# Tagged Questions

Stochastic differential equations (SDE) occur where a system described by differential equations is influenced by random noise. Stochastic differential equations are used in finance (interest rate, stock prices, …), biology (population, epidemics, …), physics (particles in fluids, thermal noise, …), ...

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### Question about using Ito Chain Rule to Understand an SDE

Given $X_t = (x^{1/3}+ \frac{B_t}{3})^3, t \ge 0$, $X_0 = x$, why does $$dX_t = \left(\frac{X_t^{1/3}}{3} dt + X_t^{2/3}dB_t\right)?$$ I am having trouble because differentiation of the given ...
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### Stochastic Differential Equation Time-Independent

I know that a generic 1-D SDE can be written in Ito form as: $dX_{t} = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$. I was curious as to how such an SDE is written when modelling time-independent processes. I ...
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### Proof about solution to SDE with Lipschitz condition:

$X_t$ follows the Ito process as described by the following stochastic differential equation $$dX_t=b(X_t)dt+dB_t\quad , \quad X_0=x$$ and $b(X)$ satisfies Lipschitz Coditions.I want to show for ...
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### How to prove a solution of SDE with Lipschitz condition is unbounded?

For Simplicity $(\Omega ,\mathbb F,\mathbb P)$ is a probability space, the S.D.E is $dX_t=b(X_t)dt+dB_t$ and it's 1 dimensional, $b(X_t)$ is Lipschitz Continuous (and let's assume with constant 1),...
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### Solving a nonlinear scalar Ito SDE

I need to solve the SDE: $$dX_t = (X_t)^3 dt + (X_t)^2 dW_t ; X(0)=1$$ Now what I found is this is an SDE of the form: $$dXt =a(X_t)dt+b(X_t)dW_t$$ where $a(x) = \frac{1}{2} b(x)b′(x)$ Using the ...
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### Solution to General Linear SDE

In order to find a solution for the general linear SDE \begin{align} dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t, \end{align} I assume that $a(t), b(t), g(t)$ and $h(t)$ ...
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### Is the generated semigroup by an elliptic operator be the transition semigroup?

I am considering the time homogeneous Ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dW_t,$$ where $b,\sigma$ currently are only assumed to be global Lipschitz for the existence of solution $X_t$. The ...
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### Why do we always consider real-valued $f$ in the Itō formula to find an expression for $f(t,X_t)$

The Itō formula (see Da Prato, Theorem 4.32) yields an expression for $f(t,X_t)$ where $${\rm d}X_t=\phi\;{\rm d}t+\Phi\;{\rm d}W_t\;,\;\;\;X_0=\xi\;.\tag 1$$ Even when $X$ takes values in a Hilbert ...
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### Chain rule for derivatives in SDE

I'm having trouble understanding applying chain rule to SDEs or actually chain rules in general. It has been a while since I took rudimentary calculus classes, so I might be slipping on the basics. ...
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### How to formulate and analyze systems of stochastic differential equations?

I'm having trouble finding reference material on how to deal with systems of stochastic differential equations. Specifically, I'm interested in ecological models. For example, consider the standard ...
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### Mathematical tools for iterative / composed function analysis

Given the following: a. x0 = known scalar b. x1 = known scalar c. x2 = unknown scalar d. an unknown iteratively applied stochastic nonlinear function $\mathbf{g}$ where     &...
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### For Ito diffusion, what is the difference between two measures $Q^x$ and $P$?

I am confused about the difference between $Q^{s,x}$ and $P$ for the following ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad t\ge s;\quad X_s=x.$$ Followings are from most books: given the ...
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### Markov property of ito diffusion [duplicate]

Most books show Ito diffusions satisfy Markov property, that is, $E[f(X_{t+h})\mid F_t]=E^{X_t}[f(X_h)]$. But I was wondering whether it's true that $E[f(X_{t+h})\mid X_t]=E^{X_t}[f(X_h)]$. In this ...
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### SDE Solution: Hull-White extension of Vasicek model

I am trying to figure out the particular ansatz (if that's all there is) for the solution to the SDE: $dr_t = [v_t - ar_t]dt + \sigma dW_t,$ where $a$ is constant and $v,t$ are, potentially, time-...
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### Applying ito integral to a specific example

Using the ito integral: $$x_t=\int_0^t b(s,x(s)) \, ds+\int_0^t \sigma (s,x(s)) \, dB(s)+x_0$$ I want to solve this: I have $$g(s,n(s))=\log(n(s))$$ Using ito's lemma taking the derivative I get ...
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### Solve the stochastic differential equation $dX_t = \alpha X_t \, dW_t + \sigma X_t \, dt$

We want to solve: $dX_t = \alpha X_t dW_t + \sigma X_t dt$ where the initial condition $X_0$ is given and $\alpha$, $\sigma$ are constant. The solution goes as follows - We rewrite the ...
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Converting a SDE from stratonovich to ito's form , Stratonovich form $$\partial X=P(X)\partial B$$ $$P(X)=I-n(X)n(X)^T$$ Ito's conversion $$dX=P(X)dB +\frac{1}{2}d(P(X))dB$$ $$dX=P(X)dB +\... 0answers 19 views ### Solve the SDE dX_t=\alpha \,dt + \sigma X_t \,dB_t, X_0=x_0 [duplicate] So I got the following SDE to solve:$$dX_t=\alpha\, dt + \sigma X_t \,dB_t, X_0=x_0$$This is what I've tried: Using Ito's I should get the following relations:$$X_t=f(s,x)\alpha = \frac{1}{...
I solved the following: SDE $dS_t=k\frac{dt}{t}+\sigma dW_t$ with the solution: $S_t=k\ln(t)+\sigma W_t$ assuming that $S_0=0$ (First, is this correct?). The mean and the variance are $k \ln(t)$ and \$\...