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

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How can we identify $\omega\in\Omega$ with a path of Brownian motion $t\rightarrow B_t(\omega)$?

In the Stochastic Differential Euqations written by Oksendal(see page 12), As we shall soon see, the paths of a Brownian motion are (or, more correctly, can be chosen to be) continuous, a.s. ...
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25 views

PDE for Brownian Bridge Expectation?

Let $\displaystyle Y(t)=\int_0^t v(s)ds+B(t)$, where $B(t)$ is the standard Brownian motion and $v(t)$ a deterministic function. Compute $m(t,y):=\mathbf E\Big[\max\limits_{s\in[0,t]} Y(s)\big|Y(t)=y\...
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2answers
51 views

Hitting times for Brownian motions

Let $B$ be a standard Brownian motion and let $\alpha, \beta > 0$. Let \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\}. \end{align} It can be shown by defining ...
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51 views

$X_t=\int_{0}^{t}(a_{0}+a_{1}\frac{u}{t}+\ldots+a_{n}\frac{u^{n}}{t^{n}})dB(u)$ is a Brownian motion for suitable non-zero constants $a_0,\ldots,a_n$

Let $B(t)$ be brownian motion. Show that for any integer $n \geq 1$, there exist nonzero constants $a_{0},\ldots,a_{n}$ such that $X_{t}=\int_{0}^{t}(a_{0}+a_{1}\frac{u}{t}+\ldots+a_{n}\frac{u^{n}}{t^{...
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1answer
33 views

Expected value of product of an ito integral and a random variable

I want to compute $$E[\int_0^t W_r dr \int_0^s W_r^2 dW_r].$$ Here $t,s$ are arbitrary. I have thought about this a lot but not sure how to proceed. I tried to apply Ito's formula to one of the ...
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1answer
62 views

Use Ito formula to compute expected value

Let $W_t$ be a standard brownian motion. I am trying to compute $E[(\int_0^t s^2 dW_S)^4]$. I applied Ito's formula and got $$t^2 W_t = \int_0^t s^2 dWs + \int_0^t 2s W_s ds$$. This gives us $$E[(\...
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1answer
38 views

Itô formula question

My question is at the end of the problem statement. Solve the following stochastic differential equation. $dX_t = (\beta - \alpha X_t)dt + \sigma dB_t$, $X_0 = x_0$ where $\alpha$, $\beta$, $\sigma$...
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14 views

Deciding which version of Ito's lemma to use

The equation above is the baby version of Ito's lemma that we are given. The equation below is the generalised version of Ito's lemma that we are given. Now from what I understand, we use the ...
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1answer
22 views

Using Ito's lemma to compute a SDE

This is the version of Ito's lemma that we are given in our notes. Now I'm just not able to understand how to begin this problem and arrive at the given solution. The g(x) integral function that ...
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16 views

Lipschitz condition for the stochastic subgradient

We know that for the subgradient method convergence, $f$ should satisfy the Lipschitz condition, i.e., $|f(x_1)-f(x_2)|\leq G\|x_1-x_2\|_2\ \ \ $ for all $x_1, x_2$ For the stochastic ...
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29 views

Multi-derivative of standard normal CDF

I am trying to solve the following $m$th-derivative of standard normal cdf, $$\frac{\text{d}^m}{\text{d}a^m}\Phi \left(\frac{a+\mu u}{\sqrt{u}}\right),$$ where $m> 0$ is an integer , $\mu>0$,...
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1answer
79 views

Following a derivation using Ito's lemma

I am trying to follow a derivation, but I get stuck could someone take me take me through the rest: We start with, $$s(t,x_t)=e^{g(t)+x_t}$$ where $$dX_t=\log (J) dq_t+\left(-\text{$\alpha $X}_t\...
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24 views

Using Ito's lemma to find a SDE

For this question I'm not able to work out how they got sigma(x)=1. If I have a SDE dxt=bdt +sigmadWt then I know the sigma(x) function is just the coefficient of dwt but in this case where I hava ...
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1answer
19 views

