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

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1answer
202 views

Integrate Brownian motion with respect to independent Brownian motion

we are wondering what can be said about the distribution of that? We are considering standard Brownian motions and the integral from 0 to 1... More precisely: What can be said about the distribution ...
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2answers
381 views

Sum of independent random variables almost surely constant

I am trying to solve the following problem: Let $(\Omega, \mathbb{A}, \mathbb{P})$ be a probability space and $X_1, X_2, \ldots, X_n$ independent real random variables. Prove that the sum $X_1 + X_2 ...
2
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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) - ...
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1answer
79 views

Readings necessary to understand Ito Integrals?

I searched for this question but couldn't find a direct answer. Basically I want to understand (and possibly compute some simple instances of) the Ito integral. I am coming from a physics background ...
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0answers
178 views

Explaining Ito formula to an analyst

From the point of view of analysis, what is Ito formula? Is it an integral by substitution, or, a radon-nikodym derivative? Define the probability space $$ \left(C\left(\Bbb ...
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1answer
36 views

Is it true that $X(t)^a > K \iff X(t) > K^\frac1a$

Let $a \in \mathbb{N}$, $K \in \mathbb{R^+}$ and $X(t)$ be a geometric Brownian Motion. Is the following true? $$X(t)^a > K \iff X(t) > K^\frac1a$$ The context of the above is that I want to ...
2
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1answer
117 views

Bounded variation and continuous local part when using Ito's Formula

When we apply Ito's Formula to a continuous semimartingale, which is the bounded variation part and which is the continuous local martingale part? Is there a general rule or does it depend on the ...
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1answer
481 views

covariance of integral of Brownian

What is the covariance of the process $X(t) = \int_0^t B(u)\,du$ where $B$ is a standard Brownian motion? i.e., I wish to find $E[X(t)X(s)]$, for $0<s<t<\infty$. Any ideas? Thanks you very ...
3
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1answer
175 views

Optional sampling exercise

I came across the following exercise in Stochastic Calculus: Let $B=(B_t)_{t\geq0}$ be a standard Brownian motion. Let also $M$ be the following process: $M_t=B^4_t-6t(B^2_t-\dfrac{t}{3})$ for ...
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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 ...
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1answer
186 views

$dt$ terms have zero quadratic variation

Why does $ds$ integral have zero quadratic variation? Even if I have a integral of the form $$\int X_s ds$$ where $X$ is a stochastic process? I know that a continuous process of finite variation ...
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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 ...
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0answers
57 views

Condition on $f$ for $e^{\int_0^t f(B_s)ds}$ to be of finite variation

Let $B$ be a standard Brownian motion, and, $$ X_t=e^{\int_0^t f(B_s)ds}, $$ for some function $f$. What are the condition on $f$ for $X_t$ to be of finite variation? Let $Y_t=\int_0^t f(B_s)ds$, if ...
2
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1answer
143 views

Quadratic variation of $X_t=\int_0^t B_s \, ds$

Let $B$ be a standard brownian motion and $$ X_t=\int_0^t B_s \, ds. $$ What is the quadratic variation $[X]_t$ of $X$? I see $dX_t$ as an sde with drift term $B_t$.
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147 views

Multivariate Stochastic Process

I am dealing with a multivariate Ornstein Uhlenbeck style SDE. Specifically $dx_{t,j}=\kappa_{j}(x_{t,j-1}-x_{t,j})dt+\sigma dW_{t,j} $ here j=1,2,...,6 , $x_{t,0}=\theta$ , ...
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0answers
625 views

Stochastic integral: Interchanging the order of expectation and integration

Let $B$ be a standard Brownian motion and $$ X_t=\int_0^t f_s ds+\int_0^t g_s dB_s, $$ where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$. Is it true that $$ ...
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0answers
76 views

Local martingale iff each component is a local martingale?

