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

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0
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0answers
15 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 ...
0
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0answers
28 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 , ...
2
votes
1answer
75 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 ...
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0answers
20 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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0answers
14 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 ...
1
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1answer
37 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 ...
2
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0answers
28 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 ...
1
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1answer
43 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 ...
0
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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 ...
1
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0answers
30 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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0answers
9 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) = ...
1
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1answer
46 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 ...
1
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1answer
50 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 ...
0
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1answer
35 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 ...
0
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1answer
37 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 ...
0
votes
1answer
26 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 ...
0
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0answers
22 views

Different methodology for maximizing entropy in continuous random variable case

Suppose we want to maximize the well-known Shannon entropy $S=-∫_{0}^{x_{max}}f(x)lnf(x)dx$ subject to the following constraints $∫_{0}^{x_{max}}f(x)dx=1$, $∫_{0}^{x_{max}}xf(x)dx=x ̅$ and so on ...
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0answers
48 views

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 ...
3
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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 ...
0
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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 ...
2
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0answers
17 views

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 = ...
0
votes
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. ...
0
votes
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 = ...
0
votes
1answer
17 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 ...
1
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1answer
34 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 ...
4
votes
1answer
121 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 ...
0
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0answers
26 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}. ...
0
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0answers
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 ...
1
vote
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 ...
5
votes
2answers
139 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 $\int_0^T W_t d W_t$. ...
1
vote
1answer
35 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 ...
2
votes
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 ...
0
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0answers
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 ...
1
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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 ...
2
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0answers
55 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 ...
1
vote
1answer
20 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\}] & = ...
1
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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, ...
1
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0answers
10 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 ...
0
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0answers
36 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
43 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 ...
0
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0answers
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 ...
0
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1answer
26 views

I want to simplify the stochastic integral by change variable

Let $f:[0,t]\rightarrow \mathbb{R^+}$ be a deterministic and integrable and $(B_t)_{t\geq 0}$ is a standard Brownian motion. If $X_t=\int_o^tf(s)dB_s$, we know that $X_t$ has normal distribution with ...
0
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0answers
25 views

Probability Reference Request

Right now I am studying probability theory and I am looking for a string of related books that will cover topics through Stochastic Calculus. More explicitly, I am hoping to self study Stochastic ...
2
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1answer
44 views

Calc II to SDEs

I am interested in learning the mathematics to understand and solve stochastic differential equations similar to those seen in quant finance journals. Currently, the extent of my mathematics ...
1
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1answer
25 views

Proof that time integral of OU process is not Markov while together with the integrand it is.

Consider an Ornstein-Uhlenbeck process $$ dX_t = \kappa( \theta -X_t) d t + \sigma d W_t, \quad X_0 = x $$ and $W$ a Brownian Motion. Consider also its time integral $$ Y(t) = \int_0^t X(s) d s. $$ ...
1
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0answers
21 views

What will be the behavior of $R(t)$ if $R(0)=\alpha / \beta$ in Vasicek model

Can someone please help explain what the behavior of $R(t) = \alpha / \beta$ would be if $dR(t)=(\alpha-\beta R(t))dt +\sigma dW(t)$ I already know the expression for R(t) and I know the mean and ...
1
vote
1answer
34 views

If given the Vasicek Interest rate model $dR(t)=(\alpha-\beta R(t))dt +\sigma dW(t)$ how do I use Ito's lemma to find $d(e^{\beta t}R(t))$?

If given the Vasicek Interest rate model $dR(t)=(\alpha-\beta R(t))dt +\sigma dW(t)$ how do I use Ito's lemma to find $d(e^{\beta*t}R(t))$ and simplify so it is a solution that does not include R(t). ...
1
vote
0answers
38 views

Bounded moments implies finite moments of supremum?

Suppose that, for all $(t,x) \in [0,T]\times \mathbb{R}$, a two-parameter stochastic process $Z(t,x)$ satisfies $$ \| Z(t,x)\|_{L^p(\Omega)} \leq C_{p,T} $$ for some constant depending on $p$ and ...
0
votes
0answers
21 views

Linear Combination of Multivariate Normal

Given $Y\in\mathbb{R}^N$ follows a multivariate Normal distribution with mean $\mu$ and variance-covariance matrix $\Sigma$. Let $\omega\in\mathbb{R}^N$ be deterministic. I know that $\omega'Y$ is ...
1
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0answers
19 views

Euler Schemes in Stochastic Differential Equations

So i am trying to understand what happens in Implicit (backward) and Explicit (forward) Euler in Stochastic Differential Equations I ll start with explicit. Say i have the following SDE known as ...