# Tag Info

### Are differentials on their own in stochastic calculus just an abuse of notation?

Differentials in stochastic calculus have very precise interpretations. The stochastic differential equation in differential form $$dX_t = \mu (t, X_t) dt + \sigma (t, X_t) dB_t$$ translates precisely ...
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### Exercise on Girsanov's theorem

There is no difference with the case your drift is non-deterministic. Your calculation shows that $X$ is a $Q$-Brownian motion; since $Q$ and $P$ are equivalent and $Q(\{X_t > M\}) > 0$ it ...
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### Help with Integration by Parts for a Markov Chain

OP here, I think I was able to figure out the problem. Part 1: Let's start with the Q Matrix: $$Q = \begin{bmatrix} B & A=-BI \\ 0 & 0 \end{bmatrix}$$ Within this matrix, we are interested ...
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### Reference request on rough paths sequence of Brownian bridges

One article where they study the stochastic Lévy area for Brownian bridges in relation to the one for Brownian motion is "BROWNIAN BRIDGE EXPANSIONS FOR LÉVY AREA APPROXIMATIONS AND PARTICULAR ...
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### Characteristic function of a random variable by Fourier transform

I'll admit that I'm not familiar with most if not all stochastic finance terminology. So, I'll assume that $K$ and $S_{0}$ are constants. See that for $f(x)=\log(x)$, by Ito's Lemma, you get, \begin{...
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### Question about convergence in stochastic integration

No here we only assume $f$ is bounded and in $L^{2}$, he just skipped a few steps. In particular, he skipped the proof that $$g(y):=\left(\int_{R} |f(x-y)-f(x)|^{p}dx\right)^{1/p}$$ is a continuous ...
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### Confused by the notation in Steele's Stochastic Calculus

The superscript $d$ on $\tau$ means the standard thing; i.e. $\tau^d$ is $\tau$ to the power of $d$.
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### Generator of the joint process $(X_t,Y_t)$ where $Y_t= e^{-t}W(e^{2t})$ and $X_t = \int^t_0 Y_sds$.

To find the generator I will look for the two-dimensional SDE that is solved by $(X_t,Y_t)\,.$ It is not hard to see that $$\tag{1} B_t:=\int_0^te^{-s}\,dW_{e^{2s}}$$ is a continuous martingale with ...
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### Proof that an Ito diffusion start with 0 will be positive immediately

1. By Girsanov's theorem, the laws of $X$ and of the solution $Y$ of $dY_t=\sigma(Y_t)dW_t$ are locally absolutely continuous, so you can assume without loss of generality that $\mu=0$. 2. When $\mu=0$...
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### Weak Uniqueness of solution of SDE

We will add a few more details, let me know if you need more. Let $X_t$ and $Y_t$ be the strong solutions constructed from $\tilde{B}_t$ and $\hat{B}_t$, respectively, as above. Here they simply ...
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### Ito's Lemma applied to the Solution to the General Linear Equation

With your finally correct definitions but with better notation $$\Phi_t=\textstyle\exp\left(\int_0^t( a - \frac{1}{2}b^2)\,\mathrm{d}s+\int_0^tb\,\mathrm{d}W_s\right)\,,$$  I_t=X_0 + \int_0^t\frac{...
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It is not very clear from Kurt's answer to Fred's answer why $\frac{\partial}{\partial f}$ is legitimate. I write a detailed explanation here. The Ito $\Leftrightarrow$ Stratonovich conversion is only ...