How to Evaluate This Stochastic Expectation Value?

In summary, the evaluation of a stochastic expectation value involves assessing the average outcome of a random process by integrating over all possible states and their probabilities. Key steps include defining the stochastic model, identifying relevant variables, calculating the probability distribution, and applying statistical methods to derive the expectation value. It is crucial to ensure that the model accurately reflects the underlying phenomena and to consider the implications of variability and uncertainty in the results.
  • #1
thatboi
133
18
Hi all,
I am not familiar with stochastic processes, but I would like to know how to evaluate the following expectation value: $$\mathbb{E}[e^{\int_{0}^{t}d\tau(V_{i}(\tau)-V_{j}(\tau))}]$$ where ##\mathbb{E}[V_{i}(t)] = 0,\mathbb{E}[V_{i}(t),V_{j}(t')] = \gamma\delta_{ij}\delta(t-t')## for some constant ##\gamma##.
Any assistance is appreciated.
 

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