# Existence of maximal solutions for the financial stochastic Stefan problem
of a volatile asset with spread

D. C. Antonopoulou#∗ ,  D. Farazakis∗ and  G. Karali†∗

###### Abstract.

In this work, we consider the outer Stefan problem for the short-time
prediction of the spread of a volatile asset traded in a financial market. The
stochastic equation for the evolution of the density of sell and buy orders is
the Heat Equation with a non-smooth noise in the sense of Walsh, posed in a
moving boundary domain with velocity given by the Stefan condition. This
condition determines the dynamics of the spread, and the solid phase
$[s^{-}(t),s^{+}(t)]$ defines the bid-ask spread area wherein the transactions
vanish. We introduce a reflection measure and prove existence and uniqueness
of maximal solutions up to stopping times in which the spread
$s^{+}(t)-s^{-}(t)$ stays a.s. non-negative and bounded. For this, we use a
Picard approximation scheme and some of the estimates of [20] for the Green’s
function and the associated to the reflection measure obstacle problem.
Analogous results are obtained for the equation without reflection
corresponding to a signed density. Additionally, we apply some formal
asymptotics when the noise depends only on time to derive that the spread is
given by the integral of the solution of a linear diffusion stochastic
equation.

###### 1991 Mathematics Subject Classification:

† Department of Mathematics and Applied Mathematics, University of Crete,
GR–714 09 Heraklion, Greece.

# Department of Physical and Mathematical Sciences, Engineering, University of
Chester, CH2 4NU, UK

∗ Institute of Applied and Computational Mathematics, FORTH, GR–711 10
Heraklion, Greece.

Keywords: Phase field models, Stefan problem, stochastic volatility, limit
order books, spreads.

AMS subject classification: 35K55, 35K40, 60H30, 60H15, 91G80, 91B70.

##  1\. Introduction

###  1.1. The Stochastic Stefan problem with spread

Let $w(x,t)$ be the density of sell and buy orders of a stochastically
volatile liquid asset with spread. The moving boundary of the outer Stefan
problem for $w$, $t\geq 0$, is the union of the curves
$x=s^{+}(t),\;\;x=s^{-}(t),$ enclosing the solid phase (or spread area)
$\overline{S(t)}$ defined at a given time $t$ by the interval
$\overline{S(t)}:=[s^{-}(t),s^{+}(t)].$ The midpoint
$s(t):=(s^{-}(t)+s^{+}(t))/2$ is the so-called mid price, and the length
$s^{+}(t)-s^{-}(t)$ of $\overline{S(t)}$ is the spread at time $t$. The asset
price $x$ has been transformed through a logarithmic scale and in general can
take negative and positive values. If $x$ is set in $\overline{S(t)}$, then
the asset is not traded, and thus the density $w(x,t)$ of sell and buy orders
is zero, otherwise the order is performed.

The Stefan problem for $w=w(x,t)$ satisfying the stochastic Heat equation is
written as follows

(1.1) $\left\\{\begin{aligned} &\partial_{t}w=\alpha\Delta w+\sigma({\rm
dist}(x,\partial
S))\dot{W}_{s},\;\;\;\;x\in\mathbb{R}-\overline{S(t)}\;\;\mbox{(`liquid'
phase)},\;\;t>0,\\\
&w=0,\;\;x\in\overline{S(t)}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{(`solid'
phase )},\\\ &V:=-\nabla w|_{\partial
S}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{(Stefan
condition)},\\\ &\partial S(0)=\\{s^{-}(0),s^{+}(0)\\}={\rm
given}.\end{aligned}\right.$

Here, $\alpha>0$ stands for the total liquidity index of the market, estimated
by the limit order book of the asset, and $\sigma\dot{W}_{s}$ is the
stochastic volatility. The noise diffusion $\sigma$ is a function of

(1.2) ${\rm dist}(x,\partial S(t))=\min\\{|x-s^{+}(t)|,|x-s^{-}(t)|\\}$

the distance of the price $x$ from the solid phase boundary $\partial
S=\partial S(t)=\\{s^{-}(t),s^{+}(t)\\}$, and

(1.3) $\dot{W}_{s}(x,t):=\dot{W}(x-s^{+}(t),t)\;\;\text{if}\;\;x\geq
s^{+}(t),\;\;\dot{W}_{s}(x,t):=\dot{W}(-x+s^{-}(t),t)\;\;\text{if}\;\;x\leq
s^{-}(t),$

where $\dot{W}(\pm x\mp s^{\pm}(t),t)$ is the non smooth in space and in time
noise defined by Walsh in [35]. The initial condition $w(x,0)$ is considered
given for all $x\in\mathbb{R}$.

