Markov Recombining Scenario Tree (MRST)

• MRST is a framework that discretizes continuous-time Markov processes and SDEs using moment-matching and a recombining tree structure to ensure computational efficiency.
• It leverages universal lattices and Carathéodory reduction to maintain sparse transition matrices, reducing exponential state growth to polynomial growth.
• MRST is applied in quantitative finance, stochastic programming, and control, offering improved scalability and weak convergence rates compared to classical methods.

The Markov Recombining Scenario Tree (MRST) framework unifies a family of efficient tree-based discretizations for Markovian stochastic processes and stochastic differential equations (SDEs). The MRST methodology is characterized by its recombining structure: the number of reachable states at each time step grows only polynomially with the time horizon, as opposed to the exponential growth inherent in classical scenario or binomial trees. This sparsity is achieved through moment-matching, recombination, and the exploitation of universal lattices or reusable empirical trajectories, supporting both simulation-based and direct model-driven applications in multistage stochastic programming, quantitative finance, and stochastic SDE approximation (Cosentino et al., 2021, Park et al., 2024, Akyıldırım et al., 2012).

1. Conceptual Foundation and Definitions

The MRST methodology addresses the approximation of continuous-time diffusion processes and multistage stochastic programs by constructing discrete-time, discrete-space Markov chains or scenario trees exhibiting full recombination. In exemplar settings—such as the approximation of a $d$-dimensional Itô SDE,

$$dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t, \qquad X_0 = x \in \mathbb{R}^d,$$

—the discrete-time Markov chain $X^n = (X^n_i)_{i=0,...,n}$ evolves on a recombining tree whose nodes lie on a fixed universal lattice $\gamma_n \mathbb{Z}^d$ with mesh spacing $\gamma_n \to 0$ as $n \to \infty$ (Cosentino et al., 2021).

Alternatively, in data-driven settings where only sample trajectories from unknown Markov processes are available, MRST yields an approximate tree by alternately reusing two long empirical trajectories at successive stages, enforcing recombination by design and enabling efficient nonparametric estimation of conditional expectations (Park et al., 2024).

2. Algorithmic Frameworks

Lattice MRST for SDE Approximation

The construction for SDEs employs the following principles (Cosentino et al., 2021):

• Universal Lattice Support: All states are constrained to the mesh $\gamma_n \mathbb{Z}^d$, with $\gamma_n = C n^{-1/2}$ (elliptic case), ensuring the number of attainable states grows polynomially.
• Local Moment Matching: At each node $x$ and time-step $h=1/n$, the increment law $Y_x$ is chosen (via Carathéodory’s theorem) so that

$$\mathbb{E}[Y_x] = \mu(x) h + O(h^{1+\alpha}), \qquad \mathbb{E}[Y_x^{\otimes 2}] = \Sigma(x) h + \mu(x)^{\otimes 2} h^2 + O(h^{1+\alpha}),$$

with $\Sigma(x) = \sigma(x) \sigma(x)^\top$.

• Sparse Transition Matrix: The transition law at each node has $O(d)$ nonzero entries, supporting efficient matrix computation and storage.

Data-Driven MRST for Multistage Stochastic Programming

For problems with unknown or nonparametric Markovian dynamics, MRST is constructed as follows (Park et al., 2024):

• Trajectory Alternation: Two independent historical sample trajectories $$\tilde{\gamma}^{(a)}_{[1:N+1]}, \tilde{\gamma}^{(b)}_{[1:N+1]}$$ are alternately reused to form stagewise node sets ($N$ nodes per stage), with all parent nodes at stage $t$ sharing the same successor set at $t+1$.
• Full Recombination: Every path sharing the same node at stage $t$ recombines into the same set of successors, yielding $O(TN)$ nodes instead of $N^T$.
• Kernel-Based Conditional Expectation: At each node, empirical conditional expectations are estimated using kernel weights over the available trajectory, facilitating a nonparametric dynamic programming recursion.

Markov Tree Construction for Stochastic Volatility

For models such as Heston’s stochastic volatility, the MRST is realized by constructing a Markov process on a multidimensional binomial lattice, with component-wise moment-matching and explicit update rules. The state includes positions and auxiliary random walk indices to preserve the Markov property and ensure recombination (Akyıldırım et al., 2012).

