A reflected moving boundary problem driven by space–time white noise

We study a system of two reflected SPDEs which share a moving boundary. The equations describe competition at an interface and are motivated by the modelling of the limit order book in financial markets. The derivative of the moving boundary is given by a function of the two SPDEs in their relative frames. We prove existence and uniqueness for the equations until blow-up, and show that the solution is global when the boundary speed is bounded. We also derive the expected Hölder continuity for the process and hence for the derivative of the moving boundary. Both the case when the spatial domains are given by fixed finite distances from the shared boundary, and when the spatial domains are the semi-infinite intervals on either side of the shared boundary are considered. In the second case, our results require us to further develop the known theory for reflected SPDEs on infinite spatial domains by extending the uniqueness theory and establishing the local Hölder continuity of the solutions.


Introduction
There are many models for the behaviour of interfaces that arise in physical, biological and financial problems. In this paper we explore interfaces in one dimension determined by competition between two types, which could be thought of as particles, species or offers to buy or sell, depending on the application. We think of a type as occupying a region on one side of the interface and evolving according to a reflected stochastic partial differential equation driven by white noise with a Dirichlet condition on the interface. The interface itself moves as a function of the profiles of the two types. One motivation is the evolution of the limit order book in a financial market. In this setting orders to buy or sell arrive at a rate determined by their distance to the best price and prices will rise or fall according to the order flow imbalance between the bid and ask sides of the book. It is also possible to build biological models in which two species interact at an interface and their behaviour is determined by the distance of individuals from the interface. We do not focus on the modelling aspects, instead our aim is to establish existence, uniqueness and some properties of solutions to equations of the form where u 1 and u 2 have spatial domains on either side of the point p(t) at any given time, and η 1 , η 2 are reflection measures which keep the profiles positive. We impose Dirichlet boundary conditions so that u 1 (t, p(t)) = u 2 (t, p(t)) = 0, with the point p(t) evolving according to the equation where h is a Lipschitz function mapping pairs of continuous functions to real numbers. The driving noiseẆ here is space-time white noise, whilst the drift and diffusion coefficients f i and σ i for i = 1, 2 depend on the spatial coordinate in the frame relative to the boundary, as well as the value of the solution itself at that point. Within the class of equations produced by this model are approximations of the Stefan problem, where the motion of the boundary would be given by for some γ ∈ R. The combination of the space-time white noise, moving boundary and the reflection measure make it difficult to find conditions which ensure differentiability of the profiles at the boundary, and so to arrive at an equation with precisely these dynamics. However, by choosing h to be a function which emphasises the mass close to the interface, so that the boundary is still being moved by the "relative pressure" of the two sides, we will have existence and uniqueness for a system which approximates the Stefan problem.

Reflected SPDE and SPDEs with moving boundaries in the literature
Reflected SPDEs of the type ∂u ∂t = u + f (x, u) + σ (x, u)Ẇ + η, (1.4) whereẆ is space-time white noise and η is a reflection measure, were initially studied in [8]. In [8], the domain for the equation is [0, T ] × [0, 1], with Dirichlet boundary conditions imposed on u. Existence and uniqueness are established for the equation in the case of constant volatility i.e. σ ≡ 1. Existence for the equation in the case when σ = σ (x, u) satisfies Lipschitz and linear growth conditions in its second argument was then established in [4] using a penalization method. The penalization approach was adapted in [9], in which the corresponding result is proved in the case when the domain for Eq. (1.4) is [0, T ] × R. Uniqueness for varying volatility σ = σ (x, u) on compact spatial domains was then proved by in [10]. The authors decouple the obstacle problem and SPDE components of the problem. This allows them to prove existence via a two-step Picard iteration, as well as uniqueness.
Similarly, there has been much recent work on moving boundary problems for SPDEs. In [6], existence and uniqueness for solutions to a Stefan problem for an SPDE driven by spatially coloured noise is proved. The corresponding problem in the case when the SPDE is driven by space-time white noise was then studied in [11] under the condition that the volatility vanishes quickly enough at the moving interface. More recent work on such problems include the models in [7] and in [5]. In these papers the focus is on essentially the same equations as (1.1) but without reflection and with coloured noise. Different boundary conditions are imposed at the interface in the two papers and in particular they are able to include a Brownian motion in the dynamics for the motion of the boundary. When thinking of the equations in [5,7] as models for the limit order book, the incorporation of a Brownian term ensures that the resulting price process is a semi-martingale.

