Probabilistic Interpretation of Solutions of Linear Ultraparabolic Equations

: We demonstrate the existence, uniqueness and Galerkin approximatation of linear ultraparabolic terminal value/infinite-horizon problems on unbounded spatial domains. Furthermore, we provide a probabilistic interpretation of the solution in terms of the expectation of an associated ultradiffusion process.


Introduction
The connection between parabolic equations and diffusion processes is well understood; the same cannot be said for ultraparabolic equations and ultradiffusion processes. Until recently, theoretical results have been fairly limited relative to the existence and uniqueness of solutions to ultraparabolic equations, deriving from two methodologies. In one, the analysis is affected along the characteristic of the first-order temporal operator, requiring that the speed of propagation varies only spatially. Such an approach was developed by Piskunov [1] in the classical case and extended by Lions [2] to the generalized sense. The second approach is based on the method of fundamental solutions and was implemented by Il'in [3] for the classical Cauchy problem and extended to more general domains via convolution by Vladimirov and Drožžinov [4], albeit at the expense of necessitating constant coefficients in the operator. Recently, however, using energic techniques Marcozzi [5] has established the well-posedness and Galerkin approximation of the generalized solution (strong and weak) to the terminal value problem for square integrable data on bounded temporal and spatial domains. We extend here the results of [5] to linear ultraparabolic terminal value/infinite-horizon temporal problems posed on unbounded spatial domains. We then provide a probabilistic interpretation of the solution in terms of the expectation of an associated ultradiffusion process.
Historically, the connection between the expectation of ultradiffusion processess and the solution to ultradiffusion equations arose from the work of Kolmogorov [6,7] and Uhlenbeck and Ornstein [8] in relation to Brownian motion in phase space-the same with respect to Chandrasekhar [9] in the context of boundary layers and Marshak [10] relative to the Bolzmann equation. A contemporary example may be found in the formulation of so-called Asian options from mathematical finance (cf. [11]), which obtains theoretical context with the present results. The paper is organized as follows. In Section 2, we consider deterministic aspects of the problem, while, in Section 3, the probabilistic interpretation is presented. Appendix A introduces certain regularity results, which, while essential for the analysis, are too extensive to prove in full. In Appendix B, we show formally that the ultraparabolic/ultradiffusion association is locally that of a parameterized parabolic/diffusion.

Approximation Solvability
We consider here the existence, uniqueness and approximation of the terminal value/infinite horizon problem on unbounded spatial domains for the linear ultraparabolic equations. To this , for some finite T > 0. The functional setting will be the weighted Sobolev spaces defined as follows. Spatially, we let with their respective norms which we equip with the norm , for all u ∈ X γ,µ . We associate with X γ,µ the dual space and the norm u * X * γ,µ , for all u * ∈ X γ,µ . In addition, let where ∇ t (u) = (∂u/∂t, ∂u/∂ϑ), which we associate with the norm for all u ∈ W γ,µ . Finally, we define We consider the ultraparabolic t-terminal value/infinite ϑ-horizon problem for u ∈ W γ,µ satisfying the evolutionary equation subject to the terminal condition where for some sufficiently large β.
Remark 1. We note that Equations (1)-(2) is an infinite horizon problem in ϑ. That is, the far-field behavior of ϑ is implicitly defined relative to the weight γ.

Remark 2.
In general, the validity of (11) will be problem dependent, predicated upon the spatial asymptotic behavior of u.
For t ∈ O T,ϑ , we define the operator A µ (t) : from which it follows that A µ (t) is linear, continuous, and strongly monotone by (11). In particular, we have and for all u ∈ V µ and t ∈ O T,ϑ .
We consider the regularization of (1)-(2) to domains of finite extent. To this end, it suffices for υ to have an extension to, or to be of compact support in, Q. Without loss of generality, we may assume that There exists a unique u m ∈ W m satisfying the ultraparabolic terminal value problem (cf. [5]) subject to the terminal conditions and boundary conditions We denote by u m the extension of u m by zero to the compliment of Q m .
Proof of Lemma 2. Taking the inner product of (16) with u m , we have Integrating the above over (0, T) × (ϑ, m), it follows that We obtain a supplementary estimate on ∇ t ( u m ).
Proof of Lemma 3. We consider the parabolic regularization with respect to ϑ of (16)-(19). To this end, The perturbation problem associated with (16)-(19) is: for any > 0, we seek u m satisfying the parabolic equation where and boundary conditions The problem (23)-(27) is well-posed, noting in particular the necessity of the auxiliary boundary condition (27).
We denote by u m the extension of u m by zero to the compliment of Q m . Taking the inner product of (23) with −n 2 γ ∂ u m /∂t, it follows that Integrating the above in time, we have that With (21), we proceed as per Lemma 2 to obtain which is valid for all u m ; passing to the limit, we obtain which holds for all m. We determine the estimate in ϑ analogously.
In order to show convergence of the regularizations u m , we have from (8) that Multiplying the above by n 2 γ and applying the Green's formula over O, we obtain and the result follows from (11) andũ m u in X γ,µ .

