Mathematics

Questions tagged [stochastic-integrals]

This tag is used for questions about stochastic integrals – especially for calculations. For questions related to more theoretical aspects of stochastic integrals such as their constructions, (stochastic-analysis) may be more appropriate.

A stochastic integral is an integral of stochastic processes with respect to stochastic processes. This may include Ito's integrals, but also variants such as Stratonovich integrals.

2432 questions
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Integral of Brownian motion is Gaussian?

Let $(W_t)$ be a standard Brownian motion, so that $W_t \sim N(0,t)$. I'm trying to show that the random variable defined by $Z_t = \int_0^t W_s \ ds$ is a Gaussian random variable, but have not gotten very far. I tried approximating the integral by…
Jonas
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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 to special processes, i.e. is $$\mathbb{E}\left[\int…
user20869
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2 answers

Could someone explain rough path theory? More specifically, what is the higher ordered "area process" and what information is it giving us?

http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, $(X,\mathbb{X})$ where $X$ is a continuous process and…
user223391
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1 answer

Why predictable processes?

So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator and admitted integrands to be progressively…
JohnSmith
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23
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4 answers

Expectation of geometric brownian motion

I was deriving the solution to the stochastic differential equation $$dX_t = \mu X_tdt + \sigma X_tdB_t$$ where $B_t$ is a brownian motion. After finding $$X_t = x_0\exp((\mu - \frac{\sigma^2}{2})t + \mu B_t)$$ I wanted to calculate the expectation…
Stijn
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22
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2 answers

Why isn't the Ito integral just the Riemann-Stieltjes integral?

Why isn't the Ito integral just the Riemann-Stieltjes integral? What I mean is, given a continuous function $f$, some path of standard brownian motion $B$, and the integral: $$\int_0^Tf(t)\;dB(t).$$ So what if we can't apply the change of variables…
Set
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3 answers

what's the difference between RDE and SDE?

what's the difference between random differential equation and stochastic differential equation? does stochastic differential equations include random differential equation?
Jim
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20
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1 answer

When is a stochastic integral a martingale?

In what follows, let the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ as well as the chosen filtration $(\mathcal{F}_t)_{t \ge 0}$ be known, and let $f$ denote an arbitrary locally bounded progressively measurable process (i.e. bounded on…
Chill2Macht
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19
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2 answers

Brownian bridge expression for a Brownian motion

Let $B_t$ be a standard Brownian motion in $\mathbb R$, then the Brownian bridge on $[0,1]$ is defined as $$ Y_t = a(1-t)+bt+(1-t)\int\limits_0^t\frac{\mathrm dB_s}{1-s} $$ for $0\leq t<1$. Here $Y_0 = a$ and $\lim\limits_{t\to 1} Y_t = b$ a.s. The…
SBF
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5 answers

Why do people write stochastic differential equations in differential form?

I am trying to teach myself about stochastic differential equations. In several accounts I've read, the author defines an SDE as an integral equation, in which at least one integral is a stochastic integral, then writes that in practice, people…
Stefan Smith
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17
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3 answers

Stochastic Integrals are confusing me; Please explain how to compute $\int W_sdW_s$ for example

I have been trying hard to understand this topic, but only failing.Reading through my lecture notes and online videos about stochastic integration but I just can't wrap my head around it. The main reason is, the notation/terminologies. I find them…
John Trail
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3 answers

Limit of a Wiener integral

How to show that $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$ where $\left (B_s \right)_{s\geq 0}$ is a real standard brownian motion…
Paul
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16
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2 answers

Stochastic Differential Equation solution for Geometric Brownian Motion

I am having difficulty in understanding the solution for a GBM given the SDE: $$dY(t)=\mu \ Y(t) \ dt + \sigma \ Y(t) \ dZ(t)$$ or $$\frac{dY(t)}{Y(t)}=\mu \ dt + \sigma \ dZ(t)$$ The solution for the above SDE is: $$\int_0^t\frac{dY(t)}{Y(t)} =…
Nehal
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2 answers

Product Rule for Ito Processes

Is it true that if $X(t)$ is an Ito process and $p(t)$ is non-stochastic, then the ordinary chain rule applies, that is, $$d(X(t)p(t)) = dX(t)p(t) + X(t)p'(t)dt?$$
user198504
16
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2 answers

Probability density function of the integral of a continuous stochastic process

I am interested in whether there is a general method to calculate the pdf of the integral of a stochastic process that is continuous in time. My specific example: I am studying a stochastic given process given by…
Gabriel
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