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9 votes

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 ...
Jose Avilez's user avatar
8 votes
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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 ...
Jose Avilez's user avatar
5 votes
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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" &...
Thomas Kojar's user avatar
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4 votes
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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 ...
Thomas Kojar's user avatar
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4 votes
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Demonstration with Itô's lemma

Note: I'll use $W(t)$ notation here instead of $W_t$. Observe that the given statement is equivalent to the following differential form: $$ {\rm d}(h(t) W(t)) = h(t) {\rm d} W(t) + h'(t)W(t) {\rm d}t....
Yalikesifulei's user avatar
4 votes
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Finishing the solution of this SDE

Here is the next steps. We begin with $$dC_t = C_tdB_t +e^{-t}B_tdB_t \tag{1}$$ Let denote $Z_t$ the solution of $$dZ_t = Z_t dB_t$$ It's easy to find that $$Z_t = Z_0\cdot e^{-\frac{1}{2}t+B_t} \tag{...
NN2's user avatar
  • 16k
4 votes

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. ...
Thomas Kojar's user avatar
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4 votes
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Ito's lemma and Stratonovich calculus

In this answer I show that for any process $Y_t$ whose stochastic integrals are defined we have $$ \tag{1} \underbrace{\int_0^tY_s\circ\,dW_s}_{\text{Stratonovich}}=\underbrace{\int_0^tY_s\,dW_s}_{\...
Kurt G.'s user avatar
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3 votes
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Same SDE with different initial conditions

It is false because you actually have dependence on the initial condition on the RHS too $$X_{t}-Y_{t}=x_{0}+ \int_{0}^{t}\mu(X_{s}^{x_{0}})ds+\int_{0}^{t}\sigma(X_{s}^{x_{0}})ds-(y_{0}+ \int_{0}^{t}\...
Thomas Kojar's user avatar
  • 3,661
3 votes

SDE with stationary Log-normal distribution

Let $F(x)$ be an arbitrary time-independent CDF and $B_t$ be a standard Brownian motion. The CDF of $B_t$ is $\Phi(x/\sqrt{t})$ where $\Phi(x)$ is the standard normal CDF. Then $$ U_t:=\Phi(B_t/\sqrt{...
Kurt G.'s user avatar
  • 14.4k
3 votes

Incorrect use of Ito's lemma

At the risk of repeating other user's comments: To add to NN2's answer: $$\tag{A} X_t=X_0\,e^{-t}+\int_0^t\sqrt{2}\,e^{(s-t)}\,dW_s $$ is the correct solution to the SDE $$\tag{B} dX_t=-X_t\,dt+\sqrt{...
Kurt G.'s user avatar
  • 14.4k
3 votes
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Solving this stochastic differential equation by variation of constants

You are already on the right track. To solve for $C_t$, do exactly like for $Z_t$. Indeed, from $$\frac{dC_t}{C_t} = B_tdB_t$$ you compute $d(\ln(C_t)$ by Ito's formula $$\begin{align} d(\ln(C_t)) &...
NN2's user avatar
  • 16k
3 votes
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Proving $X_t = 1 + \int_0^t X_s \, dN_s$ is a supermartingale

First, note that these are all about the case where $X$ is an exponential martingale, so $N$ is a local martingale. This also implies $X$ is a local martingale. For 1), let $(\tau_n) \rightarrow \...
user6247850's user avatar
  • 13.5k
3 votes

Proving $ X_t = W_t - \int_0^t \frac{X_s}{1 - s} \, ds$ is the Brownian Bridge

To show that $X$ is a Brownian bridge on $[0,1]$ it suffices to show that it is centered and Gaussian with $$\tag{1}{\rm Cov}[X_t,X_s]=\min(t,s)-ts\,.$$ (see [1] p. 35). By $$ X_t=(1-t)\int_0^t\frac{...
Kurt G.'s user avatar
  • 14.4k
3 votes
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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 ...
Thomas Kojar's user avatar
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3 votes
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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 ...
Jose Avilez's user avatar
3 votes
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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 ...
Thomas Kojar's user avatar
  • 3,661
3 votes

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 ...
Jose Avilez's user avatar
3 votes
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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 ...
Kurt G.'s user avatar
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3 votes
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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 $x$ ...
Mark's user avatar
  • 5,706
3 votes
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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{...
Mr.Gandalf Sauron's user avatar
3 votes
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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 ...
vinayak's user avatar
  • 48
2 votes

How to solve system of stochastic differential equations?

As mentioned in the comments the SDE $$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$ is decoupled and so one can solve it $$N_{1,t}=N_{1,0}\exp \left(\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{1,t}\...
Thomas Kojar's user avatar
  • 3,661
2 votes

Covariance of Stochastic Differential Equation

Here we simply apply Ito isometry. So as explained Quadratic Variations and the Ito Isometry we have $$ EX_{t}X_{r}=E[\int^t b ds\int^r b ds]+E[\int^t b ds \int^r \sigma dB_{s}]+E[\int^r b ds \int^t \...
Thomas Kojar's user avatar
  • 3,661
2 votes

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 ...
Thomas Kojar's user avatar
  • 3,661
2 votes
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Law of the multivariate Ornstein-Uhlenbeck process on path space

First to be clear the path of OU is uniquely determined by its covariance $$\begin{align} \operatorname{cov}(x_s,x_t) & = = \frac{\sigma^2}{2\theta} \left( e^{-\theta|t-s|} - e^{-\theta(t+s)} \...
Thomas Kojar's user avatar
  • 3,661
2 votes

Exit and hitting times for the Bessel process $\textrm{d}X_t=\frac{n-1}2\frac{\textrm{d}t}{X_t}+\textrm{d}B_t$

(a) Since we stopped this local martingale by $T_1$, it is a bounded martingale. (b) As you define $T_2$ as the hitting time of $0$. The real question to ask here is whether $\mathbb{P}(T_2 < \...
NagiYang's user avatar
2 votes
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When do invariant solutions of an SDE exist?

In terms of references (Stochastic Modelling and Applied Probability 66) Rafail Khasminskii (auth.) Stochastic Stability of Differential Equations. "Long-time dynamics of stochastic ...
Thomas Kojar's user avatar
  • 3,661
2 votes
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Why is Itô integration not a "pathwise solution theory" to SDEs, but rough path theory is?

The solution of an Itö-SDE $X_{t}=y+\int bdt+\int \sigma dB_{t}$ is defined outside a set of measure zero that depends on the whole equation i.e. $\Omega_{y,b,\sigma}$ with $P[\Omega_{y,b,\sigma}]=0$. ...
Thomas Kojar's user avatar
  • 3,661
2 votes

Incorrect use of Ito's lemma

It's wrong to do $$ \mathrm d X = - X \, \mathrm dt + \sqrt{2} \, \mathrm dW(t)\Longrightarrow X(t + \Delta t) \approx X(t) - X(t) \, \Delta t + \sqrt{2} \sqrt{\Delta t} \, \xi(t) \tag{1} $$ The ...
NN2's user avatar
  • 16k

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