The limit orders are instructions for trading of a portion of an asset, [26],
based on information from the limit order book. The lowest sell order
$s^{+}(t)$ defined as ask price, is the minimum price at which the investor is
willing to receive, and $s^{-}(t)$ is the highest buy order or bid price which
is the maximum price at which the investor is willing to pay. An order is
executed if the price set (the so-called spot price) lies outside the spread
interval $[s^{-}(t),s^{+}(t)]$, if not it is sorted in the order book list and
not traded, see for example in [17, 25, 32]. We also note that the Gibbs
Thomson condition on the moving boundary $\partial S(t)$ which is present in
dimensions $d\geq 2$, [29], involving the mean curvature, and the constant
value of $w=w_{0}$ in the solid phase are both replaced by the condition
$w=w_{0}:=0$ in $\overline{S(t)}$.

The velocity $V$ of $\partial S(t)$ is defined at the boundary points by the
Stefan condition

(1.4) $\begin{split}V(s^{+}(t),t):=&\partial_{t}s^{+}(t)=-(\nabla
w)^{+}(s^{+}(t),t),\\\ V(s^{-}(t),t):=&\partial_{t}s^{-}(t)=-(\nabla
w)^{-}(s^{-}(t),t),\end{split}$

for $(\nabla\cdot)^{\pm}$ denoting the derivative from the right ($x>s^{+}$)
and left ($x<s^{-}$); the Stefan condition describes the change of liquidity.
Therefore, the spread dynamics are given by

(1.5) $\partial_{t}s^{+}(t)-\partial_{t}s^{-}(t)=-(\nabla
w)^{+}(s^{+}(t),t)+(\nabla w)^{-}(s^{-}(t),t)).$

The gradients are taken along the ‘outer’ normal vector, i.e., the direction
is towards the liquid phase, so here in $d=1$ they coincide to the right and
left derivatives. Models with a.s. non-negative density $w$, when for example
a reflection measure is introduced to the stochastic heat equation, due to the
fact that $w=0$ at $x=s^{\pm}$ will result in an a.s. decreasing spread. More
specifically $(\nabla w)^{+}(s^{+}(t),t)\geq 0$ and $(\nabla
w)^{-}(s^{-}(t),t)\leq 0$ and thus by (1.5)
$\partial_{t}(s^{+}(t)-s^{-}(t))\leq 0$ for all $t\geq 0$ a.s. In contrast,
when a signed density is considered the spread is not monotone.

Motivated by the analysis of [29, 30, 7] in higher dimensions, we define the
bounded and time independent space domain $\Omega=(a,b)$ by

(1.6) $\Omega=\Omega_{{\rm Liq}}(t)\cup[s^{-}(t),s^{+}(t)],$

for a liquid phase $\Omega_{{\rm Liq}}\subset\Omega$ so that for any
$x\in\Omega_{{\rm Liq}}$

(1.7) $0\leq|x-s^{-}|,|x-s^{+}|\leq\lambda,$

for $\lambda=b-a$ a positive constant relatively very larger than the initial
spread $s^{+}(0)-s^{-}(0)$. The density $w(x,t)$ will be observed for $x$ in
$\Omega$. As $\lambda\rightarrow\infty$ the liquid phase becomes infinite as
in (1.1) and $\Omega$ will correspond to $\mathbb{R}$. The problem is one-
dimensional and the liquid phase consists of two separate bounded linear
segments. This enables the spliting of the Stefan problem equation in two
equations posed for $x\in\Omega_{\rm Liq}$ on $x\geq s^{+}$ and on $x\leq
s^{-}$ where we shall apply the change of variables

(1.8) $\begin{split}y=x-s^{+}(t)\;\;\text{if }x\geq
s^{+}(t),\;\;y=-x+s^{-}(t)\;\;\text{if }x\leq s^{-}(t),\end{split}$

and thus

(1.9) $\begin{split}y_{t}=-\partial_{t}s^{+}(t)\;\;\text{if }x\geq
s^{+}(t),\;\;\partial_{t}s^{-}(t)\;\;\text{if }x\leq s^{-}(t).\end{split}$

As we shall see the equation is transformed due to the Stefan condition into
two independent ones posed each on the fixed space domain
$\mathcal{D}:=(0,\lambda)$ with Dirichlet b.c. The value $y=0$ occurs when the
price $x$ is $s^{\pm}$, while $y=\lambda$ when the spread is zero and
$s^{+}=s^{-}$ hits the boundary of $\Omega$. These equations are of the
general form

(1.10) $v_{t}(y,t)=\alpha\Delta v(y,t)\mp\nabla v(0^{+},t)\nabla
v(y,t)\pm\sigma(y)\dot{W}(y,t)+\dot{\eta}(y,t),\;\;y\in\mathcal{D},\;\;t\geq
0,$

for $\eta$ a reflection measure keeping $v$ a.s. non-negative, while $\eta=0$
will correspond to the unreflected problem and a signed $v$. We also note that
when a system is considered in place of the Stefan problem (1.1) with buy and
sell densities observed separately and with different liquidity coefficients
$\alpha_{1}$, $\alpha_{2}$, the same equation of the above general form will
appear after the change of variables for $\alpha=\alpha_{1},\alpha_{2}$.