Main results and contributions
In this paper we combine aspects of the models discussed above and consider the system of two reflected SPDEs sharing a moving boundary, (1.1). We examine the problem in two cases-firstly when the spatial domain is restricted to a fixed distance from the moving boundary and secondly when the spatial domain is infinite in both directions. Existence and uniqueness for the system is proved in both of these cases. Our approach is similar to that of [10]. As in [10], we decouple the problem into studying a deterministic obstacle problem and applying SPDE estimates. The non-Lipschitz term created by the moving boundary is controlled by a suitably truncated version of the problem, for which existence and uniqueness is proved by a Picard argument. Consistency among the solutions of the truncated problems allows us to piece these together to obtain a solution to our original problem which exists until some blow-up time.
In the case when the spatial domain is infinite, our analysis extends the known uniqueness theory for reflected SPDEs on infinite spatial domains. In [9], uniqueness for the reflected stochastic heat equation on R is proved in the case when σ ≡ 1. We obtain uniqueness for a class of volatility coefficients which are allowed to depend on space and the solution itself at that particular point in space-time. The main condition here is that the coefficient σ is Lipschitz in its second argument, with a Lipschitz constant which decays exponentially fast in the spatial variable.
The local Hölder continuity of the solutions to our equations is also established, in both the case when the spatial domain is finite and when it is infinite. As one might expect, solutions can be shown to be up to 1 4 -Hölder continuous in time and up to 1 2 -Hölder continuous in space. In the case of the infinite spatial domain, this is a new result even for static reflected SPDEs. We argue by suitably adapting the proof in [3], in which the corresponding result was proved for solutions to static reflected SPDEs on compact spatial domains. As a corollary, we can also show that the derivative of the boundary is up to 1 4 -Hölder continuous in time, with this regularity inherited from the solution and a Lipschitz condition on the function h in (1.2).
An outline of the paper is as follows. In Sect. 2 we will discuss the deterministic obstacle problem, a key ingredient in working with reflected processes. In Sect. 3 we look at the case where the SPDEs are supported on finite intervals in the moving frame. The case of semi-infinite intervals in the moving frame is the topic of Sect. 4. The heat kernel estimates necessary for obtaining the SPDE estimates as well as the proofs for Sect. 2 are in the "Appendix".

An application: limit order books
The majority of modern trading takes place in limit order markets. In a limit order market, all traders are able to place orders of three types. Limit orders are offers to buy/sell which do not lead to an immediate transaction; they only result in a transaction when they are matched with incoming market orders. Market orders are offers to buy/sell the asset which match with an existing limit order and so result in immediate transaction. Finally, traders are able to cancel limit orders which they previously placed. The order book itself at a given time is simply the record of unexecuted, uncancelled limit orders at that time. There has been much interest in trying to model the dynamics of the book, particularly in a high frequency setting.
As in [5,7], we can think of our Eq. (1.1) as being a model for the limit order book. In this context, we would think of the spatial variable as representing the price or the log-price. The random fields u 1 (t, x) and u 2 (t, x) would then be the density of limit orders existing at price x and time t on the bid and ask sides of the book respectively, and we can choose the function h to represent an approximation of the local imbalance at the mid price. The volatility terms in our equations can naturally be thought of as the presence of high frequency trading in the model, which together with the drift term models the arrivals and cancellations of limit orders. The price process is then driven by the relative pressure of the existing orders on the two sides of the book. A potential advantage of our model over similar models in the literature is the presence of the reflection measure, which ensures that order volumes remain positive without the need for a multiplicative term in front of the noise. Figures 1 and 2 are included here for illustrative purposes. We make use of data provided by the LOBSTER (Limit Order Book System, The Efficient Reconstructor) database. Figure 1 shows the evolution of the bid price process for the SPDR Trust Series I on June 21 2012 between 09:30:00.000 and 10:30:00.000 EST. Figure 2 displays a simulated price process obtained from our moving boundary model on the compact spatial domain [0, 1]. Note that space here is on the scale of dollars i.e. a spatial interval of size 1 represents a price interval of size $1. We use a simple forward Euler finite difference scheme in order to produce the simulation. Using the data of incoming market orders, limit orders and cancellations of the SPDR Trust Series I at the different relative prices over the same time period as in Fig. 1, we fit the drift and volatility coefficients (which we assume to depend on the relative price only) of our SPDEs. When fitting these, we assume that these parameters are symmetric for the two sides of the order book, which is the case for this particular dataset up to a small error. Note that, for smooth functions k and large λ > 0, the functional g λ given by places most emphasis on the mass of k near zero. It is also the case that g λ (k) → k (0) as λ → ∞. Recall that the functional h in our equations determines the derivative of the boundary as a function of the two sides of the book, as in (1.2). For our simulation, we make the choice h(u 1 , with α = 5 and λ = 100. The boundary movement is therefore driven primarily by the local imbalance of offers to buy and sell at the mid, and approximates a Stefan condition. We also remark here that the Laplacian terms in our simulation were scaled down by a factor of 0.2 in order to ensure that the order book profiles obtained by the simulation are on the correct scale. We note that the boundary motion for our equations can be shown to be C 1,α for α < 1/4. However, by choosing h to approximate the derivative at the boundary and looking at the price process over sufficiently long timescales, we can generate price processes that are rough at the appropriate scale as can be seen in Fig. 2.

A deterministic parabolic obstacle problem
In this section we will define and state some simple results for deterministic parabolic obstacle problems on both the compact spatial interval [0, 1] and the infinite interval [0, ∞). The intuition for these equations is that the solutions solve the heat equation with a constraint that they must lie above some predetermined continuous function of time and space. It is important to note that both the obstacle and the solutions will be continuous functions here. In addition, being able to control differences in the solution by differences in the obstacles will be key in allowing us to introduce the reflection component in our SPDEs later.

