3
votes
1answer
81 views

Expectation of Square of Stopping Time

Let $B_t$ be standard Brownian motion and $a < 0 < b$. Define stopping time $T$ as follows. $$T = \min \{t \geq 0: B_t \in \{a, b\} \}.$$ The expectation of $T$ is $\mathbb ET = |a|b$ and can be ...
1
vote
0answers
26 views

Applying Ito lemma to multi dimensional semimartingales

Let $X_t=(X^1,\dots,X^d$) be a d-dimensional semimartingale. Then the formula for Ito's lemma I have found in several places, including wikipedia, is: $f(X_T)-f(X_0)=\sum_{i=1}^d\int_0^T ...
2
votes
1answer
51 views

A question about Malliavin calculus

An application of Malliavin calculus is to calculate the sensitivity of financial Greeks. However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to ...
3
votes
1answer
147 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 ...
0
votes
1answer
17 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 ...
2
votes
1answer
110 views

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 ...
1
vote
0answers
87 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
0answers
24 views

Identically distributed and same characteristic function

If $X,Y$ are identically distributed random variables, then I know that their characteristic functions $\phi_X$ and $\phi_Y$ are the same. Does the converse also hold?
1
vote
1answer
43 views

Random variables independent

We said that two random variables $X,Y$ are independent iff we have that for $Z = X+Y$: $$P_Z(B)=\int_{\mathbb{R}}P_X(B-s)dP_Y(s) = \int_{\mathbb{R}}P_Y(B-s)dP_X(s).$$ But I still don't get this ...
1
vote
1answer
24 views

Integrability and exponential integrability

I'm working on a paper, and I don't know if there is some kind of typo or if I just don't get what seems obvious to the author. Note : I'll be working with probabilities, but I guess this would be ...
1
vote
1answer
47 views

Integrate over different measures

In Probability theory the expected value of a random variables $X : \Omega \rightarrow \mathbb{R}$ is defined as $E(X) = \int_\Omega X dP$ Now, if $\Omega \subset \mathbb{R}$ and has a density ...
1
vote
0answers
31 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
1answer
41 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*} ...
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
1answer
40 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
0answers
39 views

Integrating a function of a random variable; $\int g(X) dP$

Assume a random variable $X$ on probability space $\Omega$, taking values in $\mathbb{R}$ with some known distribution $F(dX)$. Assume also a function of the random variable, $g(X)$. Does then the ...
0
votes
1answer
49 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
64 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 ...
1
vote
0answers
89 views

Tail of hitting times for Brownian motion on the circle

For $y\in \mathbb R/\mathbb Z$ and $\varphi\in C([0,\infty);\mathbb R/\mathbb Z)$ let $T_{y}(\varphi) \ := \ \inf\{t>0: \varphi_t = y \} \ \ \ $ (first time the path $\varphi$ hits $y$) ...
0
votes
1answer
43 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
68 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
0answers
30 views

Are there “necessary” conditions for a solution to the multivariate, truncated Hausdorff moment problem?

I am looking for NECESSARY conditions for a solution to the multivariate, truncated Hausdorff moment problem (i.e., conditions under which a given finite sequence of numbers is the sequence of first ...
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 ...
5
votes
1answer
230 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
1answer
48 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
105 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
70 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
80 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 $A \subseteq \partial D$ $$ ...
4
votes
1answer
96 views

What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
5
votes
1answer
211 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
1
vote
0answers
62 views

question about the sequential continuity of the set of probability measures

I have a question about the sequential continuity of the set of probability measures. Let $\Omega$ be the space of continuous functions defined in $[0,1]$ taking values in $\mathbb{R}$. Let ...
3
votes
0answers
103 views

Example of a regular strong solution of an SDE, which doesn't satisfy a Lyapunov condition?

Let $$dX_t = a(t,X_t) \, dt + b(t, X_t) \, dW_t, \quad t \in [0,T]$$ be a stochastic differential equation, where $W$ is an $m$-dimensional Brownian motion, $X_0 = x \in \mathbb{R}^d$, and the ...
1
vote
0answers
51 views

Banach space :space of all adapted processes continuous equipped wih specific norm is complete

Let $\mathbb{B}$ be space of all adapted processes continuous equipped with the norm $\lVert Y\rVert_{\mathbb{B}}^2=E\left[\sup_{t\in [0,T]} |Y_{t}|^{2}\right] < \infty $, ...
1
vote
1answer
141 views

question about conditional probability and $\sigma$-algebra

I have a question: Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and two random variables $X$ and $Y$. For a Borel-measurable set $\Gamma$, if there exists a measurable function $g$ ...
4
votes
1answer
259 views

Continuous Square integrable martingale Quadratic Variation

We know that given a continuous square integrable martingale there exists unique (up to indistinguishability) continuous, natural and increasing process which is quadratic variation process of the ...
2
votes
0answers
166 views

Good books on “advanced” stochastic analysis

Any good books suggestion for studding advanced features of stochastic analysis ? Thank's in advance
1
vote
2answers
108 views

Generalization of Doob Dynkin for Stochastic processes

Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
2
votes
2answers
920 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
vote
1answer
106 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
680 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 ...
2
votes
1answer
76 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 ...
0
votes
1answer
100 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 ...
1
vote
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 ...
6
votes
1answer
436 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
143 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 $L^2$ ...
0
votes
0answers
175 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 ...
2
votes
1answer
156 views

Is the expectation $E[\xi U'(\xi)]$ finite?

I encounter the following problem today. It seems a simple question. Let $U$ be a real function from $R^+\rightarrow \bar{R}$ satisfying the following conditions: (1) $U$ is concave, continuous, ...
3
votes
1answer
140 views

Solving SDE's on subsets of $R^n$.

It is well-known (see for instance Oskendal's text) that if $T>0$ and $$b(\cdot,\cdot): [0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^n~~~~~~\sigma(\cdot,\cdot):[0,T] \times \mathbb{R}^n ...
1
vote
1answer
194 views

basic application of Strong Law of Large Numbers

In $$ \sum_{j=0}^q {q\choose j}{1\over n}\sum_{i=1}^n X_i^j(-\bar X)^{q-j} \quad \overrightarrow{a.s.} \quad \sum_{j=0}^q {q\choose j} \mathbb{E}(X^j) (-\mathbb{E}(X))^{q-j} $$ using the Strong ...