4
votes
1answer
140 views

Stochastic inequality, true?

Consider two stochastic processes $X$ and $Y$ satisfying the following SDEs (with the same drift!): $$X_t = x + \int_0^t b(X_s)ds + B_t$$ $$Y_t = y + \int_0^t b(Y_s)ds + B_t.$$ If $0<x<y$, is ...
1
vote
1answer
27 views

Continuity problem in derivation of general ito integral

This is part of the derivation of the Ito integral. In particular extending the definition to more general functions. I cannot understand why $g(.,\omega)$ is continuous for each $\omega$. $\psi$ ...
0
votes
1answer
16 views

Extensions of the Ito integral

This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate?
1
vote
1answer
20 views

Continuity theorem in Itô integral explanation

What is the continuity theorem used here in the explanation of the Itô integral? I cannot seem to find anything that would be exactly useful in my measure and integration text.
0
votes
1answer
16 views

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 ...
1
vote
0answers
70 views

Write the Hamilton Jacobi Bellman equation

Consider the following stochastic optimal control problem. \begin{equation} V(t,x) = \max_{u}\,\, \log \left(\mathbb{E}\left[\int_{0}^{T} u^{2}(t)dt\right]\right) \end{equation} subject to the ...
0
votes
1answer
32 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
0
votes
1answer
19 views

Is there an example that shows that the optional stopping theorem fails for finite (unbounded) stopping times?

Is there a martingale $M=(M_t)_{t\geq 0}$ and finite stopping times $S,T$ with $S \leq T$ a.s. such that $\mathrm{E}(|M_T|)<\infty$, but $M_S \neq \mathrm{E}(M_T|\mathcal{F}_S)$ a.s.? I found a ...
0
votes
0answers
45 views

How to write the Hamilton Jacobi Bellman equation

We consider the following optimal control problem \begin{equation} V(t,x)=\max_{u}\mathbb{E} ( \log [\int_{0}^{T}u^{2}(t)dt + U(X(T))]) \end{equation} subject to the state process \begin{equation} ...
0
votes
0answers
36 views

BMO martingale and exponential martingale

Consider the BSDE, $$ Y_{T}-Y_{t}=\sum_{i=1}^{n} \int_{t}^{T} Z_{s}^{i}dB_{s}^{i} - \frac{1}{2}\int_{t}^{T} \left| Z_{s}\right|^{2}ds $$ where $B$ is a standard Brownian motion on a complete ...
0
votes
0answers
29 views

Power spectral density of convolution of stochastic processes

I was wondering what it is the result of convolving two WSS processes in terms of power spectral densities. I know that, the output $Y(t)$ of a generic linear time invariant system with impulse ...
0
votes
0answers
36 views

SDE with no weak solution

I'm facing the followingd d-dimensional SDE: $$dY_t=\sigma(h_t)\,dB_t$$ In addition it holds, that: $h_t\in H$ and $H$ is compact (for example the simplex of $R^n$) the proces $h_t$ is progressivley ...
0
votes
1answer
34 views

Solutions of SDE do not explode when drift term is zero.

Suppose we have $dX_t = \sigma(X_t) dW_t$ where $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is Borel and $W_t$ is a standard one-dimensional Brownian motion. I am trying to show that $X_t$ cannot ...
1
vote
1answer
58 views

Brownian motion transition density question

Let $Y_t = M_t - W_t$ where $M_t$ is the running maximum of brownian motion and $W_t$ is brownian motion. I want to show that $P^0[Y_{t+s} \in dy| Y_t = x] = p(s,x,y)+p(s,x,-y)$ where $p$ is the ...
0
votes
1answer
70 views

Sample continuity of Brownian motion

I wanted to know if the Brownian motion and the fractional Brownian motion are almost surely sample continuous or not? Many thanks.
0
votes
1answer
32 views

Is this a Brownian motion

I am learning SDE, and here is some basic things I have trouble with, Let $B(t)$ be a Brownian motion, and $F \in \mathcal L^2$ is any stochastic process and I know $\int_0^tF(s)dB(t)$ is Ito process ...
3
votes
0answers
16 views

Continuity in $x$ of $E^x \int_0^{\tau} f(X_t)dt$

Suppose I have a stochastic diffusion $X$. I am studying an expression of the form $u(x):=E^x\int_0^\tau f(X_t)dt$ where $\tau$ is the exit time of $X$ from my bounded open domain $D$. I am also ...
1
vote
0answers
29 views

Distribution of Levy driven O-U process

Is there a way to find an analytical expression for $E\left[\exp\left(-\int_0^T \gamma_s ds\right)\right]$, where $d\gamma_t=k(\theta-\gamma_t)dt+\sigma dL_t$, and $L_t$ is a symmetric alpha ...
0
votes
0answers
14 views

Simple Stochastic Measurability Question

In the proof of a Stochastic representation theorem, the author writes: $Z_t = \frac{d}{dt}<M>_t$ is progressively measurable. Here $M_t$ is a continuous local martingale and we have the ...
1
vote
1answer
32 views

Elementary Malliavin Derivative question about definition.

