Questions on the calculus of stochastic processes, or processes that have a random component.

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Expected value of stochastic process

How can i calculate the expected value of $$S_t= S_0 \exp\left( mt-\frac12\int_{0}^{t}e^{2Y_s}ds+\int_{0}^{t}e^{Y_s}dB_s\right)\quad $$ where $${Y_t}$$ is the solution of a sde and follows tha normal ...
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23 views

Brownian motion hitting probability and Martin capacity

Consider a Brownian motion $B_t$ in $\mathbb{R}^n, n\geq 3$ and the ball $B(0, r)$ of radius $r$ around the origin. Let $\overline{C}$ be a compact set inside $B(0, r)$ such that $C$ is open in $B(0, ...
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12 views

Transient Brownian motion and stopping time

Let $B(t)$ be a Brownian motion in $\mathbb{R}^n$, or on a compact Riemannian manifold $M$ of dimension $n$, $n \geq 2$. Let us consider the stopped Brownian motion at a deterministic time $T$ (in ...
3
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1answer
34 views

Minimal value of probability according to the difference of a Levy-process

Can we conclude for a Levy-Process, that for all $\epsilon>0$ it holds that $\min_{s\in [0,t]} \mathbb P\left(\left|X_t-X_s\right|\leq \epsilon\right)>0$? Stochastic continuity doesn't seem to ...
2
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1answer
25 views

Showing a “signed Markov transition density” will lead to a trivial measure on path space.

Let for all $t>0$, $x\mapsto p(t,x)$ be a Schwartz function on $\mathbb R$, satisfying $\int_{\mathbb R}p(t,x)\mathrm dx=1$ and $\int_\mathbb{R}|p(t,x)|\mathrm dx\equiv C>1$ for all $t>0$ (so ...
2
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0answers
13 views

Exponential Gaussian volterra process. Close form conditional expectation?

Asssuming a probability space $(\Omega,(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ such that $(\mathcal{F}_t)_{t\geq 0}$ is generated by a Brownian motion $W_t$. We assume that $s>0$ is fixed and $t\in[...
3
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0answers
21 views

Doob Meyer decomposition in an exercize

I have to find the Doob Meyer decomposition for the following process: $Y_t=e^{(1+B_t^2)}$ I think that the method is to derive with the Ito's formula the process and I've obtained: $dY_t=2B_te^{(...
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1answer
36 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
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0answers
68 views

Question on averages of Ito Integral: $E(\int_0^t X_sdB_s \int_0^t X_sds)=?$

Given some probability space, assume $X_t$ is a square integrable continuous process adapted to the filtration $\mathcal{F}_{t}$ generated by the standard Brownian process $B_t$. I denote by $(X.B)_t=\...
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1answer
73 views

weak L1 convergence

Given a sequence $Y_{un}$, where $Y_{1n},Y_{2n},\ldots$ have the same domain. Assume for every $u\in \mathbb{N}$ we have $e^{itY_{un}}\rightarrow \mathbb{E}[e^{it M}]$ weakly in $L_1$ as $n\rightarrow ...
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25 views

How to solve this SDE

I have been learning basic stochastic analysis, and we have only been taught about Ito formula. The professor told us how can we solve this question below using it, but I miss it. Can anyone help me? ...
1
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1answer
23 views

For a one-dimensional Brownian motion $B_t$ $Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}.$

A one-dimensional Brownian motion $B_t$ has exponential moments of all orders, i.e. $$Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}. (2.6)$$ This is given as a corollary to ...
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0answers
26 views

$M_t=N_t - \lambda t$. Find $Var[\int_{a}^{b} M_t dM_t]$.

$M_t$ is the compensated poisson process. $N_t$ is a poisson process. $M_t=N_t - \lambda t$. Find $Var[\int_{a}^{b} M_t dM_t]$. I have a doubt. I read the book and it is dealing with the left ...
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2answers
59 views

Definition of self-financing strategy

Consider a portfolio of two assets with prices $S_t$, $B_t$ and holdings $\Delta_t$ and $E_t$ respectively. So the portfolio value is $$ \Pi_t = \Delta_t S_t + E_t B_t$$ The portfolio is defined to ...
2
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0answers
73 views

Radon-Nikodym with respect to Stochastic Measure?

Question This question is now concerning stochastic processes. Let $(X_t)_{t\geq0}$ be defined on the probability space $(\Omega,\mathcal{F},P)$ with $\mathcal{F}_t=\sigma(X_s:s\leq t)$. Assume that ...
2
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0answers
83 views

Can Local Martingales be characterized only using their FV process and BM?

