# Questions tagged [stochastic-calculus]

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

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### Is optimal stopping problem a special case of stochastic optimal control problem?

Let's say all the processes are Markov. Is optimal stopping problem a special case of stochastic optimal control problem, in the sense that, to choose an admissible stopping time in an optimal ...
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### Numerical Solution of SDEs

I want to find a sample path of the following stochastic process: $$dx(t)=f(x(t))dt+g(x(t))dB(t)$$ where $B$ is the Brownian motion. Let $x_0$ be an initial condition. Can I discrete the process as ...
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### Subordinator - conditions for the Levy triple

different books write Levy Khinchnin's formula in different ways and thus I have a problem with understanding the conditions that Levy's triple must satisfy for the process to be a subordinator. In my ...
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### Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process

We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
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### Deriving Euler discretisation of Vasicek model parameters

The Euler discretisation of Vasicek model is given with: $$\Delta r = \alpha (\beta - r ) \Delta t + \sigma \epsilon_t {\sqrt \Delta t}$$ (see page 91 here) As I understand and looking at others (...
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### Sample paths of a diffusion are almost surely non-constant?

Consider a scalar diffusion $X=(X_t)_{t\geq 0}$ given by $\mathrm{d}X_t = b(t,X_t)\mathrm{d}t + \sigma(t,X_t)\mathrm{d}B_t, \quad X_0 = \xi$ for sufficiently regular coefficients $b$ and $\sigma$ and ...
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### $\operatorname{cov}\left(\int_0^tf(s,T_1)\,dW(s,T_1),\int_0^tf(s,T_2)\,dW(s,T_1)\right)=\text{?}$

Define a random field $W$ so that: $W(t,T)$ is a standard Brownian motion for every $T$. $dW(t,T_1) \, dW(t,T_2)=c(t,T_1,T_2)\,dt$ $W(t,\cdot)$ is a a continuous function for every $t$. Suppose $f$ ...
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### show $E\int^\infty_0e^{-A_t}dA_t$ is bounded above

show $E\int^\infty_0e^{-A_t}dA_t$ is bounded above by constant, where $A_t$ is an increasing stochastic process of locally integrable variation starting from 0. $A$ is possible to be purely ...
I have a stochastic process given by $$dX(t)=X(1-X)dtv+X(1-X)\sigma dW(t)$$ Once X reaches $0$ or $1$, the process stays there for ever. (the drift and diffusion terms goes to zero). I would have ...