[Paper Review] Weak and strong solutions of general stochastic models

Thomas G. Kurtz|arXiv (Cornell University)|May 29, 2013

Stochastic processes and financial applications7 references4 citations

TL;DR

This paper generalizes the Yamada-Watanabe theorems to abstract stochastic models by introducing a compatibility condition between inputs and outputs, establishing that weak solutions with pathwise uniqueness imply the existence of strong solutions on any probability space supporting the inputs. The key contribution is a unified framework linking weak and strong solutions in general stochastic settings, including SDEs, SPDEs, and backward SDEs.

ABSTRACT

Typically, a stochastic model relates stochastic "inputs" and, perhaps, controls to stochastic "outputs". A general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations is given in this context. A notion of {\em compatibility} between inputs and outputs is critical in relating the general result to its classical forebears. The relationship between the compatibility condition and the usual formulation of stochastic differential equations driven by semimartingales is discussed.

Motivation & Objective

• To extend the classical Yamada-Watanabe and Engelbert theorems to general stochastic models beyond Itô diffusions.
• To formalize the relationship between weak and strong solutions in abstract settings involving stochastic inputs and outputs.
• To introduce and analyze the concept of compatibility between input and output processes as a key condition for linking weak and strong solutions.
• To demonstrate that pathwise uniqueness and weak solution existence imply strong solution existence on any probability space supporting the inputs.
• To unify the treatment of various stochastic models, including SDEs, SPDEs, and backward SDEs, under a single theoretical framework.

Proposed method

• Formalizes stochastic models as constraints relating stochastic inputs $Y$ (in $S_2$) and outputs $X$ (in $S_1$), with $S_1$ and $S_2$ as complete, separable metric spaces.
• Defines weak solutions as joint distributions $\mu_{X,Y} \in \mathcal{S}_{\Gamma,\nu}$ with input distribution $\nu$, and strong solutions as those where $X = F(Y)$ a.s. for a Borel measurable $F$.
• Introduces the compatibility condition between $X$ and $Y$ via filtration compatibility, ensuring that the output process is adapted to the input filtration.
• Applies the Jakubowski topology and weak convergence techniques to prove relative compactness of approximating sequences in $D_{\mathbb{R}^d}[0,\infty)$.
• Uses the Meyer-Zheng conditions to establish tightness and convergence of approximating sequences under appropriate growth conditions on coefficients.
• Applies the main theorem to time-changed SDEs and martingale problems, showing that limit points of Euler approximations satisfy the original equation under compatibility.

Experimental results

Research questions

• RQ1 Under what conditions does the existence of a weak solution imply the existence of a strong solution in general stochastic models?
• RQ2 How can the concept of compatibility between input and output processes be formalized to generalize the Yamada-Watanabe framework?
• RQ3 What role does pathwise uniqueness play in ensuring the existence of strong solutions across all probability spaces supporting the inputs?
• RQ4 How can approximation schemes (e.g., Euler-type) be used to construct weak solutions and ensure convergence to a compatible solution?
• RQ5 In what general classes of stochastic models—beyond classical SDEs—does the weak-strong solution equivalence hold?

Key findings

• Weak solutions with pathwise uniqueness imply the existence of strong solutions on any probability space supporting the inputs, generalizing the classical Yamada-Watanabe theorem.
• The compatibility condition between $X$ and $Y$ ensures that the output process is adapted to the input filtration, enabling the construction of strong solutions.
• For time-changed SDEs of the form $X(t) = X(0) + \sum_k W_k(\int_0^t \beta_k(X(s))ds)\zeta_k + \int_0^t F(X(s))ds$, any limit point of Euler approximations is a compatible solution if the coefficients are continuous.
• Under appropriate growth conditions (e.g., boundedness), the sequence of Euler approximations $\{X_n\}$ is relatively compact in $D_{\mathbb{R}^d}[0,\infty)$, ensuring convergence in distribution.
• The joint distribution of the solution is determined by the input distribution $\nu$ when a strong solution exists, making strong solutions a distributional property.
• The martingale problem formulation is recovered as a special case: if $X$ is compatible with $Y$, then $X$ solves the martingale problem for the generator $Af(x) = \frac{1}{2}\sum_{i,j} a_{ij}(x)\partial_i\partial_j f(x) + F(x)\cdot \nabla f(x)$ with $a(x) = \sum_k \beta_k(x)\zeta_k\zeta_k^T$.