Ito differential of expectation with respect to a measure

How could one think of taking the Ito differntial of an expectation or measure theortic integral? In particular, I know how an Ito process $D_t$ evolves ($dD_t = \mu dt + \sigma dW_t$) and that it ...
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1answer
39 views

Brownian Motion with rescaled time as an Ito process

I have a seemingly simple question that has me stumped. Suppose $(B_t)_{t\geq0}$ is a Brownian motion, and consider its rescaled version $(B_{\alpha t})_{t\geq0}$ for some $\alpha>0$. It seems ...
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0answers
30 views

Laplace transform of survival probability for stochastic diffusion

Let $Y_t$ be a killed process defined by \begin{eqnarray} Y_t = X_t \quad \mbox{if } t<\xi,\\ Y_t = 0 \quad \mbox{if } t\geq\xi. \end{eqnarray} where $\xi$ is a random time such that $$ \xi=\inf \...
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1answer
44 views

Martingale property for two stochastic processes

Let $(\Omega,F,P)$ be a probability space with filtration $\left\{F_{t}\right\}_{t\geq 0}$ generated by one dimensional Brownian motion $(B_{t})_{t\geq 0}$ defined on $(\Omega,F,P)$, assume that $\...
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1answer
22 views

Applying ito integral to a specific example

Using the ito integral: $$x_t=\int_0^t b(s,x(s)) \, ds+\int_0^t \sigma (s,x(s)) \, dB(s)+x_0$$ I want to solve this: I have $$g(s,n(s))=\log(n(s))$$ Using ito's lemma taking the derivative I get ...
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31 views

Probability - urn problem (drawing with putting back)

In an urn are $N-1$ white and $1$ black balls. Now one draws $n \leq N$ of those balls with putting the drawn one back I'd like to find out the probability that the black ball is within the ...
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10 views

what's the relation between characteristics of $(X_1,X_2)$ and characteristics of $X_1$ and $X_2$

I am not clear how to write down of the characteristics of two-dimensional Levy process $(X_1,X_2)$ when the characteristics of $X_1$ and $X_2$ are known. More precisely, let's say $$ X_k(t) = \...
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1answer
50 views

Solve the stochastic differential equation $dX_t = \alpha X_t \, dW_t + \sigma X_t \, dt$

We want to solve: $dX_t = \alpha X_t dW_t + \sigma X_t dt$ where the initial condition $X_0$ is given and $\alpha$, $\sigma$ are constant. The solution goes as follows - We rewrite the ...
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1answer
57 views

Markov DTMC and CTMC: How to build a $Q$ generating matrix using given $2$ state $q$ probabilities? Find Steady State Probabilities

I know that $$q_{(i,j)(i+1,j)}=\lambda_1$$ $$q_{(i,j)(i-1,j+1)}=\lambda_2$$ $$q_{(i,j)(i+1,j-1)}=0.5\lambda_3$$ $$q_{(i,j)(i,j-1)}=0.5\lambda_3$$ where $\lambda_1 , \lambda_2 , \lambda_3 $ are rates ...
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1answer
37 views

Clarification on Stochastic Exponential

Consider a $d$-dimensional Brownian motion $B=\left(B_1,...,B_d\right)$ whose components are independent and let $A$ be a $d\times d$ squared matrix such that $\sum_{i=1}^dA_{ii}^2=1$. Define the $W=...
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1answer
39 views

Black Scholes SDE

This is only part of the solution I got stuck on. I don't quite understand how can the instructor got from $\rm{d} (\log S_t) = \frac{dS_t}{S_t}$. Thank you. And also where did the negative sign ...
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1answer
28 views

Stochastic Integration

I am fairly new to stochastic calculus and am having problems solving this equation.. $$X(t)=\oint_0^TL(t)(\mu \, dt + \sigma \, dW_t)$$ Now, here $L(t)$ is a constant $k$. And I have to find $X(t)$...
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Radon-Nikodym derivatives $\frac{d\mathbb{P}_1}{d\mathbb{P}_0}$ and $\frac{d\mathbb{P}_2}{d\mathbb{P}_0}$