The deterministic obstacle problem on [0, 1]
The obstacle problem on the compact interval [0, 1] was originally discussed in [8], in which the proofs can be found.

Definition 2.1
Let v ∈ C([0, T ]; C 0 ((0, 1))) with v(0, ·) ≤ 0. We say that the pair (z, η) satisfies the heat equation with obstacle v if: That is, for every t ∈ [0, T ] and every φ ∈ C 2 ((0, 1)) ∩ C 0 ((0, 1)), The following result from [8] gives us existence and uniqueness for solutions to this problem. It is also proved that we can bound the difference between two solutions in the L ∞ -norm by the difference in the L ∞ -norm of the obstacles. This will be very helpful when proving estimates later, as it will allow us to control our reflected SPDE by an unreflected SPDE.
there exists a unique solution to the above obstacle problem. In addition, if (z 1 , η 1 ) and (z 2 , η 2 ) solve the obstacle problems with obstacles v 1 and v 2 respectively, then for t ∈ [0, T ], we have the estimate 1] |u(s, x)|.

The deterministic obstacle problem on [0, ∞)
Before discussing the obstacle problem in this section, we first introduce the relevant function spaces which we will be working on.
We say that u ∈ C r if u ∈ L r and u is continuous. C r is equipped with the same norm as L r Definition 2. 4 We say that u : We are now in position to define the obstacle problem in this setting.
Let v ∈ C T r such that v(t, 0) = 0 for every t ∈ [0, T ] and v(0, ·) ≤ 0. We say that the pair (z, η) satisfies the heat equation with obstacle v and exponential growth r on [0, ∞) if: We note that the deterministic obstacle problem on the spatial domain R is considered in [9]. Here, we pose the problem in C T r for any r ∈ R, and we work on the spatial domain [0, ∞) rather than R. A proof of the following result is provided in the "Appendix". Theorem 2.6 For every r ∈ R and every v ∈ C T r such that v(t, 0) = 0 for every t ∈ [0, T ] and v(0, ·) ≤ 0, there exists a unique solution (z, η) to the heat equation on [0, ∞) with Dirichlet condition and obstacle v.
where z i is the solution to the obstacle problem corresponding to v i .

The moving boundary problem on finite intervals in the relative frame
We are interested in the following reflected moving boundary problem: where u 1 and u 2 satisfy Dirichlet boundary conditions enforcing that they are zero at p(t), with the point p(t) evolving according to the equation Here,Ẇ is a space-time white noise and h : C([0, 1]) 2 → R, satisfies certain conditions which we will introduce later. (η 1 , η 2 ) are reflection measures for the functions u 1 and u 2 respectively, keeping the profiles positive and satifying the conditions In this section, we consider the case when the functions u 1 and u 2 are supported in the sets

Formulation of the moving boundary problem
We would like to formalise what we mean by (3.1) in the compact case. Before doing so, we define what we mean for a space-time white noise to respect a given filtration. This will be useful in some of the measurability arguments which follow. Definition 3.1 Let ( , F , F t , P) be a complete filtered probability space. Suppose thatẆ is a space-time white noise defined on this space. Define for t ≥ 0 and A ∈ B(R), We say thatẆ respects the filtration Let ( , F , P) be a complete probability space, andẆ space-time white noise on R + × R. Let F t be the filtration generated by the white noise, so that is an F t -adapted process solving (3.1). Then p : R + × → R is a F t -adapted process such that the paths of p(t) are almost surely C 1 (note that, in particular, p is F t -predictable). Let ϕ ∈ C ∞ c ([0, ∞)×(0, 1)), and define the function φ by setting φ(t, x) = ϕ(t, p(t)−x). By multiplying the equation for u 1 in (3.1) by such a φ and integrating over space and time, interpreting the derivatives in the usual weak sense, we obtain the expression We now introduce a change in the spatial variable in order to associate our problem with a fixed boundary problem. Setting v 1 (t, x) = u 1 (t, p(t) − x), the above equation becomes Here,Ẇ p and η 1 p are obtained by from W and η by shifting by p(t). That is, for t ∈ R + and A ∈ B(R), 2) Note that, since the process p(t) is F t -predictable,Ẇ p is then also a space time white noise which respects the filtration F t . Also, η 1 p is a reflection measure for v, so that We can calculate It therefore follows that We can perform similar manipulations to obtain a weak form for v 2 (t, Note also that, since (u 1 , η 1 , Remark 3.2 By noting thatẆ p andẆ − p respect the filtration F t , we ensure that our solutions cannot "see the future" of the space-changed driving noisesẆ p andẆ − p . It of course makes sense intuitively that this should be the case, since they are measurable with respect to the filtration generated byẆ . We would expect that the solution is in fact adapted to the filtration generated by the noisesẆ i p . We see this indirectly, when we later prove that in any filtered space with a space-time white noise which respects the filtration, there exists a unique solution to the problem. As we can choose to take the filtration to be generated by the noise, the unique solution in an enlarged space must be adapted to the noise.

