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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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problems in exponential martingale

Let $B\left(s\right)$ be a brownian motion and $\sigma\left(s\right)$ be the nondeterministic function. The following equation then holds $$ \mathbb{E}\left(\exp\left(i\int_{0}^{t}\sigma\left(s\right)\,…
alfred chen
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Use Ito formula to compute expected value

Let $W_t$ be a standard brownian motion. I am trying to compute $E[(\int_0^t s^2 dW_S)^4]$. I applied Ito's formula and got $$t^2 W_t = \int_0^t s^2 dWs + \int_0^t 2s W_s ds$$. This gives us $$E[(\int_0^t s^2 dW_S)^4] = E[(t^2 W_t-\int_0^t 2s W_s…
S in NT
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Using Ito's lemma to compute a SDE

This is the version of Ito's lemma that we are given in our notes. Now I'm just not able to understand how to begin this problem and arrive at the given solution. The g(x) integral function that they have given is just confusing me and using…
user134785
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How to find second order derivative

Suppose we have that $dp=u dt+s dz$, where $dz$ is brownian motion. What would be $d^2 p$? that is, the second derivative under ito calculus.
he wei
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Why the set of stochastic process Ito Integrable has to be square integrable w.r.t time as well?

Ito Integral Consider a set of stochastic process $f(t)$ mainly such that a) $$ E\left(\int_0^{+\infty}f(t)^2 \,dt\right) < \infty. $$ Denote this set of stochastic process as $M^2$. Question: a) For each $w$, $f(t,$$w$$)$ is a continuous…
Yue Harriet Huang
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Question about the bounded requirement of the simple function of definition of stochastic integration

Let $W_t$ be one-dimensional Brownian motion, to calculate $\int_0^tW_sdW_s$ by the definition of stochastic integration, one way is to use the integration of $W^{(n)}_t=W_{[nt]/n}$ to approximate $\int_0^tW_sdW_s$. My question is the definition of…
Danielsen
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Calculate the variance of $\int_{t-m}^t \int_{s-m}^se^{-k(s-u)}dW(u)ds$

I need to calculate the variance of this double stochastic integration: $$ I=\int_{t-m}^t \int_{s-m}^se^{-k(s-u)}dW(u)ds $$ Note that variable $s$ is in the integrand as well as the boundaries of the integral. I have this solution by changing…
Wu David
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Stopped Ito Integrals

If $E\int_0^T X_t^2 dt < \infty$ then the Ito integral can be defined for every $0 \leq t \leq T$ such that it is a continuous process in $[0,T]$. The convention that I am working with for a simple process is that $X_t = \sum e_j…
jpv
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Novikov condition, martingale

I have a question about Novikov condition and martingale. $T>0$: fix. Let $(\Omega, \mathcal{F}, \left(\mathcal{F}_{t}\right)_{t \in [0,T]}, P)$ be a filtered probability space and $(B_{t})_{t \in [0,T]}$ be a $(\mathcal{F}_{t})_{t \in [0,T]}$…
sharpe
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Definition of Ito Integral

In Kartazas and Shreve, the integral for Bounded Progressively measurable processes is defined first. Then, for Bounded measurable processes ($f(t,\omega)$), the authors say that there exists a progressively measurable modification and show how to…
jpv
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covariance of two ito integrals

Let $X_t=\int_0^t \left(\frac{1-t}{1-u}\right)^k dW_u$. Assume $0\lt s \lt t\lt T$. Is the following the right way to compute the covariation of $X_s$ and $X_t$? $$ \begin{align} \text{Cov}(X_t, X_s) &= \mathbb{E}\left[\left( X_t - \mathbb{E}[X_t]…
Danny
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How to use Itō in this very simple case

I want to apply Ito for the following process: \begin{equation*} X_t = tW_t + \int_0^t W_u du, \end{equation*} where $W$ is a Brownian motion. I have no trouble with the part $tW_t$ This can be written $tW_t = f(t,W_t)$ with $f(t,x) = tx$. However,…
math
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Condition for the mgf of a stochastic integral to be finite

Fix $t>0$, let $B$ be a Brownian motion and let $\sigma$ be a previsible process such that $$\mathbb{E}\left[\text{exp}\left(\frac{1}{2}\int_0^t\sigma_s^2ds\right)\right]<\infty.$$ Then is…
user3341907
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Application of Ito's Lemma to stochastic integrals

From my understanding, the Ito Integral is a random variable itself. Suppose we have $X_t=\int_0^t Z_udZ_u$. To find $dX_t$, I would think we can apply Ito's Lemma. However, how would the partial derivatives with respect to $t$ and $Z_t$ work? Any…
xyz1010
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Verifying $S(t)=S(0)e^{rt} + \sigma e^{rt} \int_0^t e^{-rs} dW(s) $ satisfies $dS(t) = rS(t)dt + \sigma dW(t)$

Consider the SDE $$ dS(t) = rS(t)dt + \sigma dW(t). $$ To solve this, I let $f(t,x) = xe^{-rt}$, so $\frac{\partial f}{\partial t} = -rxe^{-rt}$, $\frac{\partial f}{\partial x} = e^{-rt}$ and $\frac{\partial^2 f}{\partial x^2} = 0$. Using Ito's…
bcf
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