SDE valuation equation (stochastic calculus)

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SUMMARY

The discussion centers on the stochastic differential equations (SDEs) governing stock price dynamics, specifically the equations $$dS_t = u_tS_tdt+\sqrt{v_t}S_tdZ_1$$ and $$dv_t = \alpha dt+\eta\beta\sqrt{v_t}dZ_2$$. The correlation between the Wiener processes, represented by $$\langle dZ_1 dZ_2\rangle = \rho dt$$, indicates that the processes are not independent, with $$\rho$$ denoting the correlation coefficient. The confusion arises from the interpretation of the inner product notation and the implications of the correlation on the joint behavior of the stock price and its variance.

PREREQUISITES
  • Understanding of stochastic calculus and SDEs
  • Familiarity with Wiener processes and their properties
  • Knowledge of financial mathematics, particularly in modeling stock prices
  • Concept of quadratic variation in stochastic processes
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  • Study the derivation and implications of the SDEs for financial modeling
  • Learn about the properties of correlated Wiener processes
  • Explore the concept of quadratic variation in depth
  • Investigate the role of volatility in financial models, particularly in stochastic volatility models
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Quantitative analysts, financial mathematicians, and anyone involved in modeling stock price dynamics using stochastic calculus.

cppIStough
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I read from a text: "suppose a stock with price ##S## and variance ##v## satisfies the SDE $$dS_t = u_tS_tdt+\sqrt{v_t}S_tdZ_1$$$$dv_t = \alpha dt+\eta\beta\sqrt{v_t}dZ_2$$ with $$\langle dZ_1 dZ_2\rangle = \rho dt$$ where ##\mu_t## is the drift of stock price returns, ##\eta## the volatility of volatility and ##\rho## the correlation between random stock price returns and changes in ##v_t##. ##dZ_1,dZ_2## are Weiner processes.

I don't really understand the third equation. Can someone help me make sense? I understand quadratic variation, but I thought ##dZ_1dZ_2 = 0## unless 1=2, which then implies ##dZ_1dZ_2 =dZ_1^2 = dt##; where does the ##\rho## come from, and I also don't understand the angled brackets (no definition from the text, is this supposed to be some inner product?)
 
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\rho is stated to be the correlation between the processes; see e.g. here.

For the meaning of angle brackets, see here.
 
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