We prove existence of unique maximal solutions $(v,\eta)$ for the stochastic
equation (1.10) for the stopping time $\sup_{M>0}\tau_{M}$ where

(1.11) $\begin{split}\tau_{M}:=&\inf\Big{\\{}T\geq
0:\;\displaystyle{\sup_{r\in(0,T)}}|\nabla v(0^{+},r)|\geq
M\Big{\\}},\end{split}$

up to which $|\nabla v(0^{+},r)|=\nabla v(0^{+},r)$ stays a.s. bounded. In the
case of the unreflected problem, $\eta$ is just replaced by zero and the
absolute value is kept. In order to return to the initial variables and to the
moving boundary problem, the stopping time will be further reduced so that the
spread stays a.s. non-negative and the spread area in the domain $\Omega$.
These restrictions will be induced by the Stefan condition and the resulting
spread dynamics (1.5) on $w$, the initial spread $s^{+}(0)-s^{-}(0)$, and the
magnitude of $\lambda$.

Deterministic parabolic Stefan problems have been so far extensively studied
when describing the phenomenon of phase separation of alloys. In [29],
Niethammer introduced the deterministic version of (1.1) in higher dimensions
in the physical context of the LSW theory for the Ostwald ripening of alloys;
there, a first order approximation was established for the dynamics of the
radii of spherical moving boundaries in dimensions $d=3$. In [2, 4, 3], the
authors considered the quasi-static problem and obtained second order
approximations by taking into account the variable in general geometry of the
solid phase. We also refer to [7] for the analysis of the parabolic Stefan
problem of [29] in the presence of kinetic undercooling and additive forcing.

Antonopoulou, Bitsaki, and Karali, in [6], derived the rigorous financial
interpretation of the parabolic Stefan stochastic model, which applies for a
portfolio of assets when $d\geq 2$; a quasi-static version thereof
approximates the parabolic one when the diffusion tends to infinity as in the
case of very large trading. In contrast to the deterministic Stefan problem
where a spherical initial solid phase or the interval $[s^{-}(0),s^{+}(0)]$ in
dimension $d=1$ are static solutions, in the stochastic case the boundary
changes as time evolves due to the random perturbation in the spde; see for
example the numerical simulations in [6] when $d=3$. When the sell and buy
orders densities are observed separately, then the evolution is described by a
system of two stochastic Heat equations with different liquidity coefficients
and volatilities depending on the distances $|x-s^{+}(t)|$, $|x-s^{-}(t)|$
respectively. Hambly, and Kalsi proved in [20] existence and uniqueness of
stochastic solutions for such two phases Stefan systems with reflection, but
under the assumption of zero spread for the asset price, i.e., for
$s^{+}(t)=s^{-}(t)=s(t)$. Considering 2-phases 1-dimensional stochastic Stefan
systems for the evolution of sell and buy orders without spread we refer also
to [16, 36], and to the more recent results of [28, 21].

###  1.2. Limit order books and spread

An asset is defined as volatile if the corresponding trading price of sell or
buy orders deviates from the mid (mean) price. The spread’s length which is
given as the difference between the actual sell price and the buy price is a
measure for the risk of investment, [5]. In particular, highly traded assets
tend to have very small spreads, while a relatively large spread indicates a
higher risk. An order is a commitment from the traders, a buyer or a seller,
to buy or sell respectively at an appropriate price at a given time $t>0$, for
which the profit of the trade is maximized for both sides, [17], also called
limit price. The spread and the density of transactions reflect asset’s
liquidity. The total volume of active limit orders in a financial market at a
given time is stored in the asset’s limit order book. The liquidity
coefficient $\alpha>0$ in the Stochastic Heat equation of the Stefan problem
measures the diffusion strength of sell and buy orders and can be
approximated, in small time periods, by the total volume of orders divided by
an average spread, [6].

