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

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0
votes
1answer
28 views

partial derivative of $f(X(t),t)$ with respect to $t$

Suppose that $f(x,t) = x^2$. Clearly, $\frac{\partial f}{\partial t} = 0$. However, let us now consider $f(X(t),t) = X(t)^2$. The book I am reading claims that $\frac{\partial f}{\partial t}(X(t),t) ...
5
votes
1answer
269 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
1
vote
1answer
980 views

Easy proof of Black-Scholes option pricing formula

I use this Book to read the option princing in Black-Scholes model in pages 93-99, The poof of the formula given by $$c(s,t)= N(d_1(s,t)- Ke^{-rT}N(d_2(s,t)))$$ where $$d_{1,2}=\frac{\ln(s/K)+(r\pm ...
3
votes
0answers
25 views

Regarding proof of converse to Girsanovs theorem

This is regarding an argument from Arbitrage Theory by Thomas Björk - Theorem 11.6, but is attempted self contained. Consider a Wiener process W on probability space ...
1
vote
1answer
48 views

Conditional likelihood of continuously-combounded returns

The simplest possible asset pricing model ist the geometric brownian motion for asset price. Here the price $S_t$ solve the familar $$dS_t = (\mu +0.5 \sigma^2)S_t \, dt + \sigma S_t \, ...
1
vote
0answers
67 views

How to derive this step in a book called Brownian motion calculus?

How to derive the step which result in the magnitude of slope of the path in section 1.8.2? I know it is not the definition itself.
0
votes
1answer
17 views

Covariance of Wiener Processes on the same Brownian Motion

I am trying to solve $Cov(Tw_T,\int^{T}_{0}tdw_t)=\mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t]$, my attempt is as below: \begin{split} \mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t] & ...
0
votes
0answers
19 views

Clarifications about SDEs, Differentials & Derivatives

A general SDE look like the following: $$ \mathrm{d}\psi=a\mathop{}\!\mathrm{d}t+b\mathop{}\!\mathrm{d}W,\tag{1} $$ where $\psi:t\mapsto y = \psi(t)$ is the solution, while $a$ and $b$ can be both, ...
0
votes
0answers
18 views

Stochastic Calculus Research topics

If I just finished taking a Stochastic Calculus course and wanted to do research on it, what are some hot topics currently going on in this area? I know there's a lot but if I wanted to investigate ...
0
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0answers
37 views

Is this stochastic differential equation wrong?

The following is an old exam question I think might be misstated. Consider the SDE $$dX(u)=(a(u)+b(u)X(u))\,du+(\gamma(u)+\sigma(u)X(u))\,dW(u)$$ where $W(u)$ is a brownian motion relative to ...
1
vote
0answers
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 ...
3
votes
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 $\Omega$ ...
2
votes
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 ...
3
votes
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 ...
0
votes
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 ...
0
votes
1answer
96 views

Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
-2
votes
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
votes
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) ...
0
votes
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 ...
0
votes
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} - ...
0
votes
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 ...
1
vote
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', ${f_X}\left( x \right)$, ...
0
votes
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 ...
0
votes
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 ...
4
votes
1answer
121 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
vote
2answers
126 views

Variance of sum of two ito integrals

I don't really understand how to solve the following problem: Var(X) where X = $\int_0^2 2t dW(t) + \int_4^6 W(t) dW(t)$ If I use $E [(A+B)^2] = E(A^2) + E(B^2) + 2E(AB)$ I get to the point where I ...
2
votes
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 ...
1
vote
1answer
62 views

Stochastic differential equation for $Y(t)=\sqrt{X(t)}$

Assume that $X(t)$ solves the stochastic differential equation $$dX(t)=\sigma(t)dW(t)+\mu(t)dt$$ with $\mu(x)=bx+c$ and $\sigma^2(x)=4x.$ Assume that $X(t)\ge 0$. Find the stochastic differential ...
1
vote
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 = - ...
0
votes
1answer
25 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
vote
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 $Y_t$ ...
1
vote
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 ...
14
votes
2answers
2k views

What are the prerequisites for stochastic calculus?

I am not a math student, and only kind of picking up something whenever I need it. After emerged in the field of machine learning, probability, measure theory and functional analysis seem to be quite ...
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 ...
1
vote
0answers
160 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 ...
2
votes
0answers
43 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 ...
0
votes
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 ...
1
vote
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 ...
-1
votes
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.
2
votes
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 $n \in \mathbb{N}$ ...
0
votes
0answers
33 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 ...
5
votes
0answers
127 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} ...
2
votes
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) ...
1
vote
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
votes
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
votes
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. ...
1
vote
1answer
52 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
votes
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 ...
0
votes
0answers
33 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 ...
1
vote
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} ...