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

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1answer
205 views

Regular matrix and regular stochastic matrix

We know that : A matrix is regular if its determinant is non zero. A stochastic matrix is regular if at a certain power all elements are positive. Question is how can I make the link between the ...
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1answer
92 views

How find stochastic logarithm of $B^2(t)+1$.

Find the stochastic logarithm of $B^2(t)+1$. I know that for find stochastic logarithm According to Theorem we must use the The following formula $$X(t)=\mathcal L(U)(t)= ...
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2answers
68 views

Is the following Itô-Integral not zero?

is the following statement true: $$\int_0^T t \, dW(t) \neq 0$$ I need it for a counter-example, that one can not change the order of integration between $dW$ and $dP(\omega)$. I thought of taking ...
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1answer
64 views

Expectation of $e^{-4B_\tau}$, where $\tau$ is an extended stopping time

This is an specific example so with a bit of luck I can get some general methodology from your answers. I have this stopping time: $$ \tau = \inf\{t \geq 0; B_t < t-2 \} $$ This is a clear ...
2
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1answer
92 views

The law of the iterated logarithm for BM and boundedness of stopping times

My question is regarding the usefulness of the law of the iterated logarithm, and its connection to stopping times. In many answers of this forum, I understand that some people often claim that some ...
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1answer
134 views

Is $t^{-\frac{1}{2}}B_{t^2}$ a Brownian Motion?

I think the title says it all. Let $X_t = t^{-\frac{1}{2}}B_{t^2}$, with $B_t$ being a brownian motion started at $0$. I think I have proved continuity at $0$ by doing the following: $$ X_t = ...
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1answer
237 views

Expectation of product of stochastic integral and brownian motion

Find the covariance: $$ COV((\int_t^T(T-s)dW_s), W_t) $$ I used the covariance formula: COV(X,Y) = E(XY) - E(X)E(Y) = E(XY) as E(X)=E(Y)=0 But I am stuck on figuring out the expectation of the ...
2
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1answer
72 views

Stochastic Integral Help

Let W(t) be a Brownian Motion. Show that the integral: $$ \int_t^T W(s)ds $$ can be written in terms of the stochastic integral: $$ \int_t^T (T-s)dW(S) $$ Is there an error with this question? I ...
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0answers
24 views

Optimal capital injection in continuous time.

Problem: Given a controlled n-dimensional linear stochastic system on $[0,T]$, let's say:$$d\underline X(t)=A\underline X(t)dt + B\underline u(t)dt + d\underline W(t) $$ $$\ \underline X(0)=x \in \Bbb ...
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1answer
59 views

Problem with Ito Isometry

I know that for one-dimensional case, $$ E \left[ \left(\int^T_S f(t,\omega)dB_t \right)^2 \right] = E\left[ \int^T_S f^2(t,\omega) \, dt \right]$$ for adapted, measurable f that satisfies that are in ...
2
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1answer
752 views

Black Derman & Toy Model

The BDT model is given by $$d(\ln\,r)=\left(\theta(t)-\frac {d(\ln\sigma(t)}{dt}\ln r\right)\,dt+\sigma(t) \, dW.$$ How can one rewrite the BDT model as $dr=A\,dt+B\, dW$, using It$\hat o$?
2
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0answers
114 views

Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
2
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2answers
49 views

A random variable $X$ with differentiable distribution function has a density

Setting: My professor defined A random variable $X: \Omega \to \mathbb{R}$ has a density $f:\mathbb{R} \to \mathbb{R}$ if for all $B \in \mathscr{B}$ $$P(X^{-1} (B)) = \int_\mathbb{R} ...
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1answer
84 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 ...
5
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2answers
503 views

Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$ dV_t = - \beta V_t dt + \sigma dW_t $$ with $V_0 = v$, where $W_t$ is a Wiener ...
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0answers
102 views

