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Questions tagged [stochastic-differential-equations]

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, …), and control and signal processing (controller, filtering, …).

SDEs are used to model various phenomena such as stock price diffusions or physical systems subject to thermal fluctuations. Typically, SDEs are often driven by Brownian motion or other continuous martingales. However, other types of random behavior are possible, such as jump processes--for instance a Poisson process.

Early work on SDEs was done to describe Brownian motion in Einstein's famous paper, and at the same time by Smoluchowski. However, one of the earlier works related to Brownian motion is credited to Bachelier (1900) in his thesis 'Théorie de la spéculation'. This work was followed upon by Langevin. Later Itô and Stratonovich put SDEs on more solid mathematical footing.

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Conditioning on initial condition in stochastic differential equation

Consider the following stochastic differential equation $$ \mathrm d X_t = b(X_t) \, \mathrm d t + \mathrm d W_t, \qquad X_0 = \xi, \qquad \xi \sim \mu $$ where $(W_t)_{t \geq 0}$ is a standard Brownian motion on $\mathbb R$ and $\mu \in \mathcal…
Roberto Rastapopoulos
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Second order SDE solutions

I have the following stochastic process: $$\ddot x(t)=a(t)+\nu(t)$$ where $a(t)$ is a deterministic function of the time and $\nu(t)$ is a random variable: $$\nu(t)\in\mathcal{N}(0,\sigma^2)$$ Obviuosly if $\nu(t)=0$, the solution would…
Riccardo.Alestra
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Solving SDE:$\frac{dy(t)}{dt}=(c+\sigma_wW(t))y(t)+\epsilon(t) $

I want so solve the following SDE. Specifically, I want to know if $y(t)$ is a Gaussian Process and if so the corresponding mean and covariance function. $$\frac{dy(t)}{dt}=(c+\sigma_wW(t))y(t)+\epsilon(t) $$ 11 where $W(t)$ is the Wiener process,…
Julian Karch
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Transforming $dX=-X^2dt+2X\circ dW$ (a Stratonovich SDE) to Ito form

As the title says, I need to transform Stratonovich SDEs to Ito form. I get similar results for some, but very different results in others. How do I do this? Thanks a lot! A) Stratonovich $dX=-X^2dt+2X\circ dW$ to Ito. Attempt: I believe that for a…
s1047857
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What's the solution of Stock price based on GBM model?

Stock price has a classic model based on GBM: $$dS = \mu S dt + \sigma S dW$$ based on this call options values could be solve -- Black-Scholes formula. But, what is the solution for the Stock price itself? is it $$S(t) = S(0) e^{\mu t + \sigma…
athos
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Different ways of solving SDE: $dX_{t} = cX_{t}dt + \sqrt{a + bX_{t}^2}dW_{t}$

I am trying to solve the following SDE: $dX_{t} = cX_{t}dt + \sqrt{a + bX_{t}^2}dW_{t}$, where a and b are positive constants, $W_t$ is a Wiener process, $X_0$ is given. I tried the trick to multiply $e^{-ct}$ on both sides and got $e^{-cT}X_{T} -…
Bandana
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Existence and Uniqueness of the solutions to SDE with locally Lipschitz coefficient and linear growth

The following theorem states the existence and uniqueness of a strong solution for a class of SDEs. Theorem. For the SDE $$ \mathrm{d} X=G(t, X(t)) \mathrm{d} t+H(t, X(t)) \mathrm{d} W(t), \quad X\left(t_{0}\right)=X_{0}, $$ assume the following…
Mark
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Solving SDE with sign function in drift term?

Consider the following SDE with $X_0 = 1$, $$ dX_t = X_t\operatorname{sign}(X_t) \, dt + X_t \, dW_t, $$ where $\operatorname{sign}(x) = \mathbb{1}_{\{x \ge 0\}} -\mathbb{1}_{\{x < 0\}}$. How am I supposed to solve this SDE?
Van Tom
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Time trend in stochastic differential equation (SDE)

(just joined, this is my first post), I've been perusing SDE’s, and a simple one is $$\mathrm{d} Y(t) = Y(t)\hspace{0.1cm}\mathrm{d}W(t)$$ where $W$ is stochastic. The solution is $$ Y = \text{exp}[W(t) - \frac{t}{2}] $$ or $\ln(Y) = W(t) -…
Daniel
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Solve Ornstein-Uhlenbeck SDE : $dX_t=(-\alpha +\beta X_t)dt+\sigma dB_t.$

I have an SDE that looks that Ornstein-Uhlenbeck SDE : $$dX_t=(-\alpha +\beta X_t)dt+\sigma dB_t.$$ I know that $$d(X_te^{\beta t})=e^{-\beta t}dX_t-e^{\beta t}X_t\beta dt=e^{-\beta t}(-\alpha +\beta X_t)dt-e^{\beta t}\sigma dB_t+e^{\beta t}X_t\beta…
John
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Is there an example of nonexistence of a weak solution to a stochastic differential equation?

There are many results on the existence and uniqueness of weak solutions to SDEs, even with discontinuous drift, and many examples of nonexistence or nonuniqueness of strong solutions. Are there examples of nonexistence, or nonuniqueness in…
Sander Heinsalu
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Solving non linear SDE

I am a little stuck in finding the solution of a non linear SDE. Hope you can help me out. The SDE has the form $dX_t = X_t^2 dt + dB_t$, where $B$ is a Brownian motion. Assuming $f(t,B_t) = X_t$ and using Ito doesnt do the trick. Thanks in…
user2736062
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A SDE with local lipschitz coefficient

Since $\exp(\cdot)$ is local lipschitz, the following sde has s strong solution $$ \mathrm{d}X_s=\exp(X_s)\mathrm{d}B_s, X(0)=1, $$ where $B$ is a Brownian Motion. I wander if the following expression…
yxyt
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Show stochastic exponential is the inverse of the stochastic logarithm

For an arbitrary semimartingale $Y_t$ show that $\mathcal{E}(\mathcal{L}(Y_t)) = Y_t/Y_0$. I saw this post and tried to do the same for a non necessarily continuous process but can not really see the equality of the terms involving the jumps of the…
Barreto
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Is stochastic differential equation a continuous mapping?

We have a well-defined SDE: $$ {\rm d}X_t=\mu(X_t){\rm d}t+\sigma(X_t) {\rm d}B_t, $$ where the initial condition $X_0$ is a known r.v., and $B_t$ is a standard Brownian motion. Can we say the above SDE is a "continuous"-mapping $f:\mathbb{R}\times…
Mike HWANG
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