The various types of financial contracts require two entities, that is the
holder of a financial asset, such as a security, commodity, or currency, who
receives the future payments, and the issuer side which has the obligation to
deliver the payments according to the initial terms and claims of transaction.
An order is a commitment to buy or sell at a given time. Market orders are
executed immediately upon submission in contrast to limit orders which remain
active until they achieve the expected ‘closing’ price; this is an automatic
execution procedure via online electronic platforms. Electronic trading
platforms offer the ability to trade upon information from historical data
like past market prices and curves of prices of stocks (old trading view). The
market orders are based on the current market prices, while limit orders
target to better future prices for maximizing profits, [26]. Limit orders are
low risk commitments since the price of execution for sell or buy is
predetermined reducing thus the odds of significant failure. However, the
process is time consuming and the order may never be executed.

In [8] the German power market liquidity was studied, we also refer to [33]
for a statistical analysis of the fluctuations of the average spread where the
relation of spread with shares volume and volatility was examined, or to [24]
for a stochastic equation model estimating the liquidity risk. In [14], the
authors analyzed how transaction costs affect the spreads while in case of
zero cost then the market price should act as a Wiener process; see also in
[27] for the liquidity risk with respect to the transaction costs and market
manipulation under a Brownian motion problem formulation, or in [13, 34, 22,
12], and in [19] for various empirical approaches on spread’s forecast. We
note that except from the bid-ask spread, there exist several other types of
spread like the asset swap spread, the yield spread, the zero volatility
spread, the option adjusted spread, the default swap spread, or the bank
spreads, see for example in [31, 9, 10, 11, 18, 23].

###  1.3. Main Results

Our analysis covers 3 versions of the Stefan problem.

  1. (1)

Let $w_{1},w_{2}\geq 0$ be the density of sell orders and buy orders
respectively. When $x>s^{+}(t)$ then only sell orders are executed
($w_{2}=0$), while when $x<s^{-}(t)$ then only buy orders are executed
($w_{1}=0$). Moreover at $x=s^{+}(t)$ $w_{1}=0$ and at $x=s^{-}(t)$ $w_{2}=0$.
The signed density $w=w_{1}-w_{2}$ is given by

(1.12)
$w(x,t)=w_{1}(x,t)\;\;\text{if}\;\;x>s^{+}(t),\;\;w(x,t)=-w_{2}(x,t)\;\;\text{if}\;\;x<s^{-}(t),\;\;w(x,t)=0\;\;\text{otherwise}.$

We introduce in (1.1) the additive term $\dot{\eta}_{s}$ defined by

(1.13)
$\dot{\eta}_{s}(x,t):=\dot{\eta}_{1}(x-s^{+}(t),t)\;\;\text{if}\;\;x\geq
s^{+}(t),\;\;\dot{\eta}_{s}(x,t):=-\dot{\eta}_{2}(-x+s^{-}(t),t)\;\;\text{if}\;\;x\leq
s^{-}(t),$

where $\eta_{1}$, $\eta_{2}$ are reflection measures so that $w_{1},w_{2}\geq
0$.

  2. (2)

We consider the reflected problem where $w\geq 0$. The Stefan condition due to
the non-negativity of $w$ which vanishes at $x=s^{\pm}$ yields an a.s.
decreasing spread. The reflection additive term on (1.1) is of the form

(1.14)
$\dot{\eta}_{s}(x,t):=\dot{\eta}_{1}(x-s^{+}(t),t)\;\;\text{if}\;\;x\geq
s^{+}(t),\;\;\dot{\eta}_{s}(x,t):=\dot{\eta}_{2}(-x+s^{-}(t),t)\;\;\text{if}\;\;x\leq
s^{-}(t),$

where $\eta_{1}$, $\eta_{2}$ are reflection measures keeping $w\geq 0$ for any
$x\in\Omega_{\rm Liq}$.

  3. (3)

The unreflected problem is analyzed with a signed density $w$ where as in (1)
the spread is non-monotone.

