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

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101 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
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1answer
62 views

SDE and Stochastic calculus

$W_t$ is 1 dimension Brownian morion. $X_t=(cosW_t,sinW_t)$ Write SDE about $X_t$ I thought that $f(t,x)=(cosx,sinx)$, but I can't how "$t$" is expressed. I heard that the hint of this question is ...
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1answer
41 views

Question regarding Notes on Strong Markov Property

I wrote the following notes from a lecture a couple of weeks ago and I don't understand a particular line. Suppose $B_t$ is a Brownian Motion. Now look at $B^x_t = x + B_t$ which is a BM starting ...
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27 views

IID implies Ergodicity

The environment space is given by $\Omega:=P^{\mathbb{Z}^{d}}$, where P contains the 2d-vectors serving as admissible transition probabilities. An Element $\omega \in \Omega$ is defined as ...
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0answers
55 views

Most probable path of diffusion process

Suppose we have an Ito diffusion $X_{t}$ on $\mathbb{R}$ given by \begin{align*} dX_{t} = A(X_{t})dt + B(X_{t}) dW_{t} \qquad (1) \end{align*} where $W_{t}$ is a standard Brownian motion. If $B = 1$, ...
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24 views

Product of Geometric Brownian motions

Let $S,P$ be geometric BMs: $$dS_t=S_t(\mu dt + \sigma dW_t^1)$$ $$dP_t=P_t(\tau dt + \beta (\rho dW_t^1+ \sqrt{1-\rho^2}dW_t^2)$$ Where $W^1$ and $W^2$ are independent standard BM. I want to show ...
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1answer
53 views

Solve the linear SDE $dX_t = aX_t \, dt +(b+cX_t) \, dW_t$

I am trying to find the solution to the SDE: $$ dX_t=aX_tdt+(b+cX_t)dW_t $$ for $t\ge0$, $X_0>0$, constants $a,b,c$ Would appreciate any hints as to how to approach this using ito's formula, I'm ...
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1answer
66 views

Showing that a certain stochastic process does not have normal distributed increments

Edit: Question Resolved. See below. As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz. In this paper he defines a stochastic process. Let ...
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2answers
44 views

Brownian Bridge conditional probability

The problem is to show that the density $P[W_{t_1} \in dx_1,...,W_{t_n}\in dx_n | W_T = b]$ is the density of a Brownian bridge from $a$ to $b$. $W$ is Brownian motion. The density of a Brownian ...
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23 views

Stochastic differential equation of a falling body

It's well known the motion of a falling body in a constant gravity model, for high speed is given by: $$m\ddot{x}(t)=g-\beta\dot{x}(t)^2$$ where $\beta$ is he drag coefficient. In a turbolent flow we ...
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2answers
28 views

Let Y be a random variable with $0\le Y\le 1.$ [duplicate]

Let Y be a random variable with $$0\le Y\le 1.$$Show that $$var(Y)\le 1/4 $$ and that $$var(Y)= 1/4 $$ if and only if P(0)=1/2=P(1).
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49 views

BMO martingale and exponential martingale

Consider the BSDE, $$ Y_{T}-Y_{t}=\sum_{i=1}^{n} \int_{t}^{T} Z_{s}^{i}dB_{s}^{i} - \frac{1}{2}\int_{t}^{T} \left| Z_{s}\right|^{2}ds $$ where $B$ is a standard Brownian motion on a complete ...
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1answer
43 views

Proving a Self Financing Portfolio

Question: Let $(S_t)_{t\ge 0}$ be a stock price process. Assume $u(.,.)$ satisfies the Black Scholes PDE with short rate $r=0$. Assume that under a risk neutral measure P: $$ dS_t=\sigma_tS_tdW_t $$ ...
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45 views

Power spectral density of convolution of stochastic processes

I was wondering what it is the result of convolving two WSS processes in terms of power spectral densities. I know that, the output $Y(t)$ of a generic linear time invariant system with impulse ...
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1answer
169 views

Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
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1answer
101 views

Problem on Solving Stochastic Differential Equation

Let $(Xt)$ be a solution to the equation $dX_t = aX_t dt + \sqrt{(1+X_t^2)} dW_t$ where $W_t$ is a Brownian motion process at time t Let $Y = F(X_t)$ for a certain function $F$. Find $F$ for which ...
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2answers
48 views

