0
votes
0answers
26 views

Product of stochastically independent random variables

Let $X, Y, Z$ be three stochastically independent random variables that are quadratic integrable (quadratintegriertbar is the German term, I didn't find a exact translation). No which statements hold ...
1
vote
1answer
49 views

is $(x-6)^2$ in $C_0^2$?

My math problem involves using a theorem that requires $f(x)=(x-6)^2$ to be in $C_0^2$. I'm trying to understand what $C_0^2$ means and how to check whether a function belongs to it. The course I'm ...
2
votes
2answers
32 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} ...
1
vote
0answers
33 views

Why does the price term in Vega disappear for a European call option?

In my course, I have been asked to prove a number of statements about "the Greeks" from the Black-Scholes model for pricing a European call option with no dividends and a strike price of $K$. One of ...
0
votes
0answers
128 views

Expectation of a Poisson Process

Cars pass a certain street location according to a Poisson Process with rate $\lambda$. An old lady and her trusty boyscout want to cross the street at this location. They wait until they can ensure ...
3
votes
0answers
44 views

Conditional expectation and coupled set of ODEs

How to find a coupled set of ODEs and initial conditions for the deterministic functions $a$ and $b$ such that $$\mathbb{E}\left[e^{-\int_{t}^{T} W^2(u)du} | \mathcal{F(t)}\right] = e^{-a(T-t) - ...
1
vote
1answer
64 views

Expectation of stochastic differential equation

I have solved the nonlinear stochastic equation $dX_t=\frac{1}{2}a(a-1)X_t^{1-2/a}dt+aX_t^{1-1/a}dW_{t}$, by reducing it to a linear one (change of variables $Y_{t}=X_{t}^{1/a}$ and applying Ito ...
0
votes
1answer
44 views

Expectation of stopping times

Let B = (Bt)t¸0 be a standard Brownian motion started at zero, let $X_t$ be a non negative stochastic process solving: $dX_t=1/X_tdt+dB_t$ Compute $E[\sigma]$ when $\sigma=\inf \{ t\ge 0 : X_t= 1 ...
1
vote
2answers
92 views

Martingale Proofs

I havent been able to find an analogous question and our textbook is lacking in good examples, so I could use a little help with this rather straight forward martingale problem: Let X=(Xn) be a ...
1
vote
0answers
39 views

is this solution correct about joint distribution?

The question is if $x,y,z$ are independent $x\sim\exp(\lambda), y\sim\exp(\mu), z\sim\exp(\gamma)$ and define $u=\min(x,y), v=\min(y,z)$ what is the probability $p(U>u,V>v)$. Consider the cases ...
1
vote
1answer
65 views

Advanced urn problem

Imagine there are two urns — urn A and urn B. Urn A contains 3 blue balls and 7 red balls. Urn B contains 7 blue balls and 3 red balls. Balls are now randomly drawn from one of these urns where the ...
1
vote
2answers
64 views

$X_1,X_2,…$ real independent random variables $\not\implies S_n = \frac{1}{n}(X_1+ \cdots + X_n)$ are independent

Is it possible to show the following Lemma: Given independent real random variables $(X_i)_{i\in \mathbb{N}}$, then $(S_n := \frac{1}{n}(X_1+\cdots + X_n))_{n \in \mathbb{N}}$ are independent as ...
2
votes
1answer
143 views

Ito differential equation

Define $$X_t := \left( \begin{matrix} \cos W_t \\ \sin W_t \end{matrix} \right).$$ where $W = \left( W_t,\mathcal F_t \right) _{t\ge0}$ is a standard Wiener process. Find the Ito differential of X ...
1
vote
2answers
30 views

Algebraic problem for satisfying a given equation

I'm trying to solve the following exercise: My backward equation looks like: $P_{i,j}'(t) = i\lambda P_{i+1,j}(t) - i\lambda P_{i,j}(t) $ So i started with differentiating $P_{i,j}(t)$: ${j-1 ...
1
vote
2answers
43 views

determine whether an integral is positive

Given a standardized normal variable $X\sim N\left(0,1\right)$, and constants $ \kappa \in \left[0,1\right)$ and $\tau \in \mathbb{R}$, I want to sign the following expression: \begin{equation} ...
0
votes
1answer
38 views

For which p>0 does $S_t=W_t+t^p$ admit an equivalent martingale measure?

Let W be a brownian motion and p>0. For which p does $S_t=W_t+t^p$ admit an equivalent martingale measure? I recently saw at my lectures that NFLVR cond: There does not exist a sequence $\{H_n\}_{n ...
1
vote
1answer
310 views

Haar basis on $L^2(0,1)$ - proof?

