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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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Doleans Dade Product Compensator

Let $X$ and $Y$ be martingales, $X_0 = Y_0 = 0$. Can the stochastic exponential $\mathcal{E}(X+Y+[X,Y]) _t$ be written on the form of $\mathcal{E}(X+Y)_t \exp (C_t)$ for some sort of "compensator" $C_t$? And if yes and not obvious what properties…
user280528
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Why this Ito integral has zero mean?

Let $W_t$ be the one-dimensional Wiener process. Does the following integral $$\int_0^t f(W_s) dW_s,$$ where $f:\mathbb{R} \to \mathbb{R}$ continuous with second derivative, have zero mean for any fixed $t>0$? The integral can be considered as a…
newbie
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How to identify an Ornstein-Uhlenbeck Process? Aquestion concerning the notes on SPDE's from Walsh.

In page 7 from the notes "An introduction to SPDE's" from Walsh, one reads: If we compute the covariance of $V_s$ we obtain (for $s\leq t$) $$ Cov (V_s, V_t) = Cov(U_{s,a +bs}, U_{t,a +bt}) = e^{-|s - t|} e^{-|b||s-t|} = e^{-(1+|b|)|s-t|}…
Conrado Costa
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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 $\lim\frac{\phi(x)}{x}=\infty, x\to\infty$ I have a proof for this…
Montaigne
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Covariance of en elliptical distributed variable?

Let $Z\stackrel{d}{=}\mu + \xi A U$ where $\xi$ is a non specified (one dimensional) random variable with $\mathbb{E}(\xi^2)=n$ independent of $U$, $U$ is uniformly distributed on the $n$-dimensional sphere $\mathbb{S^{n-1}}$ and A is a $n\times n$…
user299124
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Stochastic integral: quadratic variation and notion of convergence

I have a process $(B_t(\omega))$ with finite quadratic variation: $E[\int_{0}^{T}B_t^2dt] < \infty $ (1) According to my material, I can approximate the integral $\int_{0}^{T}B_tdB_t$ by a simple process $H_{t}^{n}$. Thereby, I build on the…
cecefuss
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What's wrong with this application of Ito's lemma?

Ito's lemma: $$\text{if } dX_t = \sigma_tdW_t+\mu_t dt, \text{ where $W_t$ is a Wiener process, then } df(X_t)=\sigma_tf'(X_t)dW_t + (\mu_tf'(X_t)+ \frac{1}{2}\sigma_t^2f''(X_t))dt$$ now I want to apply this to $$Y_t=e^{\sigma W_t+\mu t}$$ Where $W$…
user56834
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Martingale representation using Ito

If we consider a process $$X_T=e^{\int_0^TtdW_t}$$it holds that it can be expressed by $$X_T=\mathbb{E}[X_T]+\int_0^Th(t)dW_t$$but how do we derive this $h(t)$? I calculated $\mathbb{E}[X_T]=e^{t^3/6}$, and defined a new process $$Z_T=e^{-T^3/6}X_T…
user128836
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Stochastic Differential

for an exercise I need to calculate the following problem: Define $F(Y,t)=\mathbb{E}[e^{-\int_0^T r(Y(s))ds}\mid Y(t)]$ where $Y(t)=(Y_1(t), Y_2(t), ..., Y_N(t))$ follows the diffusion process: $dY(t)=\mu(Y(t))dt + \sigma(Y(t))dW(t)$ and…
math student
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What is the expected value of geometric brownian motion?

This equation solution is a geometric brownian motion $$dx_t=r.x_t.dt+\sigma .x_t.dB_t \to \\ x_t=x_0.e^{(r-\frac{1}{2}\sigma^2)t+B_t}$$ now I am asking for $E[x_t]$ I saw wikipedia ...It said that…
Khosrotash
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About repeated Ito integration.

I'm sorry for this kind of question. But I was wondering, where there is any useful formula for repeated Ito integration. Ito integration is defined for adapted processes, but Ito integral is itself adapted process. So We can define the Ito…
kolobokish
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How to show a random process is a Brownian motion?

$$dW_3 = \rho \, dW_1(t) + \sqrt{1- \rho^2} \ dW_2(t) $$ I tried applying Levy theorem. My question is should the Levy theorem argument be used for which part of the above equation?
kateBay1
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How to use Jensen's inequality to determine whether a stochastic process is supermartingale

How to find out if a stochastic process is martingale? I used Ito's formula to prove it is not a martingale. Can it be a supermartingale? How to use Jensens inequality to find if a process is martingale?
flyhigh1
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stochastic differential equation solution

I find it difficult to solve this differential equation: $dX(t)=[aX(t)+b]dt+σX(t)dW(t)$ $X(0)=x$ where $W(t)$ is a Brownian motion and $a, b, σ, x$ are real constants The thing which confuses is that $b$ is not multiplied by the $X(t)$.
Costakis M
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Ratio distribution of independent exponentially distributed variables

first things first: I am not a studied mathematician and therefore lack thorough knowledge of the topic - please consider this, even though I will of course try to express myself as accurately as possible. I want to calculate a continuous random…
mkiesner
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