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

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18 views

A problem on Ito integral [duplicate]

Let $W$ be a standard, one-dimensional Brownian motion. Let $T\in(0,+\infty)$. Then $$\lim_{\beta\to+\infty}\sup_{0\le t\le T}\left|e^{-\beta t}\int_0^te^{\beta ...
0
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1answer
16 views

ratio of two functions of finite variation

According to the Jordan decomposition, a necessary and sufficient condition for a function to have finite variation is that it can be expressed as the difference of two increasing functions. The book ...
3
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1answer
75 views

What is the explicit obstruction to almost sure convergence in stochastic integrals?

Let $B(\omega,t)$ be a Brownian motion defined on some appropriately filtered probability space $(\Omega,\mathcal{F}_{t},\mathbb{P})$, and let $f(\omega,t)$ be a stochastic process defined on $\Omega$ ...
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0answers
58 views

What is some reason that there are no book bridge the gap of these three books

I am referring to the (beginner's text- Stochastic Calculus by Mircea Grigoriu and Introduction to Stochastic Calculus by klebaner.) and the advanced texts - stochastic differential equation by ...
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0answers
17 views

Density of this quantity for a Geometric Brownian Motion?

If we define $X_T = X_t e^{(\mu-\frac{1}{2}\sigma^2 ) (T-t) + \sigma W_{T-t}}$ where $W_{T-t}$ is a classical weiner process. How would we go about deriving the density and expectation for $X_{max} - ...
-2
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1answer
65 views

Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ [closed]

Solve the following stochastic differential equations $ dX_t = \frac{1}{2 X_t} dt + dB_t$ or equivalently with a transformation $Y_t = X_t^2$ $ dY_t = dt + 2 \sqrt{Y_t} dB_t$ with $Y_0 = y_0 > ...
0
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1answer
37 views

The smallest filtration for which a sequence of random variables is adapted

Let $X_1, ..., X_n$ be a sequence of random variables. Show that $\hspace{60pt}$ $\mathcal{F}_n$ = $\sigma(X_1, ..., X_n)$ is the smallest filtration such that the sequence $X_1, ..., X_n$ is ...
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0answers
21 views

How to determine the probability density function, ${f_{\dot X}}\left( {\dot x} \right)$, for the derivative process of a stochastic process?

I would like to calculate the up-crossing rate ($\nu _a^ + $) for a stationary stochastic process, $X(t)$, given by the probability distribution function of its 'intensity', ${f_X}\left( x \right)$, ...
3
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1answer
40 views

Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

Suppose we have an sde of the form: \begin{eqnarray} dX_t=b(X_t)dX_t + \sigma (X_t)dB_t \end{eqnarray} where $b$ and $\sigma$ are Lipschitz. Then we have existence and uniqueness of the solution ...
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0answers
21 views

Solution of $dX_{t}=(sin(X_{t})+2)dB_{t}$

I am curious if $dX_{t}=(sin(X_{t})+2)dB_{t}$ has a solution i.e $X_{t}$=(stuff in terms of $B_{t}$). What about for $dX_{t}=\sigma(X_{t})dB_{t}$, where $0<\gamma^{-1}\leq \sigma\leq ...
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0answers
12 views

density of $X_{t}$ satisfying $dX_{t}=dB_{t}-V'(X_{t})dt$

find density of $X_{t}$ satisfying $dX_{t}=dB_{t}-V'(X_{t})dt$ where $V(x)=\frac{x^{2}}{2}+W(x)$ and $x_{0}$ has density $\frac{e^{-V(x)}}{\int e^{-V(y)}dy}$ and W(x) is smoothly compactly ...
2
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1answer
34 views

Density of $\int_{0}^{t}W'(B_{s})ds$ where $W'$ is smooth and compactly supported.

Only hints please Density of $\int_{0}^{t}W'(B_{s})ds$, where $B_{s}$ is 1-d Brownian motion. The density of $Y_{s}:=W'(B_{s})$ is $g_{Y}(y)=p_{B_{s}}((W')^{-1}(y))|\frac{d(W')^{-1}(y)}{dy}|$. How ...
2
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1answer
54 views

Compute almost sure limit of martingale?