Remark 3.3
The above formulation would need to be adjusted if we were anticipating rough paths for p(t). For example, if p(t) were a semimartingale with a non-zero diffusion component, we would have to apply I tô's formula for the change of variables (3.4), which would change our weak form.
We now define what we mean by a solution to a particular class of reflected SPDEs. The preceding calculation will allow us to connect the solutions to these SPDEs to our moving boundary problem. Definition 3.4 Let ( , F , F t , P) be a complete filtered probability space. LetẆ be a space time white noise on this space which respects the filtration F t . Suppose that v is a continuous F t -adapted process taking values in C 0 (0, 1). Let h : C 0 (0, 1) × C 0 (0, 1) → R and F : C 0 (0, 1) → C 0 (0, 1) be Lipschitz functions. For the F tstopping time τ , we say that the pair (v, η) is a local solution to the reflected SPDE x) = ∞ for every t ≥ τ almost surely.
(v) η is a measure on (0, 1) × [0, ∞) such that (a) For every measurable map ψ : There exists a localising sequence of stopping times τ n ↑ τ almost surely, such that for every (3.6) for every t ≥ 0 almost surely.
We say that a local solution is maximal if it cannot be extended to a solution on a larger stochastic interval. We say that a local solution is global if we can take τ n = ∞ in (3.6).

Remark 3.5 Condition (iv) above is included for the purposes of discussing uniqueness only.
Before stating the formal definition for our moving boundary problem we introduce some notation which will allow us to easily write down the profiles in suitable relativised coordinates.
We similarly define 2 .
Definition 3.8 Let ( , F , F t , P) be a complete filtered probability space. LetẆ be a space time white noise on this space which respects the filtration F t . We say that the quintuple (u 1 , η 1 , u 2 , η 2 , p) is a local solution to the moving boundary problem with initial data ( We refer to (v 1 ,η 1 , v 2 ,η 2 ) as the solution to the moving boundary problem in the relative frame.
We now introduce the precise conditions on the coefficients. We suppose that for

Existence and uniqueness
with τ > 0 almost surely.
The following notation for the Dirichlet heat kernel will be used throughout the rest of the paper. Definition 3. 10 We define H (t, x, y) to be the Dirichlet heat kernel on [0, 1], so that (3.9) We will prove that we have global existence to the problem where the moving boundary term is truncated. Before doing so, we present the following result which will be applied in the argument.
Then we have that w is continuous, and for t ∈ [0, T ], (3.10) For the first term, we have by the Burkholder's inequality that it is at most An application of Hölder's inequality then gives that this is at most By the estimate (2) of Proposition A.4, we have that this is at most By arguing similarly and making use of estimates (1) and (3) Then for every M ≥ 0, there exists a unique pair of C 0 ((0, 1))-valued processes v 1 , v 2 together with η 1 , η 2 such that Proof Note that by a concatenation argument, it is sufficient to prove existence and uniqueness on the time interval [0, T ] for any T > 0. Fix T > 0. We perform a Picard iteration in order to obtain existence. The first approximations are given by v 1 and z 2 n+1 . Writing the equation for w 1 n+1 in mild form gives the expression We deal with the three terms separately. For the first, we apply Hölder's inequality to see that it is at most Making use of the Lipschitz property of the function f 1 , this is at most For the second term, we apply Proposition 3.11 and the Lipschitz property of σ 1 to deduce that, for p > 10 and t ∈ [0, T ], Finally, we deal with the third term. Using that h is Lipschitz and bounded on bounded sets, we see that the third term is at most Applying Hölder's inequality and Proposition A.5 then gives that this is at most (3.22) For p > 10 and corresponding q ∈ (1, 10 9 ), this is at most Putting this all together, we have shown that for any t ∈ [0, T ], We can repeat these arguments to obtain similar bounds for v 2 . Together, this gives We can then argue that Iterating this, we obtain 2 and so converges to some pair (v 1 , v 2 ). We now verify that this is indeed a solution to our evolution equation. Letw 1 be given bỹ (3.27) Defineṽ 1 =w 1 +z 1 , wherez 1 , together with a measureη 1 , solves our obstacle problem with obstacle −w 1 . Then, by arguing as before, we see that . The same applies to v 2 , so it follows that the pair (v 1 , v 2 ), together with the reflection measures (η 1 ,η 2 ), do indeed satisfy our problem.
Uniqueness follows by essentially the same argument. Given two solutions with the same initial data, (v 1 1 , v 2 1 ) and (v 1 2 , v 2 2 ) (together with their reflection measures), we argue as before to obtain that, for t ∈ [0, T ], 29) The equivalence then follows by Gronwall's inequality.
We are now in position to prove Theorem 3.9. This essentially amounts to showing that the solutions to our truncated problems coincide for different M. We use this to define a candidate function, and then check the conditions for this candidate.

Proof of Theorem 3.9 For every
solves the M 1 -truncated problem in the relative frame, until the stopping timeτ We can then argue as in the proof of Proposition 3.12 to deduce that respectively, as in (3.18). Therefore, by uniqueness of solutions to the obstacle problem, agree until the random timeτ . This consistency allows us to define our candidate solution in the relative frame, is then a maximal solution to the moving boundary problem in the relative frame until the explosion time τ , with localising sequence τ M . In addition τ > 0 almost surely as, by construction, are both maximal solutions, they both satisfy the M-truncated problem until they exceed M in the infinity norm, and so we can once again argue as in Proposition 3.12 to obtain that they both agree with the unique solution of the M-truncated problem until these times. Since this holds for every M, it follows that they agree until a common explosion time. We therefore have the result.