This is probably an easy question: A local martingale is an adapted, cadlag process for which there is an increasing sequence of stopping times (going to $\infty$) such that the stopped process is a ...
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1answer
68 views

Condition for existence of a stochastic differential equation

With $B$ a standard Brownian motion, write $$ dX_t=f_tdt+g_tdB_t. $$ What are the conditions on $\left(f\right)_{t\ge 0}$ and $\left(g\right)_{t\ge 0}$ for $X_t$ to exists? I think ...
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1answer
272 views

Integration of Wiener process: $\int_{t_1}^{t_2} dB(s)$

We all know that $\int_0^t dB(s) = B(t)$, where $B(t)$ is a standard Brownian Motion. However, is the following identity true? Also, why or why not? $\boxed{ \displaystyle \ \ \int_{t_1}^{t_2} ...
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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. ...
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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) ...
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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 ...
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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 ...
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3answers
1k views

Itō Integral has expectation zero

I have a question about the following property, which I didn't know so far: Why does the Itō integral have zero expectation? Is this true for every integrator and integrand? Or is this restricted ...
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1answer
86 views

Rephrasing a Stochastic Process as a Stochastic Differential Equation

I have a continuous-time stochastic process $X$, described as follows: (1) If the process is at $x_0$ at time $t_0$, then the function $f(t_f, x_f \, | \, t_0, x_0)$ is a PDF in the parameter $x_f$ ...
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4answers
2k views

Where to begin in approaching Stochastic Calculus?

I have experience in Abstract algebra (up to Galois theory), Real Analysis(baby Rudin except for the measure integral) and probability theory up to Brownian motion(non-rigorous treatment). Is there a ...
2
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2answers
99 views

Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$

Is there a way to solve $$ \partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0? $$ This appeared as a condition for $$ X_t=u(t,B_t)e^{\int_0^tv(B_s)ds} $$ to be a martingale. With $B$ a ...
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1answer
880 views

Applying Ito formula to the Brownian bridge

Let $B$ be a standard Brownian motion and $$ W_t=(1-t)\int_0^t \frac{1}{1-s}dB_s $$ be a Brownian bridge. Calculate $dW_t$. To apply Ito formula define $$ f(t,B_t)=(1-t) \int_0^t\frac{1}{1-s}dB_s $$ ...
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1answer
346 views

superlinear and convex function

Assume $X = \mbox{random variable} X>0$, $\mathbb{E}X<\infty \implies \exists \phi:\mathbb R_+\to\mathbb R_+$, superlinar, convex with $\mathbb{E}\phi(X)<\infty$ superlinear means ...
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0answers
287 views

Analogue of Leibniz Rule for Stochastic Integrals

Suppose $$f(t,u)=f(0,u)+\int_0^t{\mu (w,u)dw}+\int_0^t{\sigma(w,u)dB_w}$$, where $B_w$ is a standard Brownian motion. I would like to calculus the drift and diffusion of $Y_t=-\int_t^s{f(t,u)du}$ ...
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1answer
71 views

Why all previsible processes are also optional?

My doubt concerns a step on demonstration of the inclusion of the set of previsible processes in the set of optional processes. The idea of the demonstration consists in: Given a filtered ...
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0answers
94 views

general semimartingale theory

Last semester I attended a course about stochastic calculus. There we constructed the stochastic integral with respect to continuous semimartingales. We restrict ourselves to the continuous case. ...
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1answer
282 views

Levy's theorem? [duplicate]

Possible Duplicate: how to show convergence in probability imply convergence a.s. in this case? Good evenig! I stumbled upon this theorem: For independent random variables ...
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1answer
30 views

Economics simplification of stochastic transition of capital

I'm taking an macro-econ paper and I can't seem to work out the following simplification. Basically somehow by combining equation 4.14 and ...
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0answers
46 views

If $S_{t}$ satisfies $dS_{t}=\frac{1}{S_{t}}1_{(S_{t}>0)}dB_{t}$, will $S_{t}$ be a martingale?