Let $Y_1, Y_2, \dots$ be nonnegative i.i.d random variables with mean 1. Let $$X_n = \prod_{m \le n}Y_m$$ If $P(Y = 1) < 1$, prove that $\lim_{n->\infty}X_n = 0$ almost surely. I feel like ...
0
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1answer
23 views

Ito integral's zero mean

My Sto Cal prof gave a long proof for the fact that $E[\int_{0}^{t} f_s dW_s] = 0$ where W is Brownian and f is Borel x $\mathscr{F}$-measurable, adapted and satisfies some integrability condition. ...
1
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1answer
32 views

Brownian motion on the circle and Itô processes

Consider the differential system \begin{cases} dX_t &=& -\frac{1}{2}X_t dt - Y_tdB_t, \\ dY_t &=& -\frac{1}{2}Y_tdt + X_tdB_t, \end{cases} $X_0 = 1$, $Y_0 = 0$. Let $X_t$ and $Y_t$ ...
4
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1answer
119 views

Solving a PDE with Feynman-Kac Formula

I'm trying to solve this PDE using Feynman-Kac formula Now i follow the regular steps Here is where I don't know how to proceed. How do I calculate this expectation?
1
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1answer
21 views

Basic question on application of Itô's formula to a stochastic process

I am working on a problem where I now find myself wanting to apply Itô's formula to: \begin{equation} X_t = \exp(W_t -W_0-\frac{t}{2}+\int\limits_0^tX_sds) \end{equation} where $W_t$ is 1D Brownian ...
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2answers
56 views

Verifying Property of Stochastic Integral

I am trying to verify this simple property for a stochastic integral. Given that f(t,w) is a bounded, nonanticipating function for a given Wiener process $W_t$ show that $E((\int_{0}^{T} f(s,w) ...
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0answers
159 views

A problem with kernels on measurable spaces

Let $(E, \mathcal{B}(E)), (F, \mathcal{B}(F))$ be two measurable spaces. A $kernel$ from $(E, \mathcal{B}(E))$ to $(F, \mathcal{B}(F))$ is an application $N : p \mathcal{B} (E) \rightarrow p ...
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0answers
119 views

Condition for a process to be a supermartingale

I am struggling in this question: Let $W$ denote a Brownian motion. Given that $ X_t = e^{- \lambda t} X_0 + \int_0^t \sigma e^{- \lambda (t-s)} \,dW_s$ solves the SDE \begin{equation} dX_t = - ...
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1answer
31 views

Prove the process is a martingale with respect to the natural filtration

Let $\{M_n\}_{n\ge 0}$ be a symmetric simple random walk. Fix a real $b$. Prove that the process $S_n = e^{bM_n} (\frac{2}{e^b + e^{-b}})^n$, $n = 0,1,2,....$, is a martingale w.r.t. the natural ...
2
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2answers
49 views

What does it mean u(dx) in the Fourier transform of a probability measure u?

Let $\mu$ be a probability on $\mathbb{R}^n$ and consider its Fourier transform $\overset{\wedge}{\mu} (u) = \int e^{i (u ,x)} \mu( dx)$, where $(u, x)$ is the scalar product of $u$ and $x$. What ...
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0answers
38 views

What's the distribution of $X = \int^{1}_{0}udB_{u}$? [closed]

Let $X = \int^{1}_{0}udB_{u}$,where $B_{u}$is the Brown motion. What's the distribution of $X$? The Stochastic integral calculate in the sense of Ito.
1
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1answer
29 views

How to differentiate the Black-Scholes formula w.r.t. volatility

The Black-Scholes-Merton formula for determining call option value is given as: $$C(S,K,\sigma,r,\tau)=N(d_1)S-N(d_2)Ke^{-rT}$$ where $N(d_i)$ is the standard normal distribution and ...
3
votes
1answer
106 views

perfectly correlated processes

I am really stuck in this question: Let $\{S_t\}$ and $\{S'_t\}$ be two stochastic processes, satisfying \begin{equation} dS_t = S_t ( \sigma_t \,dB_t + r_t \,dt), \quad dS'_t = S'_t (\sigma'_t ...
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0answers
32 views