Proposition 3.13 Suppose that h is a bounded function. Then the solution to the moving boundary problem is global.
Proof Fix T > 0. Let (v 1 , η 1 , v 2 , η 2 ) be the unique maximal solution to the moving boundary problem in the relative frame, and let τ be the blow-up time for this solution. We consider the solutions to the corresponding truncated solutions, with Dirichlet boundary conditions w 1 M (t, 0) = w 1 M (t, 1) = 0 and initial data By arguing as in Theorem 3.9, we obtain that for t ∈ [0, T ] By noting that C p,T , h ∞ and v 1 0 do not depend on M here, we can apply Gronwall's Lemma to obtain that It follows that Similarly, This can only hold if there is almost surely no blow-up before time T i.e. τ > T almost surely. Since this holds for every T > 0, we must have that τ = ∞ almost surely. We then also have that for i = 1, 2 and every t ≥ 0. This allows us to take limits in the localising sequence, so we can obtain that the solution is indeed global.

Hölder continuity of the solutions
We now prove that, as in the case of the static reflected SPDE, our equations enjoy the expected Hölder continuity-up to 1/4-Hölder in time and up to 1/2-Hölder in space. The details of the proof here are a simplification of those used in [3], where Hölder continuity is proved for the equations when there is no moving boundary term.
The following result is Lemma 3.1 in [3].
with Dirichlet or Neumann boundary conditions at zero, and zero initial data. Then We now present a slight adaptation of Lemma 3.2 in [3].
Then there exists a smooth function f p,q : (iv) Proof The proof is as in [3], replacing the use of the heat kernel on R to smooth f with the Dirichlet heat kernel on D.
We now present the result regarding the Hölder continuity of our solutions. In addition to allowing for the extra term in the equation, corresponding to the moving boundary, our proof here slightly differs from the approach used in [3] in another way. In [3], the solution to the obstacle problem is approximated by the solutions to the solutions of the penalised SPDEs By using u in the coefficients of our approximating SPDEs f and σ here, we limit the problem of uniformly controlling the Hölder coefficients to studying the deterministic obstacle problem. every γ ∈ (0, 1). Then, for every T > 0, M > 0 and every γ ∈ (0, 1)  particular, if (u 1 , η 1 , u 2 , η 2 , p) is the solution to our moving boundary problem with initial data (u 1 0 , u 2 0 , p 0 ), then (u 1 , u 2 ) enjoys the same Hölder regularity locally until the blow-up time, τ .
Proof We consider v 1 M only, since the argument for v 2 M is identical. Define w 1 M to be the C 0 ((0, 1))-valued process given by (3.42) Let r be such that γ = 1 − 12 r . By applying the inequalities from Propositions A.4 and A.6 together with Burkholder's inequality, we see that We note that it is Proposition A.6 which allows us to control the extra term arising due to the moving boundary. It then follows by Corollary A.3 in [1] that there exists a random variable X ∈ L r such that We then have (see [8]) that z +w 1 M increases to v 1 M , the solution of the reflected SPDE on [0, 1]. Let (w 1 M ) p,q be a smoothing of w 1 M as in Proposition 3.15, with respect to the random variable X , the Hölder coefficients γ /2 and γ /4, and the constants p, q which are yet to be determined. Define z p,q to be the solution of the PDE with Dirichlet boundary condition at zero and and zero initial data. We then have that (see the proof of Theorem 1.4 in [8] for details) with zero initial data and Dirichlet boundary conditions α p,q (t, 0) = α p,q (t, 1) = 0. Similarly, if we differentiate (3.46) in space with initial data z 0 = 0 and Neumann boundary conditions Applying Lemma 3.14 to Eqs. (3.48) and (3.49) controls the infinity norms of α p,q and β p,q by the infinity norms of respectively, uniformly over . By using the bounds from Lemma 3.15 and choosing p = |t − s|, q = |x − y|, we can control the γ /4-Hölder norm in time and the γ /2-Hölder norm in space of z , uniformly over . Letting ↓ 0 then allows us to conclude.

The moving boundary problem on semi-infinite intervals in the relative frame
We now consider the analogous obstacle problem, where the two sides of the equation satisfy SPDEs on the infinite halflines (−∞, p(t)] and [ p(t), ∞) respectively. That is , and on [0, ∞) × [p(t), ∞). We once again have Dirichlet conditions at the mid, p(t), so that u 1 (t, p(t)) = u 2 (t, p(t)) = 0, with the point p(t) evolving according to the equation Here, W is a space-time white noise and h is a function of the two profiles of the equation on either side of the shared boundary. As before, η 1 and η 2 are reflection measures for the functions u 1 and u 2 respectively, keeping the profiles positive and satisfying the conditions

