$-dimensional Brownian motion, and $b:\mathbb{R}^n \to \mathbb{R}^n$ a Lipschitz ...
0
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1answer
54 views

Defining an equivalent measure starting from a continuous local Martingale

Suppose we have continuous local martingal $L$ given. We define $Z=\mathcal{E}(L)$, the stochastic exponential of $L$. I am interested in finding some condition such that $Z$ defines a density, i.e. I ...
2
votes
2answers
185 views

How do I derive the Gaussian Mixture distribution of an Ito Integral?

I have a question about the distribution of an Ito Integral. Consider the integral $$ \int_0^1 B_1(r) \mathrm{d}B_2(r), $$ where $B_1$ and $B_2$ are two independent standard Brownian motions. I am ...
1
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1answer
110 views

Joint Convergence and Donsker's Theorem

I have a question about joint convergence results derived from an FCLT (i.e., a Functional Central Limit Theorem). To motivate my question, consider the following setup: Let $y_t$ be a random walk ...
2
votes
1answer
752 views

Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
1
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0answers
42 views

How do you convert an infintesimal generator of a Markov process to a transition function?

Suppose a continuous-time continuous-step Markov stochastic process $X_t$ has infinitesimal generator $\mu(x, t)$, $\sigma(x, t)$ ($\mu$, $\sigma$, and $X_0$ are known). How can we use this ...
2
votes
2answers
64 views

Prove this equation

I'm taking a course on stochastic analysis. I'm stuck on the very first problem of the lecture notes: $\lim_{n \to \infty} \left(1+\frac{\lambda}{n} + o(n^{-1})\right)^n = \exp(\lambda)$ Prior to ...
2
votes
1answer
79 views

Clarke Ocone representation formula

Let $(B_t)_{t}$ a Brownian motion and $F \in L^2(\Omega,\mathcal{F}_T,\mathbb{P})$. Then we know by Itô's representation theorem that there exist a process $X$ such that $$F=\mathbb{E}F+\int_0^T X_s ...
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2answers
320 views

Area enclosed by 2-dimensional random curve

Consider a 2-dimensional Wiener process $(W_t)_{t \in [0,1]}$. Color every area which is enclosed by the line parametrised by $W_t$ (this means that, when the Wiener process makes a loop and ...
2
votes
2answers
86 views

Question Regarding Poisson and probability.

i found this interesting question on the web but i am not quite sure if my solution is accurate. Honestly i would appreciate few opinions. Given Question: At a subway station, eastbound trains ...
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0answers
100 views

Representation theorem for continuous process of finite variation

There is a martingale representation theorem If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that $$ M_t = M_0 + \int_0^t ...
2
votes
0answers
65 views

Initial Conditions for Finite Difference of PDE

I am having trouble with figuring out what my initial conditions should be for a simple finite difference algorithm I wrote in Matlab. Specifically, I'm trying to show that the regular 1-Dimensional ...
1
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1answer
71 views

Covariation of Wiener processes, $\langle W_1,W_2\rangle_t = \rho t$.

I'm wondering why this is true: $\langle W_1,W_2\rangle_t = \rho t$. Where $W_1$ and $W_2$ are standard Brownian Motion. I know that $\langle W_1,W_2\rangle_t = 0.5\big[ \langle W_1 + W_2 \rangle - ...
2
votes
1answer
416 views

Expectation of exponential martingale and indicator function.

Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$. I want to evaluate $$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...
2
votes
1answer
82 views

Readings necessary to understand Ito Integrals?

I searched for this question but couldn't find a direct answer. Basically I want to understand (and possibly compute some simple instances of) the Ito integral. I am coming from a physics background ...
2
votes
0answers
204 views

Explaining Ito formula to an analyst

From the point of view of analysis, what is Ito formula? Is it an integral by substitution, or, a radon-nikodym derivative? Define the probability space $$ \left(C\left(\Bbb ...
0
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1answer
36 views

Is it true that $X(t)^a > K \iff X(t) > K^\frac1a$

Let $a \in \mathbb{N}$, $K \in \mathbb{R^+}$ and $X(t)$ be a geometric Brownian Motion. Is the following true? $$X(t)^a > K \iff X(t) > K^\frac1a$$ The context of the above is that I want to ...
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votes
0answers
60 views

Condition on $f$ for $e^{\int_0^t f(B_s)ds}$ to be of finite variation

Let $B$ be a standard Brownian motion, and, $$ X_t=e^{\int_0^t f(B_s)ds}, $$ for some function $f$. What are the condition on $f$ for $X_t$ to be of finite variation? Let $Y_t=\int_0^t f(B_s)ds$, if ...
1
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0answers
836 views

Stochastic integral: Interchanging the order of expectation and integration

Let $B$ be a standard Brownian motion and $$ X_t=\int_0^t f_s ds+\int_0^t g_s dB_s, $$ where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$. Is it true that $$ ...
0
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0answers
78 views

Local martingale iff each component is a local martingale?

This is probably an easy question: A local martingale is an adapted, cadlag process for which there is an increasing sequence of stopping times (going to $\infty$) such that the stopped process is a ...
0
votes
1answer
72 views

Condition for existence of a stochastic differential equation

With $B$ a standard Brownian motion, write $$ dX_t=f_tdt+g_tdB_t. $$ What are the conditions on $\left(f\right)_{t\ge 0}$ and $\left(g\right)_{t\ge 0}$ for $X_t$ to exists? I think ...
1
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1answer
342 views

Integration of Wiener process: $\int_{t_1}^{t_2} dB(s)$

We all know that $\int_0^t dB(s) = B(t)$, where $B(t)$ is a standard Brownian Motion. However, is the following identity true? Also, why or why not? $\boxed{ \displaystyle \ \ \int_{t_1}^{t_2} ...
2
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0answers
44 views

Finite p-th variation implies zero-valued q-th variation.

