Cylindrical Projections of Occupied Diffusions

1. Problem Statement

The paper addresses the computational intractability of simulating Occupied Diffusions, a class of path-dependent stochastic processes. These processes are modeled by Stochastic Differential Equations (OSDEs) where the coefficients depend on the occupation measure of the process's history.

The dynamics are given by:
$$\begin{cases} d O_t = \delta_{X_t} \lambda(O_t, X_t) dt \\ d X_t = b(O_t, X_t) dt + \sigma(O_t, X_t) dW_t \end{cases}$$
where:

• $X_t \in \mathbb{R}^d$ is the state process.
• $O_t$ is a finite measure on $\mathbb{R}^d$ representing the time spent by the process in various regions (the occupation flow).
• The dependence of coefficients ($b, \sigma, \lambda$) on $O_t$ introduces infinite-dimensional path dependence.

The Core Challenge: While the system is Markovian on the product space of measures and Euclidean space ($\mathcal{M}(\mathbb{R}^d) \times \mathbb{R}^d$), the state space is infinite-dimensional. Direct simulation is impossible because one cannot store or update the full measure $O_t$ at every time step. Existing methods struggle to approximate this infinite-dimensional object while controlling the error.

2. Methodology: Cylindrical Projections

The authors propose a numerical approximation method called Cylindrical Projections to reduce the infinite-dimensional occupation measure to a finite-dimensional system.

A. Cylindrical Structure

The method relies on identifying a finite measure $o$ with a sequence of its integrals against a countable family of test functions $(f_k)_{k \in \mathbb{N}}$.

• Cylindrical Norm: A norm is defined on the space of measures using a separating family of functions $(g_j)$:
  $$\|o\| := \left( \sum_{j=0}^\infty |o(g_j)|^2 \right)^{1/2}$$
• Projection: For a truncation level $K$, the occupation measure $O_t$ is projected onto a finite-dimensional vector $Z_t^K \in \mathbb{R}^{N_K+1}$:
  $$Z_t^K := (O_t(f_0), \dots, O_t(f_{N_K}))$$
  where $(f_k^K)$ is a partition of unity subordinate to a cover of the state space.

B. The Approximated System

The original OSDE is approximated by a finite-dimensional system $(X_t^K, Z_t^K)$:
$$\begin{cases} d Z_t^K = f^K(X_t^K) \lambda^K(Z_t^K, X_t^K) dt \\ d X_t^K = b^K(Z_t^K, X_t^K) dt + \sigma^K(Z_t^K, X_t^K) dW_t \end{cases}$$
Here, the coefficients are "lifted" back to the measure space via a reconstruction operator $o^K(z) = \sum z_k \delta_{x_k}$. This transforms the problem into a standard finite-dimensional SDE that can be simulated using standard schemes like Euler–Maruyama.

3. Key Contributions

1. Theoretical Framework: Introduction of cylindrical projections as a rigorous tool to approximate path-dependent dynamics driven by occupation measures.
2. Strong Convergence Rates: The authors prove that the approximation converges strongly to the true process. Specifically, for a fixed time horizon $T$:
   $$\mathbb{E}\left[ \sup_{0 \le t \le T} |X_t - X_t^K|^2 \right]^{1/2} \le \frac{C}{K}$$
   This establishes a linear convergence rate with respect to the truncation level $K$.
3. Weak Error Analysis: The paper extends the analysis to weak convergence (expectations of functionals), showing that for Lipschitz functionals $\phi$:
   $$|\mathbb{E}[\phi(O_T, X_T)] - \mathbb{E}[\phi(o^K(Z_T^K), X_T^K)]| \le \frac{C}{K}$$
   This is crucial for pricing financial derivatives.
4. Handling Unbounded Coefficients: The authors address cases where the coefficients might grow linearly (Lipschitz condition) by using localization techniques (stopping times) and deriving convergence rates that depend on logarithmic factors if the growth is unbounded.

4. Numerical Results and Examples

The method is validated through Euler–Maruyama simulations on three distinct examples:

• Cranston–Le Jan Self-Attracting Diffusion:
  • A process attracted to its own running average.
  • An explicit strong solution exists, allowing for direct error comparison.
  • Result: The numerical error decreases linearly with $K$, confirming the theoretical $O(1/K)$ rate.
• Raimond's Self-Interacting Diffusion:
  • A multi-dimensional diffusion where the drift depends on the direction of past positions.
  • Since no explicit solution exists, the method compares the projection against a high-resolution reference ($K=75$).
  • Result: Convergence is observed, limited only by sampling noise and discretization error.
• Local Occupied Volatility (LOV) Model:
  • A financial model where volatility depends on the exponential time occupation flow.
  • Result: The method successfully simulates path-dependent volatility, demonstrating applicability to complex financial derivatives.

5. Significance and Applications

• Computational Feasibility: The paper provides the first practical, computationally tractable method for simulating general occupied diffusions, which were previously considered intractable due to infinite dimensionality.
• Financial Derivatives Pricing: The weak error analysis directly supports the use of Monte Carlo methods for pricing path-dependent options (e.g., Asian options) in models with path-dependent volatility (LOV).
• Generalizability: The cylindrical projection approach is versatile and can be applied to other measure-valued processes, including McKean-Vlasov type dynamics and systems with delayed interactions.
• Efficiency: Unlike general path-dependent SDEs where the state dimension grows with time steps, the cylindrical projection maintains a fixed finite dimension $K$, offering significant storage and computational advantages.

Conclusion

Tissot-Daguette and Zhang successfully bridge the gap between the theoretical existence of occupied diffusions and their practical simulation. By introducing cylindrical projections, they reduce an infinite-dimensional problem to a finite-dimensional one with rigorous error bounds ($O(1/K)$), enabling the simulation of complex self-interacting systems and the pricing of sophisticated financial derivatives.