Monte Carlo Estimation of a Conditional Expected Integral

Summary

The document asks how to simulate the conditional expectation of the time integral of the squared state of a Markovian stochastic differential equation. It decomposes the target into the integral already accumulated up to the conditioning time and a future conditional expectation given the state at that time. For the past segment, the author proposes simulating a path with Euler–Maruyama and numerically integrating its squared values, for example with a trapezoidal rule.

The main unresolved issue is estimating the future conditional expectation. The author considers starting a new Euler–Maruyama path from a value of the process at the conditioning time, but is unsure how that value should be chosen. The document contains no response, derivation, or validation, so it does not establish a complete estimator. It nevertheless frames the key distinction between simulating the realized past and estimating a conditional future quantity that depends on the current state.

Key ideas

• The target is a conditional expectation of the integral of a squared stochastic process.
• The proposed decomposition separates the realized integral before the conditioning time from the expected future integral.
• Euler–Maruyama simulation and numerical quadrature are proposed for estimating the past integral.
• The future term depends on the process state at the conditioning time and requires a conditional estimate.
• The document poses the simulation question but provides no answer or evidence that the proposed procedure is valid.

Tags

# Simulation of conditional expected value


# Simulation of conditional expected value












I'd like to simulate $$V_t = \mathbb{E}\Big[\int_0^T \theta_s^2 ds | \mathcal{F_t}\Big].$$ where $\theta$ is the solution of a certain Markovian SDE. To get one sample of $V_t$ my idea is to note that $$V_t = \int_0^t \theta_s^2 ds + \mathbb{E}\Big[\int_t^T \theta_s^2 ds | \theta_t\Big].$$ Then

- Generate a sample/trajectory of $\theta$ in $[0,t]$ using Euler-Maruyama method (EM) for the first term and then trapezioid method/other method to calculate the integral



- Add both terms

Is this procedure correct? I don't know how I am supposed to do step 2.1. Maybe 4.1. Choose a random value $x$ for $\theta_t$ 4.2. Generate a sample/trajectory of $\theta$ in $[t,T]$ using $x$ as starting value with EM, then estimate integral.

But again, I'm not sure how to do 4.1, for example I just choose a random $x \in \mathbb{R}$ with some criteria or I have to take account of the structure of the SDE in some way.

Thank you for your help.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.