3. Recombination, Sparsity, and State Growth

MRST frameworks achieve full recombination and polynomial state growth:

• Polynomial State Bounds: In $d$ dimensions, the number of distinct states at time $i$ is $O(i^d)$, as paths sharing the same lattice location recombine. For $d=1$ this is linear, for $d=2$ quadratic, and in general, $O(i^d)$ or, for models with linear drift/volatility growth, $O(i^d \exp(O(i/\sqrt n)))$ (Cosentino et al., 2021).
• Sparse Representation: Both the transition matrices and scenario trees exhibit $O(d)$ nonzeros per row or $O(TN)$ nodes per stage, optimizing computational tractability for high-dimensional or long-horizon problems (Cosentino et al., 2021, Park et al., 2024).

4. Moment-Matching and Carathéodory Reduction

MRST schemes use Carathéodory’s theorem to minimize support size while realizing prescribed moment constraints at each node:

• Dimension Reduction: Matching first and second moments, the support size $r$ is at most $d+2$ for $d=1,2$ and $d^2+1$ in general, corresponding to the dimension of moment vectors plus one (Cosentino et al., 2021).
• Spectral Decomposition: In higher dimensions, covariance matrices are diagonalized (e.g., $\Sigma(x)=Q \Lambda Q^\top$), and moment-matching proceeds on rotated lattices.
• Quadratic Programs for Degenerate Cases: Where uniform ellipticity fails, local quadratic programs are solved to approximate desired moments, followed by Carathéodory reduction to obtain a sparse measure (Cosentino et al., 2021).

5. Theoretical Guarantees: Convergence and Complexity

Weak Convergence Rates

• SDE Approximation: Under standard smoothness and nondegeneracy, the MRST weakly approximates the law of the underlying SDE at rate $O(n^{-1/2})$ as $n\to\infty$. For more accurate rates ($O(n^{-1})$), higher-order moments must be matched (Cosentino et al., 2021).
• Stochastic Programming Optimality: For MRST-approximated dynamic programs with kernel-based conditional expectations, suboptimality accumulates additively, yielding overall bounds of $O(TN^{-1/(p+2)})$, and the required sample size to guarantee $\epsilon$-accuracy scales as $N = \widetilde{O}(T^{p+3}\epsilon^{-p-2})$ (Park et al., 2024).

Complexity Scaling

Context	Node or State Growth	Complexity
Lattice SDE, $d$ dim.	$O(i^d)$	$O(n^{d+1})$
Data-driven MRST, horizon $T$	$O(TN)$	$O(TN)$
Classical SAA, horizon $T$	$O(N^T)$	Exponential

MRST methods effectively surmount the curse of dimensionality with respect to the time horizon, both theoretically and empirically (Park et al., 2024).

6. Numerical Illustration and Applications

Empirical studies establish the efficacy and scalability of MRST:

• SDEs and Diffusions: In $d=2$ toy diffusions and mean-reverting Heston SDEs, MRST produces empirical moment errors decaying as $O(n^{-1/2})$, with quadratic state growth and sparse transitions ($O(10)$ nonzeros per row) (Cosentino et al., 2021).
• Option Pricing: For Heston-type models, the recombinant binomial lattice supports efficient evaluation of European and American options, with explicit backward induction and convergence (both weak and "extended weak" in Skorohod topology) to the stochastic volatility SDE limit (Akyıldırım et al., 2012).
• Multistage Control and Stochastic Programming: In linear quadratic Gaussian (LQG) control with Markovian noise, MRST delivers out-of-sample optimality gaps <1% using tree sizes increasing only polynomially in $T$, compared to the exponential sample size requirements and out-of-sample degradation in standard SAA (Park et al., 2024).

7. Comparison to Classical Methods and Impact

MRST generalizes and unifies methods for Markov chain approximations and scenario tree algorithms:

• Versus SAA: MRST requires only two long empirical trajectories for data-driven settings without knowledge of transition kernels, constructing fully recombining trees and achieving polynomial sample complexity. In contrast, SAA requires exponentially many samples and does not yield practical implementable policies for high $T$ (Park et al., 2024).
• Recombining Trees in Finance: The MRST generalizes classical binomial and trinomial approaches to higher dimensions and non-trivial volatility structures, supporting efficient pricing and risk assessment for a wide range of options (Akyıldırım et al., 2012, Cosentino et al., 2021).
• Universality: All models are handled on a single lattice, and the same recombination architecture supports multiple SDEs or Markov models on identical computational graphs, enhancing flexibility and robustness (Cosentino et al., 2021).

In summary, the MRST methodology represents a rigorously justified, computationally tractable approach to Markovian uncertainty in both model-driven and data-driven settings. Its core features—moment-matching, recombination, universal lattice support, and provable convergence—support its adoption in applications including option pricing, stochastic optimal control, and formal verification, especially where the interplay between weak accuracy and sparse computational structure is crucial (Cosentino et al., 2021, Park et al., 2024, Akyıldırım et al., 2012).