The Question: Let $X$ be a continuous process, and suppose $0 < p < q$. Prove the case $V_t^p(X) < \infty \implies V_t^q(X) = 0$. Definitions: The standard setup. $\Pi := ...
1
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1answer
173 views

Demonstrate that every martingale is a local martingale.

The Original Question: Demonstrate that every martingale is a local martingale. Attempt at a Solution: Consider the standard setup of this problem: $\mathscr{F}_t$ is the filtration that satisfies ...
1
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0answers
247 views

What kind of process is a locally bounded process?

Definition of locally bounded process is on http://planetmath.org/encyclopedia/LocallyBoundedProcess.html On that website, it says any discrete-time predictable process is locally bounded. How can I ...
2
votes
2answers
100 views

Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$

Is there a way to solve $$ \partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0? $$ This appeared as a condition for $$ X_t=u(t,B_t)e^{\int_0^tv(B_s)ds} $$ to be a martingale. With $B$ a ...
0
votes
1answer
1k views

Applying Ito formula to the Brownian bridge

Let $B$ be a standard Brownian motion and $$ W_t=(1-t)\int_0^t \frac{1}{1-s}dB_s $$ be a Brownian bridge. Calculate $dW_t$. To apply Ito formula define $$ f(t,B_t)=(1-t) \int_0^t\frac{1}{1-s}dB_s $$ ...
0
votes
1answer
88 views

another version of doob inequality

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 have $$\lambda ...
0
votes
1answer
103 views

How to show Wiener measure induces basic properties of Brownian motion?

page 19 of http://www.math.tifr.res.in/~publ/ln/tifr64.pdf gives a defintion of Wiener measure Ft1,t2,..,tk. But how can we show it is a probability measure and it satisfies the consistency condition ...
0
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1answer
50 views

Rewriting SDEs - “Multiplication on both sides”

I have a question concerning a calculus "trick" sometimes used in stochastic calculus (e.g. in the Book on Arbitrage Theory in Cont. Time of Bjoerk). There they do the following in the proof of Prop. ...
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0answers
104 views

Martingale Decomposition

Is it possible to decompose a discrete-time martingale $(M_n)$ uniquely into two processes $$M_n=M_n^I+A_n$$ where $(M^I_n)$ is a martingale with independent increments and $(A_n)$ is a martingale? If ...
0
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1answer
111 views

invertible, measurable and measure preserving

$T: [0,1)^{2}\rightarrow[0,1)^{2}$ by $T(x,y) = (2x,\frac{y}{2})$, with $0 \leq x < \frac{1}{2}$ and $T(x,y) = (2x-1, \frac{y+1}{2})$, with $\frac{1}{2} \leq x < 1$ In class we said this $T$ ...
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0answers
266 views

Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?

The following is the definition of infinitesimal generator from Oksendal. Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
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0answers
94 views

How Wiener Measure on $F(C([0,T]))$ is a Gaussian Measure

I'm looking for some simple proofs for the fact that on $(C[0,T],F(C([0,T])),P_{*})$ where $F$ represents Borel Sigma algebra , $P_{*}$ the Wiener Measure , then how to proove that $P_{*}$ measure is ...
0
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1answer
133 views

Conditional expectation of a finite variation process

A simple question: Let $H$ be a cadlag, adapted process and $A$ a process of finite variation. Then also $\int_t^T HdA_t$ is a finite variation process (see "Limit Theorems... "Jacod&Shiryaev ...
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1answer
466 views

Covariance of Gaussian stochastic process

Could someone help me to figure out solutions of following problems?: Let $X = (X_t)_{t \geq 0}$ be a Gaussian, zero-mean stochastic process starting from $0$, i.e. $X_0 = 0$. Moreover, assume that ...
0
votes
1answer
147 views

Confusion regarding Stochastic integral

I've a stupid doubt in the construction of stochastic integral of real scalar valued maps. Many times I've seen in books after the stochastic integral is defined in [$0,T$] for the integrand in $L^2$ ...
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0answers
195 views

Ito's formula for irregular functions

Let's say we have \begin{align} Y_t=h(t,X_t) \end{align} and for simplicity \begin{align} dX_t=e\,dt+f\,dW_t \end{align} then by Ito's formula we have \begin{align} dY_t=\left(\frac{\partial ...
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0answers
117 views

How to check if a process is a semimartingale?

Consider the process $X_t = \sum_{i=1}^{N_t} Y_i$. This is a Lévy process, hence Markov and so on ($N_t$ is a Poisson counting process). Now add some diffusion $D$ for each jump $Y_i$ that starts at ...
1
vote
0answers
73 views

Applicability of Itô's Lemma for $g\in \mathcal{C}^2((0,1)^2)\cap \mathcal{C}_0([0,1]^2)$

Let the domain be $[0,1]^2$. And let $W^x_t$ be the standard Brownian Motion started in $x\in [0,1]^2$ with absorbption on $\partial [0,1]^2$ and choose some $g\in \mathcal{C}^2((0,1)^2)\cap ...
1
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0answers
220 views

existence of a strong solution for a sde

Suppose we want to study a SDE of the form $$ dX_t = a(t,X_t)dt + b(t,X_t)dW_t$$ and $X_0=Y$, on a filtered probability space $(\Omega,\mathcal{F}, \mathbb{F},P)$ and where $W$ is a $(P,\mathbb{F})$ ...
1
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0answers
198 views

stochastic exponential

I was able to show the following: $X$ a semimartingale, $X_0=0$ then the SDE $$ dZ_t=Z_tdX_t$$ with $ Z_0=1$ has the unique solution $Z_t:=\exp{(X_t-\frac{1}{2}\langle X\rangle_t)}$. I was able ...