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Questions tagged [stochastic-calculus]

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

The main flavors of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most useful for general classes of , such as semimartingale processes, but the related Stratonovich integral is frequently useful in problem formulation.

The Stratonovich integral can readily be expressed in terms of the Itô integral. The main benefit of the Stratonovich integral is that it obeys the usual chain rule and therefore does not require Ito's lemma. This enables problems to be expressed in a coordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than $\mathbb{R}^n$.

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Stochastic Different Equation

Consider the stochastic differential equation $\frac{dX_t}{X_t}=adt+bdW_t$ for the diffusion $X_t$ . The parameters $a,b$ are constant.Using Ito's lemma and suitable integration over $[0,T]$, show that $X_T=X_oexp[(a-\frac{1}{2}b^2)T+b\phi\sqrt T]$…
Vladimir Nabokov
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A solution's variance and mean

I have a stochastic differential equation of this type : $$ \ dX(t) = a dB(t)+k c t^{c-1} \cos(\theta(t))dP(t). $$ with $$ \begin{align} a,k &:\text{ constants }\\ B(t) &:\text{ Brownian process}\\ P(t) &: \text{A Poisson process with parameter…
Ayoub Ennassiri
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Expected value of product of an ito integral and a random variable

I want to compute $$E[\int_0^t W_r dr \int_0^s W_r^2 dW_r].$$ Here $t,s$ are arbitrary. I have thought about this a lot but not sure how to proceed. I tried to apply Ito's formula to one of the factors in the product, but that did not seem to…
S in NT
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right continuity of martingales

I am stuck with the following problem : let $M_t$ be a right continuous martingale. Show that the map $t \to M_t$ is right continuous from $\mathbb{R}^{+}$ to $L^1$. I know that, fixed $\omega$, the map $f: t \to M_t(\omega)$ from $\mathbb{R}^{+}$…
Fred G.
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interchanging spatial integral and time integral in the Brownian context

The problem is the following My attempt is inspired in the following: Consider $$F_n(x) = \int_{-\infty}^\infty h(a) u_n(x - a)\,da $$ By Itô's formula: \begin{align} &F_n(W_t) = F_n(W_0) + \int_0^t F_n'(W_s)\, dW_s + \frac{1}{2}\int_0^t…
Conrado Costa
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What's the covariance of $B_t$ and $B_{t^2}$, where $B_t$ is the standard Brownian Motion?

What's the covariance of $B_t$ and $B_{t^2}$, where $B_t$ is the standard Brownian Motion? $B_t$ is the standard Brownian Motion, what's $\operatorname{Cov}(B_t,B_{t^2})$?
user207890
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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. What does this mean? I know that if a two…
user3503589
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non-additive noise?

I always hear about noise that is "additive" (as well as being Gaussian),and I guess I'm wondering what the opposite is - what kind of noise is not additive? What does the SDE with non-additive noise look like?
Danny W.
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$dX(t) = a(t)dt +b(t)dW(t)$ and $Y(t)=exp(X(t)$ find $dY(t)$

Having trouble with the following $$dX(t) = a(t) + b(t)dW(t)$$ $$Y(t) =exp(X(t))$$ find $dY(t)$
seraphimk
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How do you find the variance of the expression below?

I'm trying to understand the calculation of $$Var \left (\int_{a}^{b} r_{s} ds\right) $$ where $r_{s} = \sigma W_{s}$ and $W_{s}$ is a standard 1-d Brownian motion. $\sigma$ is assumed to be constant. and the solution is, $$\sigma^2…
Prateek Bhatnagar
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In stochastic calculus, what is the differentiation of $W(t)$?

I am very confused dealing $dW(t)$, what is it? $W(t)$ is nowhere differentiable, we cannot write $W'(t)~dt$, but $dW(t)$ is a notation often used in my professor's notes.
Edward Wang
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Question regarding measurable random variable

If $X_i, i=1,\dots, d$ are $F_T$-measurable random variables, is $\Pi_{i=1}^d X_i^k$ ($k$ is constant) also $F_T$-measurable?
poglhar
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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 application of the fundamental theorem of calculus, but is…
user126540
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