# 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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### In calculus, if $\frac{dy}{dx}$ is not in fact a fraction, is the equation below for geometric brownian motion technically incorrect?

As suggested in the comments, the paths of $W_t$ (and also $S_t$) are almost surely nowhere differentiable, so the notation $\frac{dW_t}{d t}$ is meaningless in this context. The SDE as written is ...
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### Stochastic integration by parts of geometric brownian motion

You can't go any further than $\int e^{B_{t}+at}dt$. This is called the integrated Geometric Brownian motion. For references on its law see "The Integral of Geometric Brownian Motion" &...
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### Existence and uniqueness of the solution to $dX_{t}=\left(2\chi_{\left\{ X_{t}>0\right\} }-1\right)\cos X_{t}dt+\cos X_{t}dW_{t}$

Due to the volatility coefficient, the solution is contained inside $I=(-\frac{\pi}{2},\frac{\pi}{2})$ and gets absorbed at the boundaries (because $X_{t\wedge \tau}$ also solves the SDE and so we ...
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### Exchangeability of an interacting particle SDE system

This follows quite quickly, as mentioned here too on page 5: the entire system is symmetric, in the sense that if we permute the “names” $i = 1, . . . , n$, we end up with the same particle system. ...
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### Stochastic Integral Evaluation

Indeed as mentioned in comments we have $$\int_{z=0}^T \int_{s=0}^z dW(s) dW(z)=\int_{0}^{T}W_{s}dW_{s}$$ and so using Itô-formula we get $$=\frac{1}{2}(W_{T}^{2}-T).$$ Generally, multiple Itô ...
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### Stochastic differential equation with Bernoulli random process as a solution

The process $X_t$ does not admit a representation as an Itô process. To see this, note that $X_t$ does not have continuous paths. However, an Itô process of the form you looked for has continuous ...
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### Covariance of Black-Scholes

In order to compute this, we need information on their joint law. If they don't form a joint-normal, then the answer can vary alot see Is it possible to have a pair of Gaussian random variables for ...
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### Proof that the stochastic exponential is a local martingale

In general, an Ito integral is a local martingale and it is a true martingale provided that the integrand is in $L^2$. This is precisely what the remark is telling you, as saying that the integrand is ...
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### How can I find a stochastic process $A_t$ s.t. $\frac{1}{S_t}=\mathcal{E}(A)_t$?

Too long for a comment: You are overthinking it. First let me remark that the solution of the SDE $dS_t=\mu S_t\,dt+\sigma S_t\,dB_t$ is $$S_t=S_0\,e^{\mu t-\frac{\sigma^2}{2}t+\sigma B_t}$$ which ...
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### Converting an Ito integral into a Backward Ito Integral

I have deciphered what Anderson meant so I can answer my own questions. The Answers to my Questions The interpretation is correct (see the derivations below). Additionally the resulting process $x$ ...
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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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### What is the solution to the SDE $X^x_t = B_t + \int_0^t \frac{x − X^x_s}{1-s} ds$

Setting $Z_t$ as what you did in your work, and then writing (2) in Differential form gives us that $$dX^x_t = xdt + dB_t - \frac{B_t}{(1-t)}dt + Z_tdt \tag{4}\label{eq4}$$ And (2) also gives us that ...
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### How to formulate and analyze systems of stochastic differential equations?

Yes, in the articles "Stochastic prey-predator system with foraging arena scheme" and "Study on Dynamic Behavior of a Stochastic Predator–Prey System with Beddington–DeAngelis ...
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