58 votes
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Difference between weak ( or martingale ) and strong solutions to SDEs

The main difference between weak and strong solutions is indeed that for strong solutions we are given a Brownian motion on a given probability space whereas for weak solutions we are free to choose ...
saz's user avatar
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42 votes
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Solution to General Linear SDE

Here is the complete solution to the problem including some special cases for an easy start. With analogy to the integrating factor method from ODEs it seems natural to rearrange \begin{align*} \...
m_gnacik's user avatar
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19 votes
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Why do we study Cameron-Martin Space and what is the motivation behind it?

The infinite-finite dimension dichotomy , and Fourier coefficients When we study (multivariate) normal random variables on $\mathbb R^d$, we often have equivalent ways of understanding them. One ...
Sarvesh Ravichandran Iyer's user avatar
14 votes
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Initial Distribution of Stochastic Differential Equations

Part 1: Weak existence In order to construct and study weak solutions to SDEs it is often very useful to take advantage of the close connection between SDEs and martingale problems. For simplicity of ...
saz's user avatar
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13 votes
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Stochastic Differential Equation solution for Geometric Brownian Motion

Your integration step is wrong. $Z$ here refers to Brownian motion, and thus you need to apply Ito Integration. One way I like to see the extra drift is needed, is that we take derivative of the ...
Jay Zha's user avatar
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12 votes

Confused about Brownian motion on Riemannian manifold

Let $(M,g)$ be a closed (compact, no boundary) Riemannian manifold with Laplace–Beltrami operator $\Delta_g$. The operator $(\Delta_g,\mathcal C^\infty(M))$ is an unbounded operator operator on $L^2(\...
AlephBeth's user avatar
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11 votes
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Conversion between solution to Stratonovich SDE and Itô SDE

You are considering the Stratonovich SDE $$ dX = h(X) \, dt + \gamma (X) \circ dW, $$ where I suppose the following hold: $h \colon \mathbb{R}^n \to \mathbb{R}^n$, $\gamma = (\gamma_{ij})_{i,j = 1}^{...
Arthur11's user avatar
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9 votes
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Why is Brownian Motion so Big in the Theory of Stochastic Differential Equations?

Actually it is a massive loss of generality. Here are two reasons why people use models based on Brownian motion: the central limit theorem. The CLT is saying that Gaussian distributions appear ...
saz's user avatar
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8 votes
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Must the strong solutions of an SDE be unique? Can there be two different strong solutions?

Example 1: Note that stochastic differential equations (SDEs) are a generalization of ordinary differential equations (ODEs); in particular, any ODE which does not have a unique solution does not have ...
saz's user avatar
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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
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7 votes
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SDE Solution: Hull-White extension of Vasicek model

The SDE can be solved similarly as in the Vasicek model. Define $F(t,r(t)) = e^{\alpha t}r(t)$, then \begin{cases} \displaystyle \frac{\partial F}{\partial t} &= \alpha e^{\alpha t} r(t) \\ \...
Cavents's user avatar
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7 votes
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Solution of SDE $dX_t = \mu(t)X_tdt + \sigma X_t dW_t$

The reason $\ln(x)$ is chosen is this: (Note: where I say $B_{t}$ I mean Brownian motion, which you denoted in your question as $W_{t}$) The SDE you provided is one of the few we can explicitly ...
layman's user avatar
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7 votes

Stochastic Differential Equation solution for Geometric Brownian Motion

I believe the answer by @Yujie Zha can be simplified substantially. Thanks to @Dr. Lutz Lehmann for providing a link to this, my solution is the same as the solution on page 15, but with more ...
baibo's user avatar
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7 votes
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Show that this stochastic process is a.s. strictly positive.

Define stopping times by $$T_k := \inf\{t \geq 0; X_t \leq k^{-1}\}, \qquad k \in \mathbb{N}$$ and $$T := \inf\{t \geq 0; X_t \leq 0\}.$$ Following the reasoning in your question we find that $$\...
saz's user avatar
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7 votes
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Solve the SDE $dX_t=\sqrt t(X_t+\sin t)dW_t$

The SDE is a particular example of a so-called linear SDE $$dX_t = (\alpha(t)+\beta(t) X_t) \, dt + (\gamma(t)+\delta(t) X_t) \, dW_t \tag{1}$$ where $\alpha, \beta,\gamma,\delta$ are deterministic ...
saz's user avatar
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7 votes

Expectation of solution to SDE $dX_t=-\tanh(X_t) dt + dW_t$

Step 1: Let's first construct a weak solution to the SDE $$dX_t = - \tanh(X_t) \, dt + dB_t, \qquad X_0 = x_0. \tag{1}$$ For a Brownian motion $(W_t)_{t \geq 0}$ on a probability space $(\Omega,\...
saz's user avatar
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7 votes
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Does a strong solution to a SDE imply lipschitz condition?