Formulation of the problem
We will be working in the spaces C r and C T r , defined in Sect. 2.2, throughout this section. This presents issues when handling both the non-Lipschitz term arising due to the moving boundary and the stochastic term. Truncating the boundary term requires more care, as we are now trying to control the C T r -norm of the process. We are also unable to suitably control the supremum of the stochastic terms using our previous arguments, as they are not well suited to unbounded domains. For this reason, we introduce extra decay for the growth of the volatility relative to the growth of the drift term. Fixing r ∈ R we take, for i = 1, 2, f i and σ i to be measurable mappings from [0, ∞) × R + → R and h : C r × C r → R to be a measurable function such that, for some C, δ > 0 Since our notion of solution here is motivated by the same ideas as in the compact case, we move straight to the definitions for solutions to non-linear SPDEs and moving boundary problems on R. Definition 4.1 Let ( , F , F t , P) be a complete filtered probability space. LetẆ be a space time white noise on this space which respects the filtration F t . Suppose that v is a continuous F t -adapted process taking values in C r . Let h : C r × C r → R and F : C r → C r be Lipschitz functions. For the F t -stopping time τ , we say that the pair (v, η) is a local C r -valued solution to the reflected SPDE r for every t < τ almost surely.
Similarly to as in Sect. 3, we say that a local C r -valued solution is maximal if it cannot be extended to a C r -valued solution on a larger stochastic interval, and we say that a local solution is global if we can take τ n = ∞ in (4.2). ( , F , F t , P) be a complete filtered probability space. LetẆ be a space time white noise on this space which respects the filtration F t . We say that the quintuple (u 1 , η 1 , u 2 , η 2 , p) is a local solution to the moving boundary problem on R with exponential growth r and initial data (

Definition 4.2 Let
with Dirichlet boundary condition v 1 (t, 0) = 0 and initial data v 1 with Dirichlet boundary condition v 2 (t, 0) = 0 and initial data v 2 We refer to (v 1 ,η 1 , v 2 ,η 2 ) as the solution to the moving boundary problem in the relative frame.

Existence and uniqueness
As in the proof of Theorem 3.9, we will use a Picard iteration in order to prove existence and uniqueness for a truncated version of this problem. There is some extra complexity introduced when trying to do this in the case of an infinite spatial domain. In particular, we should be more careful in how we truncate the problem. (u 1 , η 1 , u 2 , η 2 , p) to the moving boundary problem on R, with the blow-up time given by

Theorem 4.3 Let ( , F , F t , P) be a complete filtered probability space. LetẆ be a space time white noise on this space which respects the filtration F t . There exists a unique maximal solution
with τ > 0 almost surely.
The following notation will be used throughout the rest of the paper.

Definition 4.4
We define G(t, x, y) to be the Dirichlet heat kernel on [0, ∞), that is For r ∈ R, we also define the notation Before proving Theorem 4.3, we present here some results which will be essential to the proof.  We would like an analogous result which would allow us to control the noise term appearing in the mild formulation. The following lemmas will enable us to obtain such an estimate.
Then w is continuous almost surely and for x, y ∈ [0, ∞) and (4.11) We bound the first term only, and note that the other terms follow similarly by the estimates from Proposition A.1. Burkholder's inequality gives Hölder's inequality then gives (4.12) Applying the first bound from Proposition A.1 then gives that this is at most By making similar arguments, using the other bounds from Proposition A.1, we obtain that Continuity of w then follows by Corollary A.3 in [1].
The following result follows by adapting the proof of Lemma 3.4 in [9].
Then for every δ > 0, w ∈ C([0, T ]; C δ ) and there exists a constant C depending only on p, , T and δ, and a non-negative random variable Y such that It then follows by Lemma 4.7 that, for p > 12, So we have the result.
We are now in position to prove Theorem 4.3 with a Picard iteration. Since the ideas for the remaining arguments are similar to those in the proof of Theorem 3.9, we give an outline of the strategy only.

Proof of Theorem 4.3
Our strategy is as follows: 1. We note that, by the definition, it is sufficient to prove existence and uniqueness for maximal solutions to the coupled SPDEs with Dirichlet boundary condition v 1 (0) = 0 and initial data v 1 with Dirichlet boundary condition v 2 (0) = 0 and initial data v 2 0 = u 2 0 • ( 2 p 0 ) −1 ∈ C + r . 2. We once again consider a truncated version of the problem. That is, we find As in Proposition 3.12, h M,r is defined by applying h to suitably truncated inputs, with the truncation function here being F M,r . Formalising this, we define h M,r : with F M,r : C r → C r is given by with Dirichlet boundary condition w 1 M,n+1 (t, 0) = 0 and initial data w 1 M,n+1 (0, x) = v 1 0 (x). Writing this in mild form gives the expression ,n (s, ·), v 2 M,n (s, ·)) F M,r (v 1 M,n (s, ·))(y)dyds (4.25) n (s, y))dxds n (s, y))W(dy, ds).
We then set v 1 M,n+1 := w 1 M,n+1 + z 1 M,n+1 , where z 1 M,n+1 solves the obstacle problem with obstacle −w 1 M,n+1 . We similarly define w 2 M,n+1 and v 2 M,n+1 . Our Lipschitz conditions on h, f and σ , together with the estimates from Lemma 4.5, Proposition 4.8 and Proposition A.2 allow us to argue as in the proof of Propsition 3.12 to obtain that, for t ∈ [0, T ], (4.26) Theorem 2.6 gives that, for t ∈ [0, T ] and i = 1, 2, Plugging this into (4.26) then gives that, for t ∈ [0, T ], Arguing as in Proposition 3.12, we see that (v 1 M,n , v 2 M,n ) n≥1 is Cauchy in L p ( ; C T r ) 2 for large enough p, and the limit (v 1 M , v 2 M ) solves the truncated problem given by Eqs. (4.21) and (4.22). 4. As in Proposition 3.12, uniqueness for the truncated problem can be shown by applying the same estimates as in the proof of existence and concluding with a Gronwall argument. 5. We note the consistency of the truncated problems different truncation values M and use this to define a solution to the problem until the . C r norm blows up. 6. We observe that uniqueness of the truncated problems implies uniqueness for the original moving boundary problem. 7. To deduce that τ > 0 almost surely, consider w i M , the solution to the SPDE It follows that τ > 0.

