Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

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Finding a unique strong solution

I am brushing up on my stochastic approximation. I am having a hard time with the following problem. I have the equation dX$_t$ = ln(1+ X$_t^2$)dt + X$_t$dB$_t$ X$_0$ = x, with x ∈ ℝ I know that ...
3
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1answer
31 views

Finding Stochastic processes

I have the following differential equation dX$_t$ = (r$\mu$X$_t$ + $\frac{r(r-1)}{2}σ^2X_t$)dt + rσX$_t$dB$_t$, X$_0$ = x, with x > 0. Here, r>0. I am having trouble figuring out how to find the ...
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45 views

Why are we allowed to multiply the differential form of an SDE by a function?

Suppose for example that we have the following SDE: $$dX_{t} = a(X_{t})\,dt + b(X_{t}) \,dB_{t}. $$ What rigorous justification is there for then saying, for example, that we can multiply both sides ...
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1answer
24 views

Notation in the stochastic derivatives in the mean square sense

The stochastic limit $X$ in the mean square sense is given the definition: For a row (sequence?) of stochastic variables $X_n$ if $\displaystyle\lim_{n\to\infty}E\{(X_n-X)^2\}$ = 0 and we write $\...
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1answer
54 views

stochastic differential equation solution

I find it difficult to solve this differential equation: $dX(t)=[aX(t)+b]dt+σX(t)dW(t)$ $X(0)=x$ where $W(t)$ is a Brownian motion and $a, b, σ, x$ are real constants The thing which confuses is ...
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26 views

Brownian motion hitting probability of boundary and going outside

I was solving an exercise which asks the reader to calculate the probability that a Brownian particle $B(t) = (B_1(t),...,B_n(t))$ starting at the origin in $\mathbb{R}^n$ will strike the surface of a ...
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1answer
17 views

Estimating a parameter using the maximum likelihood-method and the method of moments

Let $X$ be a random variable that has a density function of the form $f_X(x) = (p + 1) x^p 1_{[0, 1]}(x), x \in \mathbb{R}$ where $p > 0$ is an unknown parameter. I now want to make an educated "...
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31 views

Distance between two point processes

Is there a distance metric that we can use to see how close are two point processes? If instead of point process, we deal with random variables, there are a bunch of distance metrics that we can use ...
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64 views

Ito's Formula applied to a weird equation…

EDIT: One thing I forgot to mention before is that this is all under the $\mathbb{Q}$ measure in case that changes anything I was just wondering if someone could explain how to solve this problem. I ...
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24 views

A inequality of norm of Skorohod integral

I would like to ask you a question. Is following inequality true? $\|\int_{0}^{t}u_tdW_t\|_2 \leq \int_{0}^{t}\|u_t\|_2dt$, where $\int_{0}^{t}u_tdW_t$ is Skorohod integral.
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26 views

Green's function and strong Markov property for stopped Brownian motion

Let $X(t)$ be a Brownian motion in $\mathbb{R}^n$, stopped at some fixed time $T$. Is there a notion of Green's function for such a Brownian motion? I am guessing that there is, and $G(x, y) : = \...
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0answers
29 views

Applying Ito formula to Ito process

I would like to simplify the expression $\left(\phi(s_{1})\cdot(X_{s_{1}}-X_{s_{2}})+\phi(s_{2})\cdot(X_{s_{2}}-X_{s_{3}})+\ldots+\phi(s_{n-1})\cdot(X_{s_{n-1}}-X_{s_{n}})\right)^{2}$ where $X_{t}$ ...
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0answers
24 views

Limiting behavior of sde

What can we say about the limiting behavior of Xt , as t goes to infinite , where Xt is the solution of the sde $$dX(t) = e^{-t}X(t)dB(t)$$ ?
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17 views

Expected value of stochastic process [closed]

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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53 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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17 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
40 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
30 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 ...
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29 views

Gaussian volterra process. Conditional distribution?

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[...
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23 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
49 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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72 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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27 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? ...
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1answer
27 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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27 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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68 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 ...
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74 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 ...
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118 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
25 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
47 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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40 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
53 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
33 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
52 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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14 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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18 views

System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
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1answer
28 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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43 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
55 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 (...
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29 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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1answer
47 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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131 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}\...
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1answer
98 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, ...
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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?
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2answers
67 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 ...
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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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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
21 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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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\, ...