# 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, …).

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### How to illustrate that the solution of $du=u(1+W)dt$ is ${\mathcal F}_t$ adapted?

Set a probability space $(\Omega, \mathcal F,P)$, and a one dimension wiener process $W$ on it, with filtration ${\mathcal F}_t=\sigma({W(s)|0 \le s \le t})$ which is generated by the wiener process. ...
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### Malliavin calculus for the regularity of the density of the supremum of a process

I am reading Chapter 2 from Nualart's book 'The Malliavin calculus and related topics'. Proposition 2.1.10 gives the conditions for the law of the supremum of a process to have a density. Condition (...
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### Convolution of exponential and multivariate normal

Let $i=1,2,\dots,n$. For $t \geq 0$, I have a SDE of the form: $$\frac{dX_{i,t}}{X_{i,t}} = \sigma_{i} dW_{i,t} + dJ_{t}$$ where $W_{i}$ is a Wiener process, $J_{t}$ is &...
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### References on arbitrage-free pricing and market completeness for multidimensional models with time and space dependent volatility [closed]

I am studying a multidimensional market model where the risky asset prices follow the SDE $$dS_t = \mu(t, S_t) \, dt + \sigma(t, S_t) \, dW_t,$$ along with a one-dimensional riskless bank account ...
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### Continuous time semimartingale that is not an Ito-Diffusion

Let $X_t$ be a continuous semimartingale. Does there exist $\mu, \sigma$ such that, $d X_t = \mu(X_t, t) dt + \sigma(X_t, t) dW$ If $X_t$ was only a semimartingale, then the answer is no, because I ...
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### Solution to General Linear Vector-Valued SDE w/ Diagonal Noise?

While the scalar case has a well known solution, does a closed-form solution exist for the vector-valued case where the diffusion matrix is diagonal? To be precise, $\vec{X}(t)$ is a $N$-dimensional ...
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### Showing bounds of Stochastic Process

Suppose that we have the SDE: $$dZ_t = 2Z_t(1-Z_t)dt + 4Z_t(1-Z_t)dB_t$$ With $Z_0 = \frac{1}{3}$. How can I show that $0 \leq Z_t \leq 1$. I have tried solving the 'alalogous' differential ...
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### Understanding the reverse SDE

When reading papers/blogs about diffusion models, it is stated that if $X_t$ is the unique solution to an SDE $$dX_t=f dt+g dB_t,$$ where $B_t$ is the standard Brownian motion (starting from $B_0=0$), ...
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### Connection between SDE and Doob-Meyer decomposition

Suppose that the stochastic process $(X_t)$ is the solution of the following SDE: $$dX_t = f_t dt + h_tdW_t,$$ (where $W_t$ is the stanrdard Wiener process) so that with probability $1$, it equals to ...
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### Feynman-Kac theorem of the weak solution of parabolic PDEs

Is there any reference on the Feynman-Kac theorem of the weak solution of parabolic PDEs? So far I can only find the one for classical solution.
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