In all the above cases we derive a system of independent spdes of the form
(1.10) for $v=v_{1}$, $v=v_{2}$. Then $s^{+},s^{-}$ are specified through
integration of the Stefan condition. For stopping times wherein $s^{-}\leq
s^{+}$ and $(s^{-},s^{+})\subseteq\Omega$ by applying the change of variables
(1.8), $v_{1}\rightarrow w|_{x\geq s^{+}}$, $v_{2}\rightarrow-w|_{x\leq
s^{-}}$ in (1) or $w|_{x\leq s^{-}}$ in (2) and (3), we return to the initial
Stefan problem. The suggested transformation is efficient on representing the
stochastic equation of the Stefan problem as a system of independent spdes
posed on the fix domain $\mathcal{D}=(0,\lambda)$, of the same general form.
Additionally, for the reflected equations, we impose the non-negativity of
$v_{1,2}$ by proving existence of the measures $\eta_{1,2}$ on the fix domain
which then define the additive reflection term in the initial equation. Our
novel approach on transforming first the problem to an spde of reference and
then establishing maximal solutions to the initial one by using the Stefan
condition for the stopping times is also applicable for various other one-
dimensional versions with financial interest being analyzed for example in
[20, 21, 28] without spread. Note that our model permits zero spread. The
noise diffusion and the noise depend on the distance of $x$ from the spread
area boundary and not on the position of $x$. This yields, since the velocity
is given by the standard Stefan condition of (1.1), to spdes in the $y$
variables where $s^{\pm}$, that belong to the initial problem unknowns, are
absent. We also mention that variables of the form $y=-x+s(t)$ when $x\leq s$,
$y=x-s(t)$ when $x\geq s$ for the zero spread model where $s$ is the sell/buy
price, are used in [20, 28]. In [20] the problem is not transformed, a weak
solution formulation for proper test functions compactly supported in $[0,1]$
induces somehow a relevant (not splited) system posed on the fixed domain
$(0,1)$ that seems to facilitate the authors proof of maximal solutions.

In Section 2, we present analytically the change of variables $y=x-s^{+}$,
$y=-x+s^{-}$ for $x\in\Omega_{\rm Liq}$, use the Stefan condition and derive
per case the Stefan problems as systems of two independent spdes of the form
(1.10) for $v=v_{1},v_{2}$, cf. (2.6), (2.11), (2.14). Section 3 is devoted to
the existence of unique weak maximal solutions $(v,\eta)$ of the Dirichlet
problem on $\mathcal{D}$ for (1.10) with reflection, and then of maximal
solutions to the initial variables with stopping times restricted by the
Stefan condition, the non-negativity of spread and the boundedness of the
liquid phase. In detail, we write the spde in an integral form using the
Green’s function of the negative Dirichlet Laplacian and construct an
approximate Picard scheme for the truncated problem. In Theorem 3.1 we prove
existence and uniqueness a.s. for the Picard approximations, and on the limit
existence and uniqueness of the truncated solution. For this, we use some of
the Green’s estimates of [20] and a proper Banach space introduced therein.
The reflection measure $\eta$ is associated to the obstacle problem estimated
in [20]. In Theorem 3.2, using the consistency of the truncated solutions we
prove existence of a unique maximal solution $(v,\eta)$ a.s. in the maximal
time interval $[0,\displaystyle{\sup_{M>0}}\tau_{M})$ for $\tau_{M}$ given by
(1.11). Given the maximal solution $(v,\eta)$, for $v=v_{1,2}\geq 0$,
$\eta=\eta_{1,2}$, in Theorem 3.3 we prove existence of unique maximal
solutions $(w_{1},\eta_{1})$, $(w_{2},\eta_{2})$ to the reflected Stefan
problem (2.3)-(2.15)-(2.16) corresponding to (1), and of $w|_{x\geq
s^{+}}=w_{1}\geq 0$, $w|_{x\leq s^{-}}=-w_{2}\leq 0$, in the maximal interval
$\mathcal{I}_{1}:=[0,\hat{\tau})$ for
$\hat{\tau}:=\min\\{\displaystyle{\sup_{M>0}}\tau_{1M},\tau_{1s},\tau_{1}^{*}\\}$,
with $\tau_{1M},\tau_{1s},\tau_{1}^{*}$ given by (3.38), (3.39), (3.40) for
which the spread exists and stays a.s. non-negative for any
$t\in\mathcal{I}_{1}$. An analogous result for the case (2) is proven in
Theorem 3.4 but in a different maximal interval
$\mathcal{I}_{2}:=[0,\hat{\tau})$ for
$\hat{\tau}:=\min\\{\displaystyle{\sup_{M>0}}\tau_{1M},\tau_{2s}\\}$, with
$\tau_{1M},\tau_{2s}$ given by (3.38), (3.41). There, the decreasing property
of the spread is used.

In Section 4 we consider the Stefan problem without reflection, i.e., (3), and
the Dirichlet problem on $\mathcal{D}$ for the spde (4.1) that $v$ satisfies.
Theorem 4.1 establishes existence and uniqueness a.s. of the truncated
equation, and Theorem 4.2 existence of a unique maximal solution $v$ in
$[0,\displaystyle{\sup_{M>0}}\tau_{M})$ for $\tau_{M}$ as in (1.11). Then the
existence and uniqueness of maximal solution in the initial variables is
proven in Theorem 4.3 for the resulting stopping time. We also present some
formal asymptotics for very large liquidity coefficient $\alpha$, when the
noise is only time dependent (for example the formal derivative of a Brownian)
and for constant noise diffusion. Under the assumption that $w$ approximates a
mean-field value $w_{\infty}(t)$ when the distance from the spread boundary is
very large, we derive that the spread is given by the integral of the solution
of a stochastic linear diffusion equation, see the dynamics in (4.10), and the
linear sde (4.11).