Prove or Disprove: A process $(X_t)_{t \ge 0}$ is a (continuous) local martingale if and only if it can be represented in the form: $$\int_0^t \xi dB = \large B_{\int_0^t \xi_s^2 ds} $$ where the ...
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2answers
22 views

Question about Langevin equation

The Langevin equation is given by: $dq=pdt,\ dp=-\nabla V(q) dt-pdt+\sqrt{2}dW$ I want to know what does the variables $p,\ q,\ t,\ V,\ W$ represent . Can someone help me ? Thanks.
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1answer
39 views

Distribution of a exponetial Random Variable

i have a stopping time $T$ of an Poisson Process $N$ with rate $\lambda$. Somehow this stopping time is exponential distributed. So we have $ T \sim exp(\lambda)$. I want to know the distribution of ...
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34 views

If a process is previsible, is the stopped process previsible? [closed]

Assume we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ is an $\{\mathscr F_n\}_{n \in \mathbb N}$-...
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0answers
31 views

Calculation probability of dynamic process model of capacity

I found this place really helpfull and now I got my first own question I cant solve. I want to unterstand the calculation of an Article im reading. Therefore we define a capacity process $C$ in a ...
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1answer
50 views

Correlation between stochastic processes

Question: if $W(t)$ is a standard Brownian motion with $W(0)=0$, what is the linear coefficient between the stochastic processes $W(t)$ and $I(t)=\int_0^t W(s)ds$? I argued as follows: what we want ...
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1answer
30 views

Thinning a Renewal Process - Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a ...
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1answer
48 views

Distribution Convergence of an Random Variable

I need to show that $$\frac{\sqrt{2n}}{\theta +1}\left(\frac{1}{\bar{X}_n}-1-\theta \right) \to^{d} N(0,1)$$ where $\bar{X}_n = \frac{1}{n} \sum_{i=1}^nX_i$ and iid random variables $X_i$, $X_1 \...
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0answers
12 views

Maximum property, Resolvent, Markov process

I have a question about Markov processes and related topics. Let $E$ be a locally compact separable metric space and $(X_{t},P_{x})$ a Markov process on $E$. For a bounded measurable function $f : E \...
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0answers
18 views

System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
0
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1answer
27 views

Let $Z_t=\int{W_s }ds $. Show that $Z_t=\int (t-s) dW_s$

Let $Z_t=\int_{0}^{t} W_s ds$. Use integration by parts to show that $Z_t=\int_{0}^{t} (t-s) dW_s$. I have tried and i can't get the answer.
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41 views

Help with change of measure and martingales

Consider two three stochastic processes $X$, $Y$ and $Z$ in probability space $(\Omega, (\mathcal F_t)_{t \geq0},\mathbb P)$ such that $$ X_t = \exp\left(\int_0^t f_s ds\right), $$ $$ Y_t = \exp\...
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2answers
54 views

brownian noise and stochastic differential equations

Consider the SDE $$dx=3x(t)dt+dW(t)$$ Where we're dealing with Brownian noise. Now, dW comes from $$dW(t)=\int_0^{dt}ds\ \eta (s)$$ As far as I understood, $\eta$ is the noise distribution (...
2
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0answers
28 views

PDE with Stochastic Coefficients

Does anyone have reference suggestions for pde's with stochastic coefficients? I've seen many papers on more advanced problems, but it would be great to have a reference discussing the basic theory ...
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32 views

System of Stochastic Differential Equations (SDEs) from Diffusion on Manifold

I am looking at a system of SDEs due to Brownian motion on a 3d Riemannian manifold (see e.g. Ito, 1962, The Brownian Motion and Tensor Fields on Riemannian manifolds). I have reduced the associated ...
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124 views

Inequality for Lévy SDE

Let $X_{s}^{t,x}$ denote the solution at time $s$ of an Ito SDE whose coefficients are Lipschitz continuous with initial condition $X_t=x$. Let $t\leq s\leq T<\infty$. The inequality $$ \mathbb{E}\...
3
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1answer
94 views

Itô-isometry in the extended case?

It is shown when constructing the Itô-integral that if: $E[\int_0^T X_t^2dt]< \infty$. Then we have that Itô-isomtry: $E[\int_0^T X_t^2dt]=E[(\int_o^TX_tdB_t)^2]$. In the extended Itô integral, ...
3
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0answers
37 views

unbounded variation of $\sin(x)/x$

How can I show that the variation of $sin(x)/x$ is unbounded? Could you please help me. I know that I have to use but how can I rough estimate that this is bigger than infinity?
3
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2answers
63 views

What's the variance of the following stochastic integral?

