# Questions tagged [sde]

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### Price a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$

Two stocks are modelled as follows: $$dS_{1t}=S_{1t}(\mu_1dt+\sigma_{11}dW_{1t}+\sigma_{12}dW_{2t})$$ $$dS_{2t}=S_{2t}(\mu_2dt+\sigma_{21}dW_{1t}+\sigma_{22}dW_{2t})$$ with $dW_{1t}dW_{2t}=\rho dt$....
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### Modelling support and resistance using sde

This initiative was sparked by the identification of cointegrated pairs, fitting them to an OU process, and devising an optimal strategy based on the OU process—areas that have already been well ...
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### State space equation of CARMA(p,q) processes

Thanks for visting my question:) I am currently working on Carma(p,q) processes and do not understand how to derive the state equation. So the CARMA(p,q) process is defined by: for $p>q$ the ...
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### Maximizing the expected log utility

Let's assume that we have a self-financing portfolio made by $\delta_t$ shares and $M_t$ cash, so that its infinitesimal variation is: $$dW_t = rM_t \, dt + \delta_t \, dS_t$$ We define $\alpha_t$ ...
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### Derivation of option pricing PIDE: Why does the drift need to be zero?

I started studying PIDE methods for option pricing and am struggling to understand or find the necessary theory that shows why the PIDE is obtained by the condition that the drift term has to be zero. ...
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### What is the difference between "stochastic" heat equation and just heat equation?

I am trying to understand the difference between the "stochastic" heat equation and the heat equation. Will i be wrong to say the stochastic heat equation is just the heat equationg with the ...
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In a few papers I have seen the following notation: $$\langle X_t \rangle$$ Also, in Bergomi's book, at page 8, we have the following equality: $$\biggr\langle \int_0^T e^{-rt}s^2 \frac{d^2P_{\hat{\... 0 votes 0 answers 93 views ### Alternatives to Lognormality for negative Prices If I would want to use a different type of distributions (i.e. to allow for negative prices) f.e. a beta distribution how would I have to start to proceed to apply it f.e. to a SDE of the type of a ... 2 votes 0 answers 52 views ### Solution to Stock Price SDE with mean reversion [duplicate] Suppose S_t follows the process (notice the S_t term in the diffusion part):$$ S_t := S_0 + \int_{h=t_0}^{h=t}\alpha(\mu -S_h)dh + \int_{h=t_0}^{h=t}\sigma S_h dW(h) $$. I actually don't know how ... 3 votes 2 answers 699 views ### SDE Jump-Diffusion If you combine the compound Poisson process with the Brownian motion you obtain the simplest case of a Jump-diffusion. Let’s define$$X_t = \mu t + \sigma W_t + J_t$$where W_t is a Wiener process ... 2 votes 0 answers 32 views ### Expression for the expectation of Integrated variance in case of GARCH(1,1) process I have the following SDE (GARCH(1,1)) for the instantaneous variance:$$ d\sigma_t^2 = \kappa (\theta - \sigma_t^2) dt + \psi \sigma_t^2 dW_t $$I would like to find an expression for IV_t = E[\int_{... 3 votes 1 answer 385 views ### Application of Ito's Lemma in expected utility theory An investor with utility curve U(.) has wealth X_t at time t. He invests A proportion p of his wealth in a risky asset that follows a geometric Brownian motion, with parameters \mu and \... 0 votes 0 answers 228 views ### Dynamic programming and Bellman equation to obtain the maximum This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM" Suppose an endowment economy where the representative ... 2 votes 1 answer 882 views ### Idea of using logarithm for solving SDE in Black-Scholes model In the Black-Scholes model they consider that the stock follows this stochastic differential equation:$$ dS = \mu S dt + \sigma S\ dW  I was wondering, was it common at the time they work on this ... 