Using Ito's Lemma with more than one brownian motion term

Question : Let $$ dY_t=c_tdt+d_tdW^1_t+e_tdW^2_t $$ Where $W^1_t,~~W^2_t$ are standard independent brownian motions. I am trying to apply Ito's formula to this, say for example trying to find ...
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1answer
30 views

Pricing a claim dependent on two stock processes

QUESTION Consider two stock processes: $$ dS^1_t=S^1_t(r\,dt+\sigma_1\,dW^1_t) $$ $$ dS^1_t=S^2_t(r\,dt+\sigma_2\,dW^2_t) $$ $$ t,S^1_0,S^2_0\ge0 $$ and $$ W^1_t,W^2_t $$ are standard independent ...
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38 views

SDE with no weak solution

I'm facing the followingd d-dimensional SDE: $$dY_t=\sigma(h_t)\,dB_t$$ In addition it holds, that: $h_t\in H$ and $H$ is compact (for example the simplex of $R^n$) the proces $h_t$ is progressivley ...
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1answer
85 views

Difference between Borel Sigma algebra and Cylindrical sigma algebra?

I see that there are two differen concepts for Sigma Algebras on cartesian products over the real numbers. The first one is the Borel Sigma Algebra created by the product topology. The other one is ...
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43 views

Differentiating Exponential matrix Expression

To give the scalar version first: For the well known Ornstein-Uhlenbeck process: $dr(t)=\alpha(b-r(t))dt+\sigma dW(t)$ It is well known that the variance is: $\sigma_r^2=\sigma^2 \int_u^t\exp^{-2 ...
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1answer
62 views

The Vacisek Model and the short rate process

I am trying to do some calculations related to the Vacisek model, but I think I am mixing up concepts and I'm not getting to any solution. Let me explain what the problem is. The Vacisek model ...
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31 views

Plot histogram and density function

I need to plot a histogram for the data: ...
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1answer
32 views

I want to show $\operatorname{Cov}(X(t),X(s))=\min(s,t)- \frac{st}{T}.$

i have this Equation with Condition $X\left(0\right)=a $ and $ 0\le t \lt T$ $$dX\left(t\right)=\frac{b-X\left(t\right)}{t-T}dt+dB\left(t\right)$$ I solved and got $$X\left(t\right)= ...
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0answers
51 views

Black Scholes Pricing of a claim

Question: Let H(x)=1/x be the payoff function for a European style derivative security. Find a closed form expression for the price: $$ u(t,x)=e^{-r(t-t)}E[H(S_T)|S_t=x] $$ for this claim using Black ...
2
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1answer
97 views

Solving a Stochastic Differential Equation (SDE)

Question: Solve the stochastic differential equation: $$ dX_t=X^3_t\,dt-X^2_t\,dW_t $$ where: $$ X_0=1 $$ My Attempt: Using Ito's with: $$ f(x)=\log(x) $$ I get that: $$ ...
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1answer
37 views

Prove a P Martingale

If: $$ \sigma_t $$ is a bounded function of both time and sample path, show that: $$ dX_t=\sigma_tX_tdW_t $$ is a P Martingale. *Does this question make sense, that is, should the question be: is ...
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1answer
47 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
52 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
64 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
51 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
49 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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37 views

Girsanov Measure Question.

If $Z_t = exp^{\int_0^t X_s dW_s - \frac{1}{2} \int_0^t (X_s)^2 ds}$ is a martinagle then by Girsanov's theorem, the measure $P_T$ defined by $P_T(A) = E^P(AZ_T)$ is mutually absolutely continuous ...
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96 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
68 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 ...
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1answer
42 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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21 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
31 views

Problem with Ito Isometry

I know that for one-dimensional case, $$ E[ (\int^T_S f(t,\omega)dB_t)^2] = E[ \int^T_S f^2(t,\omega)dt]$$ for adapted, measurable f that satisfies that are in $L^2(dt \times dP)$. For $f = ...
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1answer
203 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$\hat o$?
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59 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 ...
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2answers
35 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
43 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 ...
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1answer
140 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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60 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
90 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
57 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 ...
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1answer
31 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
70 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
40 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
32 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 $M_t^n$ ...