I am reading a book that defines the Malliavin derivative $D_tF$ as follows: If $F = \sum_{n=0}^{\infty} I_n(f_n)$ is the Wiener Chaos expansion. $F$ is in the brownian filtration and $F \in ...
0
votes
1answer
37 views

Reflected process - Brownian motion

I am still new to stochastic processes and I tried to do this exercise. I don't know how to go on. Define the maximum process \begin{align*} M_t = \max_{0 \leqslant s \leqslant t} W_s, \end{align*} ...
0
votes
0answers
11 views

Computing the probability that a stock process is more valuable than the bond process

I am currently revising for my exam and I cannot really deal with the following problem (I am a beginner in terms of stochastic processes): $W_t$ is the standard Brownian motion. Consider a stock ...
1
vote
0answers
37 views

Construct an arbitrage opportunity in a multi-period model

I am currently revising for my exam in Financial Mathematics, and I could not solve this question: For $T > 1$, consider a $T$-period model with a single risky asset and a bank account which pays ...
0
votes
0answers
37 views

Simple Stochastic Control Problem

Consider $dX_t = \pi_t X_t dt + \pi_t X_t dW_t, X_0 = x$, where $W_t$ is a standard brownian motion, and $\pi$ is some real valued process. Let T>0. How can we calculate $P[X_T\geq 2x]$, where ...
0
votes
0answers
29 views

Stochastic PDE representation

I am trying to find a pde which $u$ satisfies when $u(x) = E^{x}[\cos(X_1)]$ where $dX_t = \sin(nX_t)\,dt + dW_t$ and $X_0 = x$. I have tried using Feynman-Kac but I can't seem to get it into the ...
0
votes
1answer
36 views

Application of Ito's Lemma to integral expression

I have a problem applying Ito's lemma. I know that if: $dX_t= \mu_t \, dt + \sigma_t \, dB_t$ then for $f(t,x)$: $df(t,X_t) =\left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial ...
0
votes
1answer
43 views

Representation of Markov process adapted to given Filtration

Let $X$ be continuous Markov process adapted to a filtration generated by Brownian motion $B$. Does there exist a function $f$ such that $X_t = f(t,B_t)$? My guess is that it should have such ...
2
votes
1answer
71 views

About the increasing process in the Doob-Meyer decomposition

As we know, a RCLL submartingale on [0,T], $Y$, in class D can be decomposed as: $$Y_t=Y_0+M_t+A_t,\ a.s.,$$ where $M$ is a martingale and $A$ is an increasing previsible process. In my question, I ...
2
votes
1answer
55 views

Skorohod convergence (space of right continuous functions with left limit)

If $f_n$ is a sequence of functions of the Skorohod Space $D([0,\infty),E)$, where $E$ is a separable Banach space, such that $f_n \to f$ in the Skorohod topology. Is it possible that there exists a ...
0
votes
1answer
40 views

Quadratic variation - Semimartingale

We know that any Semimartingale has Quadratic variation. I am interested to know if the converse is also true i.e. if a process has quadratic variation then it is semimartingale. Can some one ...
4
votes
1answer
64 views

Transforming semimartingale to local martingale by change of measure

Consider a continuous $\mathbb{P}$ - semimartingale X which can be decomposed as M+A (M is local martingale and A is bounded variation process). Is it possible to change measure to $\mathbb{Q}$ s.t. ...
0
votes
1answer
41 views

Is any FV-Process a special Semimartingale?

Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special ...
0
votes
1answer
22 views

Autocorrelation of Radial Stochastic Process with Planar Derivatives

I have a random field $h(\vec{r})$ that depends on $\vec{r}=(x,y)$, such that \begin{equation} \langle h(\vec{r})h(\vec{r}+\vec{r}') \rangle \sim \exp(-||\vec{r}-\vec{r}'||/a^2) \end{equation} where ...
0
votes
1answer
109 views

Poisson Process Change of Measure

I have seen the following result stated in the literature: Let $N(t)$ be a (finite time horizon) Poisson process defined on a probability space $(\Omega, \mathbb{P})$ with constant intensity ...
1
vote
1answer
93 views

Limit of stochastic integrals?