$\Omega$- is the interval [0,1], $\mathbb{P}_0$ is Lebesgue measure, $\mathbb{P}_1$ is the probability measure given by $\mathbb{P}_1([a,b])=\int_a^b 2\omega d\mathbb{P}_0(w)$ and $\mathbb{P}_2([a,b])=...
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1answer
41 views

Do $C([0,1];\mathbb{R}^2)$ and $C([0,1];\mathbb{R}^3)$ differ as measure spaces?

Do $C([0,1];\mathbb{R}^2)$ and $C([0,1];\mathbb{R}^3)$ with the respective Borel sigma algebras under the uniform topology differ as measure spaces? The question is simply a curiosity with no ...
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1answer
34 views

Stopping time and random variable measurability

Let $\sigma$ be a stopping time and $Z$ a $\mathcal{F}_{\sigma}$-measurable random variable. Now, I want to show that for any $A \in \mathcal{B}_{[0,t]}$, $\mathbb{1}_{\{\sigma \in A\}} Z$ is $\...
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Solve the SDE $dX_t=\alpha \,dt + \sigma X_t \,dB_t$, $X_0=x_0$ [duplicate]

So I got the following SDE to solve: $$dX_t=\alpha\, dt + \sigma X_t \,dB_t, X_0=x_0$$ This is what I've tried: Using Ito's I should get the following relations: $$X_t=f(s,x)$$ $$\alpha = \frac{1}{...
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1answer
56 views

Why this set does not belong to the product $\sigma$-algebra (Borel)

Let $\mathcal{F}_t$ be the $\sigma$-algebra generated by singletons on $\Omega=[0,1]$. And let \begin{align} X_t(\omega)=\mathbb{1}_{\{\omega=t\}} \qquad \text{for} \qquad 0 \leq \omega, t \leq 1. \...
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1answer
21 views

Calculate moments of the solution to the SDE $dX_t = -\frac{k}{2} X_t \, dt + \frac{\beta}{2} \, dZ_t$

I have this process: $dx_t = -\frac{k}{2}x_t \, dt + \frac{\beta}{2} \, dz_t$ and must prove it's normally distributed with first two moments: $\mu = e^{-\frac{1}{2}kt}x_0$ $\sigma^2 = \frac{\...
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1answer
18 views

Equivalent definitions of $\mathrm{BMO}_p$ martingales

I'm working through exercise 3.16 in Revuz and Yor. Assume $Y$ is a continuous UI martingale and $1\leq p<\infty$. Then these are equivalent $\exists C\ \forall T$ stopping time $E[|Y_\infty-Y_T|^...
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1answer
35 views

Change of variable in $\varphi(s) = t$, effect in $\mathbb{d} W_t$

I'm a little confused here. If I have the stochastic integral $$ \int_0^T f(t)\,\mathbb{d} W_t $$ and perform the change of variables $t = \varphi(s)$, how will $\mathbb{d} W_t$ transform (where the ...
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1answer
123 views

Solution for SDE: $dF_t= \beta_t\left(F_t - \alpha\right)dW_t$

I am trying to derive the solution for the following stochastic differential equation, but I must be doing something wrong in my calculations because I can't arrive to the correct solution. The SDE ...
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28 views

Showing the Haar wavelet is a complete and orthonormal sequence within $L_2[0,1]$

I define the mother Haar wavelet to be: \begin{align} \phi(t) = \begin{cases} 1 &\mbox{if } 0 \leq t < 1/2 \\ -1 & \mbox{if } 1/2 \leq t \leq 1 \\ 0 &\mbox{otherwise}. \end{...
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53 views

Variance of Brownian Integral when the end point is specified

Consider the Brownian $W_u$. Suppose you are only considering realizations of this brownian that verify both $W_0=0$ and, for a specific (given) $t$, $W_t=a$. Under these specific conditions, what is ...
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1answer
49 views