Proposition 4.9 Suppose that h is bounded. Then the solution to the moving boundary problem on R is global.
Proof We argue as in Proposition 3.13, replacing bounds on H with the corresponding bounds on G r (t, x, y) and G r +δ (t, x, y).

Remark 4.10
We note that our uniqueness result here extends the existing theory for uniqueness for reflected SPDEs on infinite spatial domains. Until now, uniqueness had only been shown for equations in the case when σ is constant. This was proved in [9], where the spatial domain was R (this makes no difference to the arguments here). Choosing h = 0 in our equations i.e. a static boundary, we obtain uniqueness for solutions to these equations in the spaces C r , provided that the dependence of the volatility on the solution itself decays exponentially, as in conditions (iii) and (iv) in the formulation of the problem.
The proof of this is similar in spirit to that of Theorem 3.16. There are, however, some intricate differences which arise due to the infinite spatial domain. We first introduce here a modified version of Lemma 3.14 which is suitable for this context.
with Dirichlet or Neumann boundary conditions at zero, and zero initial data. Then there exists a constant C r ,T such that Proof We will prove the result for the Neumann boundary condition-the argument for the Dirichlet condition is essentially the same. Let (B x t ) t≥0 be a Brownian motion on [0, ∞) with reflection at 0, started at x. Then by arguing as in Lemma 3.6 in [2], we have that (4.36) Hence (4.40) Therefore, we have that (4.41) Plugging this into (4.37), we obtain that So we have the result.

Proof of Theorem 4.11
The argument broadly follows the steps in the proof of Theorem 3.16 and consequently those in [1]. Fix some y))W(dy, ds).
with Dirichlet boundary condition at zero and and zero initial data. By Proposition B.2, we obtain that The boundary condition is Dirichlet, since z does not change at 0, which means the time derivative is zero there. The initial data is z 0 = 0, since z 0 is identically zero. Note that g is negative, so we can use Proposition 4.12 to deduce that (4.56) Differentiating (4.52) with respect to x, we obtain, that y ,α,β := ∂z ,α,β ∂ x satisfies (4.57) with initial data z 0 = 0 and Neumann boundary condition at zero. Proposition 4.12 then gives (4.58) Another application of Proposition B.2 gives We clearly have that This is at most Making the choices α = |t − s| and β = |x − y|, this is at most (4.62) Since X and w 1 M C T r +μ are in L p , they are finite almost surely. Letting ↓ 0 and noting the inequality (4.49) gives that, for every x, y ≥ 0 and every s, t ∈ [0, T ] (4.64) Then, for every γ ∈ (0, 1), the derivative of the boundary is locally γ /4-Hölder and v 2 (t, x) = u 2 (t, p(t) + ·). For M > 0, define We have, by the Lipschitz property of h, that

A Heat kernel estimates
We present here some of the simple estimates for the heat kernels on [0, 1] and [0, ∞) which were used throughout, and details of their proofs.

A.1 Heat kernel on [0, ∞)
Recall that the Dirichlet heat kernels on [0, ∞) is given by and that G r (t, x, y) := e −r (x−y) G(t, x, y). We define the functions In this section, the proofs focus on the F 1 component of G. The arguments for the F 2 components are similar.

Proof of 1 Note that
If p > 4, we have that − p 2( p−2) > −1, and so this is equal to The result follows.

Proof of 2
We again make use of Eq. (A.2). This gives that The integral in the second term is integrable if p > 4. Therefore, the second term is equal to C p,r ,T |t − s| p for p > 4. For the first term, we have that it is equal to du. (A.7) By making the substitution v = u |t−s| , we see that this integral is at most The substitutionz = z √ t−s then gives that this is equal to The integral in (A.9) converges provided that p > 4, giving the result.

Proof of 3 By (A.2), we have that
du.
(A.10) Making the change of variables w = h |x−y| and v = u (x−y) 2 , we see that this is equal to dv. (A.11) The integral here converges provided that p > 4, and only depends on p. We therefore have the result.
Proof It is sufficient to prove the result for F 1 and F 2 separately. Calculating gives that (A.14) The proof for the F 2 part of the heat kernel is similar. By noting that and arguing as before, we obtain the result.

