# Tagged Questions

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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### Finding a unique strong solution

I am brushing up on my stochastic approximation. I am having a hard time with the following problem. I have the equation dX$_t$ = ln(1+ X$_t^2$)dt + X$_t$dB$_t$ X$_0$ = x, with x ∈ ℝ I know that ...
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### Finding Stochastic processes

I have the following differential equation dX$_t$ = (r$\mu$X$_t$ + $\frac{r(r-1)}{2}σ^2X_t$)dt + rσX$_t$dB$_t$, X$_0$ = x, with x > 0. Here, r>0. I am having trouble figuring out how to find the ...
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### Why are we allowed to multiply the differential form of an SDE by a function?

Suppose for example that we have the following SDE: $$dX_{t} = a(X_{t})\,dt + b(X_{t}) \,dB_{t}.$$ What rigorous justification is there for then saying, for example, that we can multiply both sides ...
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### Applying Ito formula to Ito process

I would like to simplify the expression $\left(\phi(s_{1})\cdot(X_{s_{1}}-X_{s_{2}})+\phi(s_{2})\cdot(X_{s_{2}}-X_{s_{3}})+\ldots+\phi(s_{n-1})\cdot(X_{s_{n-1}}-X_{s_{n}})\right)^{2}$ where $X_{t}$ ...
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### Limiting behavior of sde

What can we say about the limiting behavior of Xt , as t goes to infinite , where Xt is the solution of the sde $$dX(t) = e^{-t}X(t)dB(t)$$ ?
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### Expected value of stochastic process [closed]

How can i calculate the expected value of $$S_t= S_0 \exp\left( mt-\frac12\int_{0}^{t}e^{2Y_s}ds+\int_{0}^{t}e^{Y_s}dB_s\right)\quad$$ where $${Y_t}$$ is the solution of a sde and follows tha normal ...
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