Stopping times, open sets and Brownian Motion

Let $B_t$ be a brownian motion started at 0. I am trying to prove that $\tau$, defined as: $$ \tau = \inf\{t > 0 \mbox{ }|\mbox{ } \left|B_t\right| \geq \frac{1}{1+t} \} $$ is a stopping time with ...
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2answers
133 views

Use Ito's formula to determine the stochastic differential equation satisfied by $V_t$

A stochastic process $V_t$ is defined by $$V_t =\sqrt{t(t+W_t^2)}$$ $W_t$ is the Wiener process and $t$ denotes the time ($t > 0$). Use Ito's formula to determine the stochastic differential ...
3
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1answer
81 views

Extended (or augmented) stopping times

I am trying to prove that $\tau$, defined as: $$ \tau = inf\{t > 0 \mbox{ }|\mbox{ } B_t < t-1 \} $$ is a stopping time with respect to the filtration $(\mathscr{F}_{t+}^B)_{t\geq 0}$ where ...
0
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1answer
38 views

Covariance of a random function

Suppose $X(s)=\int_0^1 G(s,t)\, dW(t)$, where $W(t)$ is Brownian motion, then what is the variance of $X(s)$ and the covariance of $X(s)$ and $X(r)$. Note that this is not the usual Ito integral ...
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1answer
101 views

Mean and Variance of Gaussian Process

Let $B = (B_t : t \geq 0)$ be a standard Brownian Motion. Fix $0 \leq s \leq t$. How can I prove that, conditionally on $\{B_s = x, B_t = z\}$, the intermediate value $$B_{\frac{t+s}{2}}$$ has ...
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1answer
47 views

Lifetime of a spaceship run by three computers

A spaceship is controlled by three independent computers. The ship can function as long as at least two of the three computers are functioning. Suppose the lifetimes of the computers are i.i.d. ...
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1answer
42 views

Sufficient condition for time-changed quadratic covariation to vanish in probability

Let $(M_t^n)_{t \geq 0}$ be a sequence of continuous martingales of the form $M^n_t = \int_0^t X^n_s \, dB_s$ where $B_s$ is a Brownian motion. Let $\tau^n(t)$ be the time change associated to $M_t^n$ ...
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1answer
91 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 ...
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1answer
38 views

Fourier transform of n-th power of autocorrelation of a random process

I'm having troubles in understanding how Fourier transform of the n-th power of a time function is obtained. In particular I came across to a particular result with respect to the calculation of the ...
0
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1answer
186 views

Ornstein-Uhlenbeck process and Markov property

There isn't a similar question in the forum, so here it goes. Firstly, let the O-U velocity process be defined as $$ dV_t = -\beta V_t dt + \sigma dB_t $$ with $V_0 = v$, and $B = (B_t), t \geq 0$ a ...
2
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1answer
212 views

Running average of Brownian motion

Question : Let us define the cumulative sum (Brownian motion): $$x_k = \sum_{i=1}^k y_i$$ and the running average : $$ \overline{x_k} =\frac{1}{W}\sum_{i=k-W+1}^k x_i$$ for $ k>W $, $W$ ...
0
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1answer
17 views

Relation between two stochastics

I got stuck trying to figuring out how to show the following question in probabilistic theory: "We say that A and B (with P(A), P(B) > 0) attract each other when P(A|B) > P(A)." I've shown that ...
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1answer
128 views

Resource for Stochastic Calculus and Ito processes

May someone please recommend a book or website where one can learn Stochastic Calculus and Ito processes from scratch.
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123 views

Poisson integral and discontinuous martingale (Ito-Levy formula)

Consider compounded Poisson process $P$ given by $P_t = \int_0 ^t \int _{\mathbb R}z~ N(dr,dz)$ where $N$ is a Poisson random measure of intensity $dt \otimes \nu$ and $\nu $ is a Levy measure. Why ...
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0answers
22 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 ...
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0answers
42 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
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1answer
68 views

Relation between autocorrelation and mean of a stochastic process

It is said that if the autocorrelation approaches zero as $\tau$ tends to zero, then the mean of the stochastic process is also zero. I am having trouble understanding the above concept. Say we have ...
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1answer
63 views

What does it mean by “mean of a process”?