I have the following problem. We defined $\mathbb{H}=\{f_0,\quad f_{j,n} \quad j=1,...,2^{n-1} \quad n=1,2,...\}$ where for all $t\in[0,1]$ we put $f_0(t)=1$ and setting $K=2j-1$, $$f_{j,n}(t)=\left\{ ...
1
vote
1answer
392 views

Finding the distribution of an Ito integral. $\int_0^t sB_s \, \mathrm{d}s$

I'm a little baffled by this, I'm supposed to find the distribution of $X_t$ where, $X_t=\int_0^t sB_s \, \mathrm{d}s$. What I can think of is to consider the process $$\begin{align} Y_s &= ...
3
votes
1answer
53 views

two r.v sharing the same law

I have a question: Let $X=B^{+}$ or $X=|B|$ where $B$ is the standard Brownian motion. Set $$J_p=\sup_{t\geq 0}(X_t-t^{\frac{p}{2}})$$ where $p>1$ and $q$ its conjugate ...
1
vote
1answer
31 views

a homework question about Levy air

I have a question in my homework: Let $X_t$ and $Y_t$ be two Brownian motions issue de $0$ and define $$S_t=\int_0^tX_s\,dY_s-\int_0^tY_s\,dX_s$$ Show that $$E[e^{i\lambda S_t}]=E[\cos(\lambda ...
2
votes
1answer
372 views

Expectation of exponential martingale and indicator function.

Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$. I want to evaluate $$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...
1
vote
0answers
165 views

Independent Exponentially Distributed Random Variables - Athletes Problem??

Q) At a javalin competition two athletes (1 & 2) are competing against each other. Each has one attempt to throw the javalin. Assume the acheived distance of a throw ($L$1 & $L2$) [note these ...
1
vote
0answers
189 views

Integral with respect to Wiener process.

Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process. My First Question What is ...
3
votes
1answer
450 views

Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$

Solve the SDE: $dX(t) = udt + \sigma X(t)dB(t)$ Provided Question The SDE is $dX(t) = udt + \sigma X(t)dB(t)$. Find $X(t)$, where $X(t)$ is some stochastic process and $B(t)$ is a Wiener process. ...
1
vote
1answer
32 views

Integrating $d(e^{-ut}X(t))$, where $X(t)$ is stochastic.

Given that $\sigma e^{-ut}dB(t) = d(e^{-ut}X(t))$, where $X(t)$ is a stochastic process and $B(t)$ is a Wiener process, we have that: $$ \int_0^t d(e^{-ut}X(s)) = X(0) + \sigma \int_0^t e^{-us}dB(s) ...
1
vote
1answer
90 views

Derive an SDE for $B^2(t)$, where $B(t)$ is standard Brownian Motion

Original Question: Derive an SDE for $B^2(t)$, where $B(t)$ is standard Brownian Motion. Attempt at an answer: Apply Ito's calculus over $f(t,b):= B^2(t)$. $$df(t,b) = \frac{\partial ...
1
vote
1answer
468 views

Min of two stopping times is also a stopping time.

Preface: I'm having trouble with the correct solution. The Original Question: Given that $\mathscr{F}_t$ is a filtration that satisfies all the usual conditions, and given ...
1
vote
2answers
172 views

Generating function of the stopping time

Let $X_t$ be a generalized Wiener process with drift rate $\mu$ and variance $\sigma^2$, and let $\tau$ be the stopping time $$\tau:=\inf \left\{ t\geq0: X_t= b\right\}, \quad b\geq0 $$ Can anyone ...
1
vote
1answer
142 views

proof of a lemma on stopping times

Hi I got some problem with the proof of this lemma which is left as excercise Let $(F_t)$ be a filtration and S,T be stopping times, then show $ \{S=T\},\{S≤T\},\{S<T\} \in F_S\bigcap F_T$. Could ...
0
votes
1answer
93 views

How to solve the expected value of such integral

$$ V_t = E^\mathbb{Q} \left[\int_t^{+\infty} e^{-r(u-t)}X_u \, du|X_t\right] $$ with a given process $ X_t $ satisfied: $dX_t = (\mu-\sigma^2 \gamma) X_t \, dt + \sigma X_t \, dW_t^\mathbb{Q}$
2
votes
0answers
126 views

Finding an SDE which satisfies $X(t)$

I am attempting the following problem, and was hoping if you guys could provide any feedback on whether my approach is valid. Thank you in advance for your time! The question is as follows: "Let ...
2
votes
0answers
148 views

Solving a SDE and finding its related moments

I am attempting to answer this multi-part question, and hope you can provide any feedback on any of my workings. My apologies for the length and thank you in advance for any help! i) Let $g$ be a ...