##  2\. The Stefan problems

###  2.1. Change of variables

We consider $\Omega$ given by (1.6), $\Omega_{\rm Liq}$ by (1.7), and $y$
defined by (1.8) for any $x\in\Omega_{\rm Liq}\cup\\{s^{-},s^{+}\\}$. Let
$\tilde{w}_{1}(x,t)$ be defined in $\\{x\in\Omega_{\rm
Liq}\cup\\{s^{-},s^{+}\\}:\;x\geq s^{+}\\}$ and $\tilde{w}_{2}(x,t)$ be
defined in $\\{x\in\Omega_{\rm Liq}\cup\\{s^{-},s^{+}\\}:\;x\leq s^{-}\\}$ and
set for $y:=x-s^{+}$

$\tilde{w}_{1}(x,t):=\tilde{v}_{1}(y,t)\;\;\forall\;x\in\Omega_{\rm
Liq}\cup\\{s^{-}(t),s^{+}(t)\\}:\;x\geq s^{+}(t),$

while for $y:=-x+s^{-}$

$\tilde{w}_{2}(x,t):=\tilde{v}_{2}(y,t)\;\;\forall\;x\in\Omega_{\rm
Liq}\cup\\{s^{-}(t),s^{+}(t)\\}:\;x\leq s^{-}(t).$

If $x\geq s^{+}(t)$ we get

(2.1)
$\begin{split}&\tilde{w}_{1}(x,t)=\tilde{v}_{1}(x-s^{+}(t),t)=\tilde{v}_{1}(y,t),\;\;\;y=x-s^{+}(t),\;\;y_{x}=1,\;\;y_{t}=-\partial_{t}s^{+}(t)\\\
&(\tilde{w}_{1})_{t}(x,t)=(\tilde{v}_{1})_{y}(y,t)y_{t}(y,t)+(\tilde{v}_{1})_{t}(y,t)=-\partial_{t}s^{+}(t)(\tilde{v}_{1})_{y}(y,t)+(\tilde{v}_{1})_{t}(y,t),\\\
&(\tilde{w}_{1})_{x}(x,t)=(\tilde{v}_{1})_{y}(y,t)y_{x}=+(\tilde{v}_{1})_{y}(y,t),\\\
&(\tilde{w}_{1})_{xx}(x,t)=(\tilde{v}_{1})_{yy}(y,t)(y_{x}(y,t))^{2}=(\tilde{v}_{1})_{yy}(y,t),\end{split}$

and if $x\leq s^{-}(t)$

(2.2)
$\begin{split}&\tilde{w}_{2}(x,t)=\tilde{v}_{2}(-x+s^{-}(t),t)=\tilde{v}_{2}(y,t),\;\;y=-x+s^{-}(t),\;\;y_{x}=-1,\;\;y_{t}=\partial_{t}s^{-}(t)\\\
&(\tilde{w}_{2})_{t}(x,t)=(\tilde{v}_{2})_{y}(y,t)y_{t}(y,t)+(\tilde{v}_{2})_{t}(y,t)=\partial_{t}s^{-}(t)(\tilde{v}_{2})_{y}(y,t)+(\tilde{v}_{2})_{t}(y,t),\\\
&(\tilde{w}_{2})_{x}(x,t)=(\tilde{v}_{2})_{y}(y,t)y_{x}=-(\tilde{v}_{2})_{y}(y,t),\\\
&(\tilde{w}_{2})_{xx}(x,t)=(\tilde{v}_{2})_{yy}(y,t)(y_{x}(y,t))^{2}=(\tilde{v}_{2})_{yy}(y,t).\end{split}$

###  2.2. Case 1

Let for any $x\in\Omega$ the signed density $w$ be given by

$w(x,t)=w_{1}(x,t)-w_{2}(x,t)=\begin{cases}w_{1}(x,t)&\text{if}\;\;x>s^{+}(t),\\\
-w_{2}(x,t)&\text{if}\;\;x<s^{-}(t),\\\ 0&\text{otherwise},\end{cases}$

for $w_{1},w_{2}$ the densities of sell orders and buy orders respectively. We
then have $w(x,t)|_{x\geq s^{+}}=w_{1}(x,t)$, $w(x,t)|_{x\leq
s^{-}}=-w_{2}(x,t)$.

