# Tagged Questions

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### The Lévy-Khintchine formula and integrability conditions of a random measure

I am trying to see the connection between the Lévy-Khintchine and the integrability conditions of a Lévy measure. The literature seems to always connect both, but I cannot make sense of this relation ...
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### 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 ...
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### 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 $\omega$. $\psi$ ...
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### 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?
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### 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.
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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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### Write the Hamilton Jacobi Bellman equation

Consider the following stochastic optimal control problem. $$V(t,x) = \max_{u}\,\, \log \left(\mathbb{E}\left[\int_{0}^{T} u^{2}(t)dt\right]\right)$$ subject to the ...
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### 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?
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### 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 ...
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### How to write the Hamilton Jacobi Bellman equation

We consider the following optimal control problem $$V(t,x)=\max_{u}\mathbb{E} ( \log [\int_{0}^{T}u^{2}(t)dt + U(X(T))])$$ subject to the state process ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 $$\langle h(\vec{r})h(\vec{r}+\vec{r}') \rangle \sim \exp(-||\vec{r}-\vec{r}'||/a^2)$$ where ...
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### 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 ...
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Let $(W_t)t$ be a Wiener process. I want to find the limit for $\epsilon\to 0$ of $$\frac{W_t^2}{2\epsilon}\chi_{(-\epsilon,\epsilon)}(W_t)-\int_0^t ... 1answer 95 views ### Representing a stochastic integral as product of a unknown random variable and a standard normal random variable Consider a probability space (\Omega,\mathcal F, (\mathcal F_t)_{t\geq0},\mathbb P) where \mathbb F=(\mathcal F_t)_{t\geq0} is generated by B=(B_t)_ { t \geq 0} a standard brownian motion ... 0answers 51 views ### Stochastic control problem Suppose we have the following stochastic optimal control probelm V(t,x) = \sup_{u} \mathbb{E}[ g(X_{T}) +\int_{0}^{T}f(t,X_{t},u_{t})dt] + (\mathbb{E}[ ... 1answer 134 views ### Inequality for Euclidean norm Let:| | be Euclidean norm on \mathbb{R}^{n} and b : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n} and \sigma : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n\times m} two continuous functions. ... 2answers 54 views ### Expectation Geometric Brownian Motion Can someone help show me a simple way to show:$$\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: ... 1answer 153 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 ... 1answer 206 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 ... 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 ... 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 ... 2answers 134 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 ... 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: ... 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 ... 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.
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)} = ...