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

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43 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 ...
0
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0answers
5 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} - ...
-1
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1answer
34 views

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

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 > ...
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1answer
20 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
12 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', ${f_X}\left( x \right)$, ...
2
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0answers
23 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
12 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
7 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 ...
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0answers
22 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 ...
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0answers
12 views

Apply Ito's formula to Bessel prosess [on hold]

Let $X_t=\sqrt{(B_t^1)^2+(B_t^2)^2+(B_t^3)^2}$ be a Bessel process (starting from 0) with respect to a 3-dimentional standard Brownian Motion $B_t=(B_t^1,B_t^2,B_t^3)$. How to apply Ito's formula to ...
2
votes
1answer
48 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
16 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. ...
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0answers
21 views

Calculating expectation function and covariance function

Let $E_n(t)$ denote the empirical cdf based on iid uniform $u[0,1]$ random variables $U_1,...,U_n.$ The corresponding uniform empirical process $(e_n(t),0\leq t\leq 1)$ is given by ...
1
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1answer
28 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
101 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
15 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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0answers
13 views

Ito integral and Riemann integral : Order of integration [closed]

I am trying to simplify this quantity : $$ J_{a}(t) = \int_{s=0}^{t}\left(\int_{u=0}^{s}e^{-a(s-u)}dW(u)\right)ds $$ Where $W$ is a standard brownian motion. Any help ? Thanks in advance,
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2answers
48 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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1answer
26 views

How can I calculate the integral of W(t)^2dt t from 0 to 1 [closed]

How to calculate $$\int_0^1[W(t)]^2\ dt$$
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0answers
69 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
15 views

Express S_t as a solution of a SDE [closed]

I've seen many examples of using Ito's Lemma to solve an SDE but I'm not sure how to proceed in the reverse direction. The question I'm looking at in particular is $S_t=50e^{\mu t+\sigma W_t} $ and ...
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0answers
94 views
+50

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 = - ...
0
votes
1answer
26 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
votes
1answer
40 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
33 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
19 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 ...
-1
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0answers
26 views

Solve SDE $dX_t = tX_tdt + e^{t^2/2}dW_t$

Solve $dX_t = tX_tdt + e^{t^2/2}dW_t, X_0 = \alpha$ by considering $X_t = a(t)(X_0 + \int_0^t b(s) dW_s)$, where a(t) and b(t) are not random. I don't think I quite understand stochastic ...
3
votes
1answer
104 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
25 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
20 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
118 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
38 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
37 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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0answers
16 views

Girsanov's Theorem - Change of Measure

I have trouble understanding Girsanov's theorem. The Radon Nikodym process $Z$ is defined by: $$Z(t)=\exp\left(-\int_0^t\phi(u) \, dW(u) - \int_0^t\frac{\phi^2}{2} \, du\right)$$ Now $\hat P$ is a ...
1
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1answer
46 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
31 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
32 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} ...
2
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0answers
20 views

Finding a stochastic process that satisfies a few constraints

In a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, let $ \{ \mathcal{F}_t \} $ be a filtration generated by the Brownian motion $W$. Let $\mu$ and $\sigma$ be predictable processes such that ...
0
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0answers
27 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
63 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
23 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} ...
0
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1answer
27 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
33 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 ...
0
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0answers
21 views

Textbook recommendations for stochastic dynamical systems

There are many excellent introductory books on analysing nonlinear dynamical systems such as obtaining stability and bifurcations, e.g. Strogatz or Hirsch, Devaney and Smale or Wiggins. I'm finding ...
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2answers
29 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
30 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
votes
1answer
22 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
16 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
votes
1answer
30 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
votes
1answer
42 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 $(J_{t})$ ...