Equality of two spaces of stochastic processes

Let $(\Omega, \mathcal F, P, \mathcal F_t)$ be a filtered probability space. Consider two spaces $M$ and $S$ defines as follows: $M$ is a collection of all continuous $\mathcal F_t$-adapted processes ...
2
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0answers
29 views

On Borell's Theorem (Gaussian processes)

Let ${X(t):t \geq 0}$ be a Gaussian process with mean $0$ and bounded (with probability $1$) sample paths. Borell's Theorem states then that for all $u>0$ we have $$P(\sup_{t \geq 0} X(t)>u) ...
5
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0answers
126 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
1
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0answers
43 views

Ito's Product Formula?

I'm asked to consider three Ito processes $(X(t), t \ge 0)$, $(Y(t), t \ge 0)$, and $(Z(t), t \ge 0)$. I am asked to show: $$d(X(t)Y(t)Z(t)) = X(t)Y(t)dZ(t) + X(t)Z(t)dY(t) + Y(t)Z(t)dX(t) + ...
2
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0answers
42 views

Why is the pathwise integral of $\alpha_s$ w.r.t the Lebesgue measure continuous?

My class notes on stochastic calculus say that the if $(\alpha_s(\omega))_{s\in \mathbb{R_+}}$ is progressive then $\int_0^t \alpha_s ds$ is a pathwise continuous process? How does the joint ...
1
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1answer
51 views

Uniform integrable proof

Lets be $E[\sup_{t\in[0,T]}e^{-rt}\Psi(S_{t})]<\infty$. I want to show that $J_{t}$ defined by \begin{align} J_{t}:=\mathrm{ess sup}_{\tau \in \mathcal F_{t,T}}E[e^{-r\tau}\Psi(S_{\tau})|\mathcal ...
0
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0answers
37 views

Girsanov's formula for an Ornstein-Uhlenbeck process

This is homework so no answers please. Question:If I know that for an OU process $X_t\stackrel{d}{=}e^{-t} B_{e^{2t}}$, can I use that for the Radon-Nikodym derivative of $X_t$? Context and Attempt ...
2
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0answers
33 views

Challenging CDF of $\sup_t|B_t|$ ($B_t$ is a Brownian Bridge)

Question 1: Let $W_t$ be a Brownian motion. Then how could we prove that $$\Pr\left\{\sup_t|W_t|<b\right\}=1-\frac{4}{\pi}\sum_{j=1}^\infty \frac{(-1)^j}{2j+1} ...
0
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0answers
32 views

Proof of equality in Expectation with the Help of a Brownian Motion (Put-Call-Symmetry)

Hey I want to reproduce a proof of Damien Lamberton; proof begins at page 14. Under some assumptions i want to show that \begin{align} \sup_{t\in \mathcal T_{0,T}}\mathbb ...
3
votes
1answer
71 views

conditional expectation of some solution of SDE

Let $(M_t)$ be a nonnegative martingale in a probability space $(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, \mathbb{P} )$ given by \begin{equation} dM_t = M_t \sigma_t dW_t \end{equation} for some ...
1
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1answer
26 views

deterministic expression of stochastic integral

Let $(M_t)$ be a non-negative martingale on a probability space $(\Omega, \mathcal{F}, \{ \mathcal{F}_n \} , \mathbb{P})$. Let $dM_t = M_t dW_t$. How can we write the following \begin{equation} ...
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1answer
32 views

Approximation of stochastic integrals by Riemann sums

I know that for $f:[0,1]\to \mathbb{R}$, the Riemann Integral converges in the sense that $$\sum_{k=1}^Kf(t_k)(t_{k} - t_{k-1})\longrightarrow \int_0^1f(t)dt$$ as the grid becomes smaller and smaller. ...
2
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0answers
36 views