If the process $S=S_{t}$ satisfies the SDE: $$dS_{t}=\frac{1}{S_{t}}1_{(S_{t}>0)}dB_{t}, \ S_{0}=1.$$ will $S_{t}$ be a martingale? It seems reasonable to say so because $S_{t}$ is clearly ...
3
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1answer
55 views

$E \left\{ \left( \sum_{i=0}^{n-1} \left[ B_{c_i} \left( B_{t_{i+1}} - B_{t_i}\right)\right] \right)^2 \right\}$, where $c_i \in [t_i, t_{i+1}]$

Let $B$ be a standard Brownian motion and $\{t_i\}_{i=0}^n$ a partition of $[0,t]$. Define $c_i= (1-c)t_{i+1}+ct_i$, for some $c \in [0,1]$. Write $B_i$ for $B_{t_i}$ and $$ S_n=\sum_{i=0}^{n-1} ...
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0answers
67 views

an example indicating the relation between Brownian motion and PDE

I have a question: Let $(B_t)_{t\geq 0}$ be a brownian motion. Consider the following function $u(x)$ defined by ...
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1answer
85 views

Expectation as an integral

I wish to express as a Lebesgue integral the following expectation, $E[\varphi(B_t)\varphi(B_s)]=\int ?$ for $0\leq s\leq t$, where $B_t$ is a Brownian motion with law $N(0,\sigma^2 t)$. Any ideas? ...
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0answers
108 views

How to construct the strong solution to the SDE $dX_{t}=\sqrt{X_{t}}dW_{t}$?

Given the SDE: $dX_{t}=\sqrt{X_{t}}dW_{t},$ $\ X_{0}=1$ , where $W_{t}$ is a 1-d Brownian motion. I was told that this SDE has a unique strong solution, but I don't know how to construct it. I know ...
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0answers
146 views

how to show that the price process is a martingale

Suppose I have an $d$-dimensional semimartingale $S=\{S_t\}$ with $t\in[0,T]$ under $P$. $S $ need not to be continuous (RCLL can be assumed). Suppose $Q$ is an equivalent measure w.r.t. $P$ such that ...
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1answer
77 views

Formulae to about Moment and Cross-moments of Stratanovitch Iterated Integrals

The title is a bit long but quite explicit, I am looking for a reference where the moments and cross moment Stratanovitch Iterated Integrals defined as : $E[J_n(1).J_p(1)]$ with $p\not=n$ With : ...
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1answer
50 views

Rewriting SDEs - “Multiplication on both sides”

I have a question concerning a calculus "trick" sometimes used in stochastic calculus (e.g. in the Book on Arbitrage Theory in Cont. Time of Bjoerk). There they do the following in the proof of Prop. ...
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0answers
160 views

Correlated diffusion processes and covariance matrix

I'm really noob in maths topics so I hope you will excuse me if I use terms which aren't correct. I would like to simulate $n$ dimensional diffusion processes with $n$ noises. Each process has its ...
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241 views

Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?

The following is the definition of infinitesimal generator from Oksendal. Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
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1answer
395 views

Stochastic integral : $\int_0^T (W(s))^2dW(s)$

How to evaluate this integral $$\int_0^T(W(s))^2 \, dW(s)$$ where $W(s)$ is random variable associated with brownian motion. I am new to this .Thanks in advance.
2
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1answer
244 views

Prove Yor's formula not using Itô?

In a book is a exercise to prove Yor's formula for stochastic exponential, i.e. $$\mathcal{E}(X+Y)\exp{(\langle X,Y\rangle)}=\mathcal{E}(X)\mathcal{E}(Y)$$ where ...
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48 views

Stochastic differential equations and experimental data

If we have a set of experimental data: $$X=\{x_1,x_2,\ldots,x_N\}$$ is it possible to write down an equation of the kind: $$dx(t)=b(x(t))\,dt+\sigma(x(t))\,dB(t)$$ describing the process from which ...
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1answer
128 views

Conditional expectation of a finite variation process

A simple question: Let $H$ be a cadlag, adapted process and $A$ a process of finite variation. Then also $\int_t^T HdA_t$ is a finite variation process (see "Limit Theorems... "Jacod&Shiryaev ...
6
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1answer
426 views

Covariance of Gaussian stochastic process

Could someone help me to figure out solutions of following problems?: Let $X = (X_t)_{t \geq 0}$ be a Gaussian, zero-mean stochastic process starting from $0$, i.e. $X_0 = 0$. Moreover, assume that ...
0
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1answer
41 views

Payoff function continuity

$dS=μSdt+σSdB$ $P(S,T)=[(1/n)∑S(t\{i\})-K]⁺$ is the asian option payoff. Which is also clearly pathwise continuous. How can i mathematically show that it is continuous?