The equation (1.1) by introducing the additive term $\dot{\eta}_{s}$ given by
(1.13) takes in $\Omega_{\rm{Liq}}$ the form

$\partial_{t}w=\alpha\Delta w+\sigma({\rm dist}(x,\partial
S))\dot{W}_{s}(x,t)+\dot{\eta}_{s}(x,t),\;\;\;\;\;\;\;\;x\in\Omega_{\rm{Liq}},\;\;t>0,$

or equivalently for $x\in\Omega_{\rm Liq}$

(2.3) $\begin{split}\partial_{t}w_{1}=&\alpha\Delta
w_{1}+\sigma(x-s^{+}(t))\dot{W}(x-s^{+}(t),t)+\dot{\eta}_{1}(x-s^{+}(t),t),\;\;\;\;\;\;\;\;\;\;\;x>s^{+}(t),\;\;t>0,\\\
\partial_{t}w_{2}=&\alpha\Delta
w_{2}-\sigma(-x+s^{-}(t))\dot{W}(-x+s^{-}(t),t)+\dot{\eta}_{2}(-x+s^{-}(t),t),\;\;x<s^{-}(t),\;\;t>0,\end{split}$

while
$w(x,t)=w_{1}(x,t)=w_{2}(x,t)=0,\;\;\;\forall\;x\in[s^{-}(t),s^{+}(t)],\;\;\forall\;t>0,$
and $w_{1}(x,0)=w(x,0)$ for any $x\geq s^{+}(0)$, $w_{2}(x,0)=-w(x,0)$ for any
$x\leq s^{-}(0)$. We shall assume that $w_{1}(x,0),w_{2}(x,0)\geq 0$. The
reflection measures $\eta_{1}$, $\eta_{2}$ if exist will keep $w_{1},w_{2}\geq
0$ for all $t$ a.s. Using the Stefan condition (1.4), we obtain

(2.4) $\begin{split}V(s^{+}(t),t)=&\partial_{t}s^{+}(t)=-(\nabla
w)^{+}(s^{+}(t),t)=-(\nabla w_{1})^{+}(s^{+}(t),t),\\\
V(s^{-}(t),t)=&\partial_{t}s^{-}(t)=-(\nabla w)^{-}(s^{-}(t),t)=(\nabla
w_{2})^{-}(s^{-}(t),t),\end{split}$

and so the spread dynamics are given by

(2.5) $\partial_{t}s^{+}(t)-\partial_{t}s^{-}(t)=-(\nabla
w_{1})^{+}(s^{+}(t),t)-(\nabla w_{2})^{-}(s^{-}(t),t)).$

We apply the change of variables $w_{1}(x,t)=v_{1}(y,t)$ for $y=x-s^{+}$ and
so $(\nabla w_{1})^{+}(s^{+},t)=\nabla v_{1}(0^{+},t)$, and
$w_{2}(x,t)=v_{2}(y,t)$ for $y=-x+s^{-}$ and so $(\nabla
w_{2})^{-}(s^{-},t)=-\nabla v_{2}(0^{+},t)$, use (2.1), (2.2), and (2.4) which
yields that $\partial_{t}s^{+}(t)=-(\nabla w_{1})^{+}(s^{+}(t),t)=-\nabla
v_{1}(0^{+},t)$, and that $\partial_{t}s^{-}(t)=(\nabla
w_{2})^{-}(s^{-}(t),t)=-\nabla v_{2}(0^{+},t)$, and derive the system of two
independent initial and boundary value problems

(2.6) $\begin{split}\partial_{t}v_{1}(y,t)=&\alpha\Delta
v_{1}(y,t)+\partial_{t}s^{+}(t)\nabla
v_{1}(y,t)+\sigma(y)\dot{W}(y,t)+\dot{\eta}_{1}(y,t)\\\ =&\alpha\Delta
v_{1}(y,t)-\nabla v_{1}(0^{+},t)\nabla
v_{1}(y,t)+\sigma(y)\dot{W}(y,t)+\dot{\eta}_{1}(y,t),\;\;y\in(0,\lambda)=:\mathcal{D},\;\;t>0,\\\
&v_{1}(0,t)=v_{1}(\lambda,t)=0,\;\;t>0,\;\;v_{1}(y,0)=w(y+s^{+}(0),0)\geq
0,\;\;y\in\mathcal{D},\\\ &{\rm and}\\\ \partial_{t}v_{2}(y,t)=&\alpha\Delta
v_{2}(y,t)-\partial_{t}s^{-}(t)\nabla
v_{2}(y,t)-\sigma(y)\dot{W}(y,t)+\dot{\eta}_{2}(y,t)\\\ =&\alpha\Delta
v_{2}(y,t)+\nabla v_{2}(0^{+},t)\nabla
v_{2}(y,t)-\sigma(y)\dot{W}(y,t)+\dot{\eta}_{2}(y,t),\;\;y\in(0,\lambda)=:\mathcal{D},\;\;t>0,\\\
&v_{2}(0,t)=v_{2}(\lambda,t)=0,\;\;t>0,\;\;v_{2}(y,0)=-w(s^{-}(0)-y,0)\geq
0,\;\;y\in\mathcal{D},\\\ \end{split}$

where we used the Dirichlet b.c. $v_{1}(\lambda,t)=v_{2}(\lambda,t)=0$.