Let $(W_t)t$ be a Wiener process. I want to find the limit for $\epsilon\to 0$ of $$\frac{W_t^2}{2\epsilon}\chi_{(-\epsilon,\epsilon)}(W_t)-\int_0^t ...
1
vote
1answer
95 views

Representing a stochastic integral as product of a unknown random variable and a standard normal random variable

Consider a probability space $(\Omega,\mathcal F, (\mathcal F_t)_{t\geq0},\mathbb P)$ where $\mathbb F=(\mathcal F_t)_{t\geq0}$ is generated by $B=(B_t)_ { t \geq 0}$ a standard brownian motion ...
0
votes
0answers
49 views

Stochastic control problem

Suppose we have the following stochastic optimal control probelm \begin{equation} V(t,x) = \sup_{u} \mathbb{E}[ g(X_{T}) +\int_{0}^{T}f(t,X_{t},u_{t})dt] + (\mathbb{E}[ ...
1
vote
1answer
130 views

Inequality for Euclidean norm

Let:| | be Euclidean norm on $\mathbb{R}^{n}$ and $b : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n}$ and $\sigma : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n\times m}$ two continuous functions. ...
1
vote
2answers
53 views

Expectation Geometric Brownian Motion

Can someone help show me a simple way to show: $$\mathbb{E}(S_t)= S_0e^{\mu t}$$ for $$ S_t = S_0\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right) $$ from this page: ...
4
votes
1answer
150 views

“Continuity” of stochastic integral wrt Brownian motion

I'd like to prove a nice property of a stochastic integral with respect to Brownian motion. Let $(H_t)_{t\geq0}$ be a progressive and bounded process that is continuous at $0$ and $B$ a standard ...
5
votes
1answer
201 views

Holder continuity of Ito integral

Let $\sigma(t,\omega)$ be a progressively measurable function and $\mathbb{E}[\int_0^T \sigma_t^2\mathrm dt] < \infty$. Can we say that the Ito process $\int_0^t \sigma_s \mathrm dW_s$ is Hölder ...
0
votes
1answer
32 views

Probability Space and proof of existence for my specific problem involving stochastic differential equations

I have a question regarding the probability space for my problem. This deals with radiation therapy. If X(t) and Y(t) represent the number of two types of cancer cells. X(t) and Y(t) satisfy the ...
0
votes
0answers
14 views

Reference request: The use Lyapunov-type functions in the analysis of diffusion processes

I've been told that there exists a series of Lyapunov type results for diffusion processes that are used to establish things like the existence and uniqueness of solutions, existence of invariant ...
0
votes
2answers
129 views

A book/text in Stochastic Differential Equations

Somebody know a book/text about Stochastic Differential Equations? I'm in the last period of the undergraduate course and I have interest in this field, but my university don't have a specialist in ...
0
votes
1answer
47 views

Locally Nondeterministic Property of Brownian Bridge

Could anyone please give ideas or point me out references where I can find any result concerning the locally nondeterministic (LND) property (in the sense of Berman: ...
1
vote
0answers
104 views

Forming a local martingale with continuous increasing process

If $M_t$ is continuous martingale, we know that there exists quadratic variation process which is continuous and increasing. I am interested to know if the converse is also true. To make it precise ...
1
vote
1answer
69 views

limit of sup of a stochastic integral

Let $W$ be a standard, one-dimensional Brownian motion and $0 < T < \infty$. Show that $$\lim_{\beta \to \infty} \sup_{0\leq t \leq T} |e^{-\beta t }\int_0^t e^{\beta s } dW_s| = 0$$ a.s.
0
votes
1answer
92 views

2 dimensional Brownian motion but not 3 dimensional Brownian motion

Let $W_t = (W_t^{(1)},W_t^{(2)},W_t^{(3)})$ be 3 dimensional Brownian motion. Let $X=sgn(W_1^{(1)})sgn(W_1^{(2)})sgn(W_1^{(3)})$. Define a 3 dimensional process $M_t$ as follows : $M_t^{(1)} = ...
1
vote
1answer
132 views

Finding the joint distribution of a random process with memory

I'm modeling a digital system as a random process and attempting to solve for the autocorrelation in order to arrive at the power spectral density of the process. The system is as follows: At any ...
0
votes
1answer
78 views

exit time and indicator function

let $D$ open set of $\mathbb{R}^{n}$ and $T_{D}=\inf\{t\geq 0 : X_{t}\notin D\} $ be the first exit time from the $D$ and $1_{A}$ is Indicator function of $A \subseteq \partial D$ $$ ...