Pricing/Valuation of American Options

Hi i'm a litte bit confused by the pricing valuation of American options. For simple Assumtions on the Blacksholes Model and no dividends, and constant rates else one can show, that for a given ...
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2answers
40 views

Change of measure on Brownian motion

I have a small doubt as I am currently self-studying stochastic calculus. In Brownian motion part, the author talked about change of probability measure over Brownian motions. Now we we know that ...
1
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1answer
33 views

Definition of Itô process

Let $\lambda_t$ and $r_t$ be predictable processes and suppose that $\int_{0}^t | \lambda_s |^2 \,ds < +\infty$, for all $t>0$. We define \begin{equation} Y_t = Y_0 \,\text{exp} \bigg\{ ...
2
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1answer
26 views

Quadratic covariation of Martingales

I was succesful at showing that the quadratic covariation $\langle\cdot ,\cdot \rangle_t$ is a positiv semidefinit, symmetric and bilinear form for each $t$ on the set of local martigales. So the ...
2
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0answers
20 views

Question on perpetual American put

Define $u(x):=\sup_{\tau \in T_{0,\infty}}E[e^{-r\tau}(K-S_{\tau})_{+}1_{\tau<\infty}$]. $T_{0,\infty}$ the set of stopping times taking values in $[0,\infty)$ and ...
2
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1answer
32 views

higher moments of a r.v., combinatorical problem

I'm studying the book of Rick Durrett, I want to understand the proof of the Erdös Kac central limit theorem, so I also need to understand the Lindeberg-Feller theorem: for every $n \in \mathbb{N}$ ...
2
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1answer
48 views

Requirements for square integrable in the Doob-Meyer-Decomposition

Hey i have given a non negative supermartingale $(J_{t})_{t\in[0,T]}$ of Class D. So there exists a Doob meyer decomposition $J_{t}=M_{t}-A_{t}$ where $M_{t}$ is uniformly integrable since $(J_{t})$ ...
1
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1answer
43 views

Inequality of an expectation (here: perpetual put of an american option)

for a given function $u(x):=\sup_{\tau \in T_{0,\infty}}E[(Ke^{-r\tau}-xe^{\sigma B_{\tau}-(\sigma^{2}\tau)/2})_{+}1_{\tau <\infty}]$ and $x \in [0,\infty)$, K a positive real number, $(B_{t})$ a ...
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1answer
31 views

When does $\int_0^t dX_s = X_t-X_0$ hold for a stochastic process?

So I am learning stochastic calculus and I have seen this relationship be used many times: $$ \int_0^t dX_s = X_t-X_0 $$ where $X_t$ is some stochastic process. It looks like some sort of ...
0
votes
0answers
7 views

Is $f \in \mathbb{C(R)}$ in $\Lambda^2_{\text{loc}}$

I know that $\Lambda^2_{\text{loc}}=\{\Phi $ progressive $\mid \int_0^t \Phi^2 ds <\infty$ , $\forall t \geq0 \}$ Now since any process which is right/left continuous and adapted is progressive ...
1
vote
1answer
29 views

show that the solution is a local martingale iff it has zero drift

Most financial maths textbook state the following: Given an $n$-dimensional Ito-process defined by \begin{equation} X_t = X_0 + \int_0^{t} \alpha_s \,d W_s + \int_0^{t} \beta_s \,d s, \end{equation} ...
0
votes
0answers
18 views

Why are these processes indistinguishable?

My class notes on Stochastic Calculus says that processes in $\mathbb{M_c}^{loc}, \mathbb{A}_c $ and $\mathbb{V}_c$ where they have their usual meaning, are indistinguishable of continuous processes. ...
0
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0answers
23 views

Basic question about application of Ito's formula

I am a complete beginner in stochastic calculus, and I am looking at a calculation of $d(W_t^2)$ where $W_t$ is a Brownian motion, using Ito's formula $$ df(W_t) = f'(W_t)dW_t+ \frac{1}{2}f''(W_t)dt ...