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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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prove $\frac{e^{iuX_t}} {\mathrm{E}{[e^{iuX_t}]}}$ is martingale

...knowing that $X_t$ has independent increments and is adapted to its natural filtration, $u \in \mathrm{R}$ My problem is in particular how to show this process has finite mean...(can I use the fact that $e^{iux}$ is bounded $\forall…
mg91
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Is it a martingale

I am working on problem V.3.21 in Revuz Yor which claims that $\mathbf{1}_{B\geq0}B$ is a martingale. Indeed $\mathbf{1}_{B_t\geq0}B_t=\int_0^t\mathbf{1}_{\{B_s\geq0\}} dB_s$ and since $\int_0^t\mathbf{1}_{\{B_s\geq0\}} ds\leq t<\infty$, it is a…
Onstack01
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If $Y_t=\int_0^t X_sdW_s$ where $\int_0^t X_s^2ds<\infty $ a.s., do we have that $\mathbb EY_t=0$?

Let $Y_t=\int_0^t X_sdW_s$. I know that if $\mathbb E\int_0^tX_s^2ds<\infty $, then $(Y_t)$ is a martingale, and $\mathbb EY_t=0$ for all $t$. But what if we only have $\int_0^t X_s^2ds<\infty $ ? I know that in this case, $(Y_t)$ is a local…
Bruce
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Example of application of stochastic calculus to discrete probability

Integrals extend the concept of sum in the continuous domain, and the usefulness of such extension with respect to the generalized concepts (i.e. discrete mathematics) can be easily illustrated with some examples, such as providing bounds on the…
Immanuel Weihnachten
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Ito's formula with $\log X_t$

I want to use Ito's formula on the following SDE: $$ dX_t= - \gamma (\log X_t - \theta) X_t d t + \sigma X_t d W_t $$ to obtain an expression for $ \log X_T $ where $T > t $ is some fixed time. pretty new to this, I tried applying Ito's with $…
Christopher Turnbull
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Ito's formula for non smooth functions like Tanaka's formula

Does there exist an Ito's formula for function of Brownian Motion which are once differentiable but not twice differentiable like Tanaka's formula?
Lost1
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Conditional expectation and the Dirac delta function

I'm looking for a rigorous proof of an identity I came across many times (in the context of Gÿongy's lemma). Suppose $X$ and $Y$ are two r.v. We know that $\mathbb{E}\left[X|Y\right]$ is $\sigma(Y)$-measurable so there exist a function $\varphi$…
Aguelmame
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Finding the distribution of an Ito integral. $\int_0^t sB_s \, \mathrm{d}s$

I'm a little baffled by this, I'm supposed to find the distribution of $X_t$ where, $X_t=\int_0^t sB_s \, \mathrm{d}s$. What I can think of is to consider the process $$\begin{align} Y_s &= s^2B_s \\\\ dY_s &= 2sB_sds+s^2dBs+sds \\\\ \int_0^t \,…
ActuariallyImpaired
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Stochastic differential equation solution suggestion

Any suggestion on solving the stochastic differential equation \begin{align} dW(t) = d\widetilde{W}(t) + \left(\frac{\kappa - W(t)}{\tau-t} - \frac{1}{\kappa - W(t)}\right)dt \end{align} where $\kappa,\tau\in\mathbb{R}$ are known and…
Peter5
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intuitive explanation of Doleans exponential of Brownian motion

A simple SDE is given as: $$dX_t=\sigma X_tdW_t$$ Ito's lemma confirms that the following is a solution: $$X_t=X_0e^{\sigma W_t-\frac{1}{2}\sigma^2t}$$ I understand how this result is correct according to Ito's lemma, but I don't see why it makes…
user56834
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SDE and Ito's product rule

In some financial application, we are given that $$X(t)=B(t)\times[{X(0)+\int_{0}^{t}\lambda(s)dW(s)}]$$ and need to calculate the $dX(t)$ using Ito's product rule. Assuming $X(0)$ is given, then $d[X(0)+\int_{0}^{t}\lambda(s)dW(s)]$ is simply…
user424828
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Stochastic calculus integral

How can I evaluate, or at least find an upper bound for, the following integral without the Hölder inequality, is there an alternate way anyone knows of: $$\mathbb{E}\left[\sup\left|\int_0^t\mu X(u)du\right|^2\right]?$$ Here $dX = \mu X dt + \sigma…
Becky D
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stochastic integral of ou process with respect to itself

Let $X_t$ follows a Ornstein-Uhlenbeck process $dX_t = -\theta X_t dt + dW_t$ for $\theta < 1$ and $X_0 = 0$. I am interested in the stochastic integral $Y_T = \int_0^T X_s dX_s$ and I wonder if what I do to solve it is the right approach. I am…
Steve Yang
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Itô formula question

My question is at the end of the problem statement. Solve the following stochastic differential equation. $dX_t = (\beta - \alpha X_t)dt + \sigma dB_t$, $X_0 = x_0$ where $\alpha$, $\beta$, $\sigma$ are constants, and $\alpha > 0$, and $B$ denotes a…
Ptru
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$Y(t)=W^2(t)\cdot e^{aW(t)}$, find $dY$

The answer given in a textbook is $dY=e^{aW}[t+\frac{1}{2}a^2W^2+2Wa]dt+ae^{aW}W^2dW$, but mine is $dY=e^{aW}[1+\frac{1}{2}a^2W^2+2aW]dt+e^{aW}(2W+aW^2)dW$. Which is correct? My solution: $$\begin{aligned} dW^2 &= 2WdW+dt \\ de^{aW} &=…
Joe Li
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