Probabilistic Interpretation
In order to provide a probabilistic interpretation of the solution to (1)-(2), we make the additional assumptions that a 2 , a 1 , a 0 , b ∈ C 1 Q ; a 2 bounded, f , We likewise suppose the existence of a function such that and satisfies the same assumptions as f . From Proposition 2 and Appendix A, we allow that there exists a unique solution u ∈ C 2,1 ( Q) ∩ C 0 Q to the problem (1)-(2).
Moreover, by extending the functions a, b, and σ outside of Q, we may assume that as well as |σ| ≤ K o and |a(t, ϑ, for all t ∈ O T,ϑ . We now seek a probabilistic interpretation of the function u satisfying (1)-(2) by constructing a stochastic differential equation for which the trajectories (Θ(t), X(t)) are the characteristics of −∂(b ·)/∂ϑ + A. To this end, we take a probability space (Ω, A, P), an increasing family of sub-σ-algebras F t of A, and a R-valued standardized Wiener process w(t), which is an F t martingale. We can then consider, on an arbitrary finite interval, the stochastic differential equation where x and ϑ are fixed and non-random; the solution of (51)-(54) is unique.

Proposition 4.
The assumptions of Proposition 2, as well as (34) through (46); the solution of (1)-(2) is given by Proof of Proposition 4. The proof relies on the existence and uniqueness of the regular solution to the ultraparabolic terminal value problem (1)- (2). With this exception, the result is standard such that we will provide only a brief exposition, deferring to e.g., ([12], Chapter 2, Theorem 7.4). For the process (Θ(t), X(t)), we have that for all s ∈ [t, T] and all k ∈ N, in which case the right-hand side of (55) is well-defined.
Differentiating the functional Ψ · Z, applying Ito's formula to Ψ, and integrating from t to T, we obtain Ψ(T, Θ(T), X(T)) = Ψ(t, ϑ, x) From (56) with k = m and the assumptions (44) on the growth of ∂Ψ/∂x, we have that the expectation of the stochastic integral is defined and is equal to zero. We therefore have that

s, Θ(s), X(s))] Z(s)
and so the problem reduces to proving (55) with υ = 0, with f replaced by f − g, and with u replaced by u − Ψ and a solution to (1)- (2) corresponding to data f − g and 0.
We therefore assume υ = 0; we prove that We start by considering the bounded case. We approximate f by f N M defined by and We note that u MN is bounded. This follows as per ( [5], Prop. 2'). We show: To this end, let O R = ξ ∈ R 2 | |ξ| ≤ R and τ R be the exit time form O R of the process (Θ(t), X(t)). We can suppose that the (fixed) initial data (x, ϑ) of (51)-(54) belongs to O R , for R that is sufficiently large. That is, we have, from the continuity of the process a.s. τ R ≥ T for some for R ≥ R 0 (ω). As above, with the use of Ito's formula applied to u N M between the instants t and τ R ∧ T − , taking → 0, and using the continuity of u N M on Q, we have However, from (63), we have as R → ∞. Application of Lesbesque's theorem then provides the result (62). From the estimates it follows that u N M lies in a bounded subset of L 2 ( O T,ϑ ; H 1 µ ) and we obtain (59) by proceeding to the limit successively in M and N.

Conclusions
We have demonstrated the existence and uniqueness of the solution to linear ultraparabolic equations on unbounded domains, both spatial and temporal, as well as the strong convergence of the regularized problem, providing a basis for the subsequent application of a Galerkin approximation. Furthermore, we present a probabilistic interpretation of the solution in terms of the expectation of an associative ultradiffusion process. In practice, the usefulness of this result often stems from the converse formulation; that is, one often wishes to obtain the discounted expectation associated with an ultradiffusion process, e.g., the valuation of an Asian option in mathematical finance (cf. [11]). To this end, the regularity assumptions of Section 3 are necessary for the existence of the solution to the ultradiffusion process (3.5). With respect to a simple regular transformation, the associated ultraparabolic problem maintains the approximation solvability of Section 2, for which efficient and general numerical procedures are readily available (cf. [13][14][15]).
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflicts of interest.

Appendix A. Regularity
There exist two approaches to obtaining regularity; for parabolic differential equations of the second order, we note that Wloka [16] has provided regularity theorems based on raising the differentiability assumptions of the data while Ladyženskaja et al. [17] have taken the approach of increasing the p-power summability of the data. The results stated below follow the latter approach, the demonstration of which lies outside the scope of this manuscript. As regularity theorems are local, they do not require assumptions on the boundary or the boundedness of the domain .
We let W 1,2,p ( Q) denote the space of functions u such that for 1 ≤ p ≤ ∞. Here, the "1" refers to the order of temporal derivatives, and "2" refers to the number of spatial derivatives. If p = 2, we write W 1,2 ( Q), which we equip with the natural Banachand Hilbert-space norm. We denote by W 1,2,p loc ( Q) the space of functions u such that, for all test functions ϕ ∈ D( Q), the set of infinitely differentiable functions with compact support in Q, we have ϕ u ∈ W 1,2,p loc ( Q). We suppose that a 2 , a 1 , a 0 , b ∈ C 1 ( Q) .
Moreover, for v ∈ L p loc ( Q), we denote by Lv the following distribution on Q: for all ψ ∈ D( Q).
Proposition A1. Local Regularity. The assumptions of Proposition 2, as well as (65). Let u ∈ L p loc ( Q) be such that then u ∈ W 1,2,p loc ( Q), for p > 1.
In order to obtain results on the boundary, we set  Increased smoothness of the data may then be translated into smoothness of the solution.
Proposition A2. The assumptions of Proposition A1, as well as In particular, u ∈ C 1,2 ( Q).
Finally, the key regularity result is: Proposition A3. The assumptions of Proposition A2. If f ∈ L p ( Q) and υ = 0, then the unique solution to (1)-(2) also satisfies u ∈ L p loc ( Q).