No, Lipschitz continuity is not a necessary condition for the existence of a strong solution. There is, for instance, the following general result which goes back to Zvonkin: Let $(B_t)_{t \geq 0}$ ...
saz's user avatar
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7 votes
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Do there exist diffusions that do not solve any SDE?

Yes, not all strong Markov processes with continuous sample paths solve an SDE. Here is an example: Let $(B_t)_{t \geq 0}$ be a Brownian motion. The process $X_t := |B_t|^{1/3}$ is not a ...
saz's user avatar
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7 votes
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A question about SDE and geometric Brownian motion.

Typically associativity of the integral is proved early on. If $X$ is a semimartingale and the integral $K \cdot X = \int K dX$ makes sense then the integral $(HK) \cdot X = \int HK dX$ makes sense if ...
nullUser's user avatar
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7 votes
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Why does not noise accumulate in a stochastic differential equation as it would in a random walk?

The chaotic attractor of the Lorenz system is, as the name says, attracting. Thus any deviation introduced by the stochastic term is corrected towards the double spiral. One still gets a trajectory ...
Lutz Lehmann's user avatar
7 votes
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How was the Ornstein–Uhlenbeck process originally constructed?

The O-U process was introduced as a solution to a Langevin equation $dX_t = - X_t \, dt +dB_t$ by Ornstein & Uhlenbeck (1930), and the solution of that equation was made rigourous by Doob in 1942,...
John Dawkins's user avatar
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7 votes

A Gauss-Markov process is always a solution to a Langevin type SDE?

What the book you are quoting is almost true. It has in fact become very popular in applied mathematics to identify solutions to the Langevin SDE with Gauss-Markov processes. The exact story goes as ...
Kurt G.'s user avatar
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6 votes
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Find closed-form solution to the SDE $dX_t = dt + 2 \sqrt{X_t} \, dW_t$

Hints: Suppose that $(X_t)_{t \geq 0}$ is a solution to the SDE. Using Itô's formula show that $Y_t := \sqrt{X_t}$ satisfies $$dY_t = dW_t.$$ (See the second part of this answer to get a better ...
saz's user avatar
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6 votes
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What does the drift and diffusion coefficent of a solution to an SDE tell us in general?

Let $(X_t)_{t \geq 0}$ be a solution to the SDE $$dX_t = \sigma(X_t) \, dW_t + b(X_t) \, dt, \qquad X_0 =x. \tag{1}$$ Throughout my answer, I will assume that the drift coefficient $b$ and the ...
saz's user avatar
  • 120k
6 votes
Accepted

Show that a SDE has no solution

You didn't apply Itô's formula correctly; it should read $$-X_t^3 1_{\{X_t<0\}} = -3 \left( \int_0^t X_s^2 1_{\{X_s<0\}} \, dX_s + \int_0^t X_s 1_{\{X_s<0\}} \, d\color{red}{\langle X \...
saz's user avatar
  • 120k
6 votes

Show that $dX_t=\frac{X_t}{1-t}dt+dW_t$ can be written as $X_t=(1-t)\int_{0}^{t}\frac{1}{1-s}dW_s$

Let $Y_{t} = \int_{0}^{t} \frac{1}{1-s}\, dW_{s}$. Next take a look at $$X_{t} = (1-t) \int_{0}^{t} \frac{1}{1-s}\, dW_{s} = (1-t)Y_{t}$$ and differentiate using It^o's lemma \begin{align*} dX_{t} &...
Christopher K's user avatar
6 votes

Computing expected value of a function along a Brownian Path

Write $$E[\exp\big(t + \frac{1}{2}W(t)\big)]=e^{t} \cdot E[\exp\big( \frac{1}{2}W(t)\big)]$$ at this point remember the fact that $W(t)$ is a centered Gaussian r.v. with variance equal to $t$, and use ...
Chaos's user avatar
  • 3,327
6 votes

A question about SDE and geometric Brownian motion.

I find notation such as $\frac{dN_t}{N_t}=rdt+\alpha dB_t$ incredibly unfortunate, it leads to the type of confusion you have encountered (and that I used to encounter myself). That is why I advise ...
Jan Stuller's user avatar
  • 1,219
6 votes

How was the Ornstein–Uhlenbeck process originally constructed?

The answer by John Dawkins already covers most of the bases; I'll add two points: The Langevin equation was considered independently by Ornstein in a 1919 paper 1; Ornstein attributes it to the PhD ...
Jose Avilez's user avatar
  • 12.7k
6 votes

Expected value of $S_t$ where $dS_t=(\mu S_t+a)dt + (\sigma S_t +b)dW_t$

Following @KurtG's comment, you may observe that $\beta_t M_t = \int_0^t \beta_s dW_s$, being the Itô integral of an $L^2(\Omega \times [0,t])$ process, is a martingale, so that $E(\beta_t M_t) = 0$. ...
Jose Avilez's user avatar
  • 12.7k

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