Proof of 1 First note that
We bound the terms corresponding to these two components separately. Let q = p/( p − 2). Then, by applying Hölder's inequality, we have that (A.17) We split the time integral into two parts-the integral on [0, |x − y|] and the integral on (|x − y|, ∞). For the first of these domains, we have To bound on (|x − y|, ∞), we note that Therefore, outside the region u ∈ [−1, 0], we have that In the region u ∈ [−1, 0], we have that (2 p−4) . We have therefore deduced inequality (1) for the first component on the right hand side of expression (A.16). For the second component of (A.16), we have that that Putting this together, we obtain that (A.28) It follows that Inequality (1) for the second component of (A.16) is then a simple consequence of this. The manipulations required to prove inequalities (2) and (3) are similar, and we therefore omit these lengthy calculations.

A.2 Heat kernel on [0, 1]
Recall that (A.30) We make the observation here that this expression can be written as where L is a smooth function of t, x, y which vanishes at t = 0. Consequently, we are able to prove the estimates for H in this section in analogous ways to how we proved the corresponding results for G. We therefore omit the proofs here.
(B.2) We know that Testing our equation for y with the positive part of y, we obtain that We note that testing against the last two terms gives a negative contribution, since when y ≥ 0, we have that z 1 ≥ z 2 and so (( Putting this together, we see that Gronwall's inequality then gives that y + T = 0 i.e. that z 1 ≤ z 2 . The following bound will allow us to control solutions of our obstacle problems by the obstacles themselves. Proposition B.2 Let r ∈ R and v 1 ,v 2 ∈ C T r such that v 1 (t, 0) = v 2 (t, 0) = 0. Fix > 0. For i = 1, 2, let z i be the solution to the PDE with boundary condition z i (t, 0) = 0 and zero initial data. Then there exists a constant C r ,T such that Proof Let w be given by w(t, x) = e r x+r 2 t φ(t), (B.7) where we define φ(t) := v 1 − v 2 C t r . Then we have that ∂w ∂t = w + e r x+r 2 t dφ dt . (B.8) We note here that φ is positive and increasing, and that we interpret dφ dt in a weak sense in the equation above. From the definition of w we see that w ≥ v 2 − v 1 . Let δ > max(0, −r ) and definẽ z(t, x) := e −(δ+r )x (z 1 (t, x) − z 2 (t, x) − w(t, x)).
Note also that the last term on the right hand side of (B.9) is negative. Therefore, testing the equation withz + we obtain that for t ∈ [0, T ] As we did in order to construct z, we define the functions z n to be the solutions to the equations ∂z n ∂t = z n + 1 arctan(((z n − v n ) ∧ 0) 2 ). (B.18) We can then argue as in the proof of Theorem 4.11, differentiating the equation in space and time respectively, and applying Proposition 4.12 to see that ∂z n ∂ x C T r +δ ≤ C r +δ,T ∂v n ∂ x C T r +δ , and ∂z n ∂t C T r +δ ≤ C r +δ,T ∂v n ∂t C T r +δ Note that we use the condition that v(0, ·) ≤ 0 for this step, in order to ensure that the initial data for ∂z n ∂t is zero. Therefore, as (s, y) → (t, x), and so z is continuous. Therefore, z ∈ C T r . We now want to verify that z solves the obstacle problem. Clearly, z(t, 0) = 0 and z(0, x) = 0 for all x. Let (t, x)) η(dt, dx) ≤ 0 (B.23) almost surely. By applying the DCT, noting that η assigns finite mass to compact sets in (0, ∞) almost surely, we obtain that Since z − v ≥ 0, this integral must be zero. So we have the result, and (z, η) is a solution to the obstacle problem.
We now turn to the problem of uniqueness. The following lemma is an adaptation of the result from Section 2.3 in [8].
Lemma B.4 Let (z 1 , η 1 ) and (z 2 , η 2 ) be two solutions to the obstacle problem with obstacle v ∈ C T r . Set ψ(t, x) := z 1 (t, x) − z 2 (t, x). Then, for φ ∈ C ∞ c ([0, ∞)) with φ(0) = 0, and t ∈ [0, T ], we have that Proof Fix some t < T and φ ∈ C ∞ c ((0, ∞)). The result would follow if we could test the equation for ψ with the function ψ(t, x)φ 2 (x). Since this isn't possible, as ψ isn't regular enough, we must test with a smooth approximation of this function and take a limit. Let be a non-negative function supported on [−1, 1] which is symmetric, smooth, positive definite and so that 1 −1 (x)dx = 1.
We then obtain approximations of the identity, given by n (x) := n (nx). We now define the function of two variables, n,m to be where * here denotes the convolution on R 2 . That is, we define (B.27) We take the limit for each term separately. The first term is In order to bound (B.36), we first suppose that ψ is smooth. Integrating by parts and making use of the positive definiteness of , we obtain that (using angle bracket notation to indicate integration over [0, ∞)) Letting ↓ 0, we obtain that This concludes the proof.