The stochastic integral is defined as $$u_t = \int_{t-1}^t e^{-\kappa(t-s)}\int_0^s e^{-c(s-r)} \, dW(r) \, ds.$$ where $W(t)$ is a standard Brownian motion, $\kappa$ and $c$ are both positive. I ...
2
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0answers
18 views

Preservation of ui condition [closed]

I have a stopping time $\tau_n$ with $\mathbb{P}(\tau_n=\infty)\rightarrow 1$ for $n \to \infty $. With this stoppingtime $M^{\tau_n}$ is a uniformly integrable martingale. I deduced that $M$ is a ...
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0answers
23 views

How is the martingale property of Brownian motion used in the construction of the Ito integral?

I am trying to learn about Fractional Brownian Motion (and eventually integration with respect to FBM) and keep running into references to the fact that the construction of the Ito integral with ...
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1answer
18 views

Conditions for limiting distribution to equal stationary distribution of SDE

I have SDE of the form $$dX_t=a\mathopen{}\left(X_t\right)dt+b\mathopen{}\left(X_t\right)dW_t,$$ where $W$ is Brownian motion. If the stationary distribution of $X$ exist is it equal to the limiting ...
3
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0answers
64 views

Show that $\hat{Y}$ is an optimal linear estimator of Y

Relevant Information. Let $X(t)$, $t \in T$ be a second order process. Let $M_0$ be the set of random variables of the form $a + b_1X(s_1)+ \cdots + b_nX(s_n)$ for a positive integer $n$ and constants ...
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1answer
41 views

Girsanov theorem calculations help

I need help to understand a couple of calculations in this Girsanov theorem related SDE problem. I have five questions as stated below. Let $X_t$ solve the Ornstein-Uhlenbeck equation $$dX_t = X_t\, ...
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31 views

Changing the order of integration for Brownian motion (outer integration over the range of inner integration)

$X_t$ is bounded Brownian motion and it can be even standard Brownian motion if you wish. I want to express $E[\int_{0}^{T}\int_{0}^{t}X^{n}dsdt]$ as a function of $E[\int_{0}^{T}X^{n}dt]$ For ...
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2answers
53 views

Quadratic covariation of two Itô processes

If $dX(t)=\Delta_x (t) + \ominus_x(t) dt$ and $dY(t)=\Delta_Y(t) dW(t)+ \ominus_Y(t) dt$, where $X(t), Y(t)$ are two Ito processes. I need show that $d[X,Y](t)=\Delta_x(t)\Delta_Y(t)dt$, where $\...
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1answer
36 views

Decision theory references for advanced undergrad/early grad students?

I'm studying measure theoretic stochastic calculus, and I was hoping to pick up some knowledge of decision theory along the way. I'm very happy with Rudin or Karatzas in level of rigor, and I was ...
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0answers
39 views

Strong existence of solutions to SDE and continuity in the time variable

I recently come across some literature in stochastic analysis that uses the following result: Consider the one-dimensional SDE $$dX_t= a(t, X_t) \, dt + b(t, X_t) \, dW_t, $$ where $a, ...
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1answer
94 views

When is the compensated Poisson random measure a martingale ? (extensions to sets not bounded from 0)

Assume you have a Lévy process X. Let $N(t,A)$ be defined as the number of jumps in the interval $(0,t]$, such that the jumps size $\Delta X_s \in A$. It can be shown that if $0 \ne \bar{A}$, then $...
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55 views

Convergence of a process

this may be viewed as a duplicate of this post. However i have put in much effort in the shared link and donated it with reputation, to check the proof considered there. Here however i want to argue ...
3
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1answer
43 views

Stochastic differential equation substitution reasoning?

I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble ...
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0answers
45 views

Time scaling birth process in Poisson process

Given a birth process $\{B_t:t\geqslant0\}$ with $\lambda >0$, define $$K_t=\int_{0}^{t}B_s ds=\sum_{i=1}^{n}B_{t_{i}}(t_{i+1}-t_i)$$ if there were $n$ births in $[0,t]$ and let $t_{i}$ be the ...
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1answer
25 views

Question about Ito integration in SDE in Stochastic optimal control

Here is my question statement. I cannot understand the last equality. Let $U=[-1,1]$. \begin{equation} \mathcal{U}[0, T] = \left\{ u:[0,T] \rightarrow U \mid u \text{ is } \{\mathcal{F}_t\}_{t\geq0}\...