Calculate $E[\exp(iu\int_0^ts \, dB_s)]$ for a Brownian motion $(B_t)_{t \geq 0}$

Since $X_t:=\int_0^ts \, dB_s$ is a process with independent increments, its distribution is infinitely divisible and its variance is $c_t=\frac{1}{3}t^3$. I think, its characteristic function $E[\...
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2answers
142 views

The stochastic integral $\int W_t dW_t$

I'm reading an introduction to Stochastic Calculus. I'm at the point where Ito integrals are developed and constrasted with the Stratonovich integral. Below is a calculation of $\int_0^T W_t d W_t$. ...
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1answer
36 views

Itô integral probability distribution

I know in general this must not have an analytical expression in terms of common functions, but how do you (at least in theory) get the probability distribution of $X_t$ for a given $t$ in the ...
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0answers
23 views

Itô's lemma and Feynman-Kac theorem for Lévy processes?

I'm facing the problem to try to extend some financial way of reasoning in the case we do not live in the platonic Brownian Motion world. I come from an economic background so I'm stuck on this: the ...
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25 views

infinitesimal generator of a pair of Ito's process

I have a question concerning the infinitesimal generator of a pair of two process. Let say, I have two processes : $dS_t = (\mu - r) S_tdt + \sigma S_tdW_t$ and $dX_t = (\pi_t (\mu - r ) + rX_t )dt ...
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1answer
36 views

How to formally proof this measurability?

I have $Y = \int_0^t X_s ds$ It is intuitive that $Y$ is $\{\sigma(X_s), s \le t, \}$-measurable, since it onl y depends on the values of the process $X_s$ up until $X_t$ but how to write it ...
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58 views

How to derive the Radon-Nikodym derivative?

Denote by $\mu_x^c$ and $\mu_x^{\lambda}$ the measures induced by the process $x^c$ and $x^{\lambda}$ generated by the following SDE: $$dx^c(r) = cx^c(r) dr + dW(r)$$ $$dx^{\lambda}(r) = \lambda x^{\...
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1answer
23 views

Expectation equals to Black-Scholes Equation

Let $S_t$ be ageometric brownian motion with parameters $\sigma$ and $r$ and fix $T,K\in (0,\infty)$. How can I show that: \begin{align} \mathbb{E}[e^{-rT}max\{(S_T-K),0\}] & = x\Phi(d_+(T-t,x))...
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1answer
34 views

Proof that $\mathbb{d}\,\mathbb{E}(f(X_t)) =\mathbb{E}(\mathbb{d}f(X_t))$

I've read in a paper that, if $f$ is continuous, then $$ \mathbb{d}\,\mathbb{E}(f(X_t)) =\mathbb{E}(\mathbb{d}f(X_t)) $$ where $X_t$ is a stochastic process and $\mathbb{d}$ is a differentiation, ...
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0answers
12 views

Positivity of density function of an SDE

I am interest in proving that the transition probability density function $p(t,x,y)$ for the process $(X_t,Y_t)$ is STRICTLY positive on all of $\mathbb{R}^2$, where $(X_t,Y_t)$ is the solution to the ...
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0answers
37 views

Wick renormalization of stochastic integral

I am trying to understand a paper that summarizes some results concerning Wick renormalization of some stochastic integral. In the last few lines of the paper the authors say: In Euclidean ...
3
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0answers
44 views

The limit of the ratio of two stochastic integrals

I am just wondering how to calculate the limit of stochastic integrals. Here is one example: $$ \lim\limits_{N \rightarrow \infty}\dfrac{\int_{0}^{N}B(s)dB(s)}{\int_{0}^{N}B^2(s)ds}$$ where $B(s)$ is ...
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26 views

Brownian motion through gates

Exercise 1.9.6 of Ubbo F. Wiersema's textbook "Brownian Motion Calculus" (Wiley 2008, p. 27) is titled "Brownian motion through gates" and begins: "Consider a Brownian motion path that passes through ...