Say $X(t)$ is a stochastic process. Now when it says that the mean of the process, does it mean that the mean of $X(t)$. Elaborating further, a process is an collection of random variables - ...
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1answer
45 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 ...
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1answer
230 views

Variance of Ito Integral

I want to find the variance of the Ito integral: $X(t)=\displaystyle \int_0^t\sqrt{s}WdW$ where W is a Brownian motion and s is the variable of integration. This is what I have done so far: ...
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1answer
61 views

Determining $dX_t$ for stochastic equations, and which of these are $\mathcal{F} $ - martingales?

I want to write down an expression for $dX_t$ for both: i. $X_t=t^2W_t^2-2\int_0^t(sW_s^2+s^2)ds$; and ii. $X_t=W_t^2-tW_t$ What is the process I would use for differentiating these stochastic ...
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1answer
49 views

Path solution for a SDE

I would like to get help in solving an Ito stochastic equation: $dY_t=-dW_t \, (Y_t^2+1)$ The process $W_t$ is the standard Brownian motion. Is it possible to get a path solution solution in terms ...
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1answer
69 views

Question on complex valued local martingales

So I was reading and found that the following was given as an example of a complex valued local martingale: $M_t = e^{\int_0^t f(\omega,s)dB_s - \frac 12\int_0^tf(\omega,s)^2ds}$ with $f(\omega,s) = ...
2
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1answer
48 views

$\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
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1answer
71 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 ...
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1answer
65 views

Ito's Lemma for Integral

Let $S$ follow GBM with $dS=(r-q)S\,dt+\sigma S\,dW$ where $W$ is a standard Brownian motion. Define $I_t=\int_0^t qe^{r(t-u)}S_u \,du$, then how can I determine $dI_t$? The answer should be ...
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0answers
29 views

Does this Stochastic Differential Equation have a name?

I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times. The SDE is $$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$ ...
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1answer
94 views

Ornstein-Uhlenbeck process written explicitly

I need to show that the Ornstein-Uhlenbeck process, $$ dX_t = -\theta X_tdt + dB(t) $$ Where $X_0=0$, $B(t)$ is Brownian motion and $\theta>0$ can be written explicitly as: $$ X_t=B(t) - \theta ...
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1answer
108 views

Solving the SDE $dX_t=bdt+cX_t dW_t$

I want to solve the SDE $dX_t=bdt+cX_t dW_t$, $X_0=0$ for $b,c\in\mathbb R$. I start by rewriting this as $$dX_t=(\mu_1+\mu_2 X_t )dt+(\sigma_1+\sigma_2 X_t )dW_t$$ where $\mu_1=b, \mu_2=0, ...
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1answer
128 views

Solution to a stochastic differential equation

I could really do with some help on this question, have no idea where to start. Any advice would be much appreciated, thank u in advance. I am given $$\begin{align}dx(t)&=(1+x(t))dt + x(t) ...
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1answer
86 views

Ornstein-Uhlenbeck operator and divergence operator

So I'm still struggling with Malliavin calculus, and this time about the divergence operator. We are working in the classical Wiener space $(W,H,\mu)$ where $W$ is the Wiener space ...
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0answers
33 views

Weak stochastic integral

I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prévôt/Röckner [PR07]: $$ \int_0^T \langle \Psi \,\mathrm dW(t), \Phi(t)\rangle $$ A ...
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1answer
34 views

Do stochastic processes form a Banach space?

I'm interested in solving a particular integral equation: $$g(X) = \int_0^1 K(X,p)f(p) \ dp$$ where $f(p)\in L^1([0,1])$ and $X$ is a stochastic process of finite length; i.e. a collection of random ...
3
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0answers
80 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
0
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0answers
111 views

Second Fundamental Theorem of Asset Pricing

It seems that there is a step missing in the proof of the second Fundamental Theorem of Asset Pricing in Shreve's Stochastic Calculus for Finance II: Does anyone know how to show the following: If ...