By using (2.5), the spread evolution is given by

(2.7) $\partial_{t}(s^{+}(t)-s^{-}(t))=-\nabla v_{1}(0^{+},t)+\nabla
v_{2}(0^{+},t).$

###  2.3. Case 2

Let for any $x\in\Omega$ the density $w$, and define

$w_{1}(x,t):=w(x,t)\;\;x\geq s^{+},\;\;w_{2}(x,t)=w(x,t)\;\;x\leq s^{-},$

and so

$w(x,t)=\begin{cases}w_{1}(x,t)&\text{if}\;\;x>s^{+}(t),\\\
w_{2}(x,t)&\text{if}\;\;x<s^{-}(t),\\\ 0&\text{otherwise}.\end{cases}$

In this case, we have $w(x,t)|_{x\geq s^{+}}=w_{1}(x,t)$, $w(x,t)|_{x\leq
s^{-}}=w_{2}(x,t)$.

We introduce in (1.1) the additive term $\dot{\eta}_{s}(x,t)$ given by (1.14),
for $\eta_{1}$, $\eta_{2}$ there reflection measures keeping $w_{1},w_{2}\geq
0$ and thus $w\geq 0$. The equation on $\Omega_{\rm Liq}$ takes the form

$\partial_{t}w=\alpha\Delta w+\sigma({\rm dist}(x,\partial
S))\dot{W}_{s}(x,t)+\dot{\eta}_{s}(x,t),\;\;\;\;\;\;\;\;x\in\Omega_{\rm
Liq},\;\;t>0,$

or equivalently for $x\in\Omega_{\rm Liq}$

(2.8) $\begin{split}\partial_{t}w_{1}=&\alpha\Delta
w_{1}+\sigma(x-s^{+}(t))\dot{W}(x-s^{+}(t),t)+\dot{\eta}_{1}(x-s^{+}(t),t),\;\;\;\;\;\;\;\;\;\;\;x>s^{+}(t),\;\;t>0,\\\
\partial_{t}w_{2}=&\alpha\Delta
w_{2}+\sigma(-x+s^{-}(t))\dot{W}(-x+s^{-}(t),t)+\dot{\eta}_{2}(-x+s^{-}(t),t),\;\;x<s^{-}(t),\;\;t>0,\end{split}$

while
$w(x,t)=w_{1}(x,t)=w_{2}(x,t)=0,\;\;\;\forall\;x\in[s^{-}(t),s^{+}(t)],\;\;\forall\;t>0,$
and $w_{1}(x,0)=w(x,0)$ for any $x\geq s^{+}(0)$, $w_{2}(x,0)=w(x,0)$ for any
$x\leq s^{-}(0)$. We shall assume that $w_{1}(x,0),w_{2}(x,0)\geq 0$. The
reflection measures $\eta_{1}$, $\eta_{2}$ if exist will keep
$w_{1},w_{2},w\geq 0$ for all $t$ a.s. Using the Stefan condition (1.4), we
obtain

(2.9) $\begin{split}V(s^{+}(t),t)=&\partial_{t}s^{+}(t)=-(\nabla
w)^{+}(s^{+}(t),t)=-(\nabla w_{1})^{+}(s^{+}(t),t),\\\
V(s^{-}(t),t)=&\partial_{t}s^{-}(t)=-(\nabla w)^{-}(s^{-}(t),t)=-(\nabla
w_{2})^{-}(s^{-}(t),t),\end{split}$

and so the spread dynamics are given by

(2.10) $\partial_{t}s^{+}(t)-\partial_{t}s^{-}(t)=-(\nabla
w_{1})^{+}(s^{+}(t),t)+(\nabla w_{2})^{-}(s^{-}(t),t)).$

We apply the change of variables $w_{1}(x,t)=v_{1}(y,t)$ for $y=x-s^{+}$ and
so $(\nabla w_{1})^{+}(s^{+},t)=\nabla v_{1}(0^{+},t)$, and
$w_{2}(x,t)=v_{2}(y,t)$ for $y=-x+s^{-}$ and so $(\nabla
w_{2})^{-}(s^{-},t)=-\nabla v_{2}(0^{+},t)$, use (2.1), (2.2), and (2.4) which
yields that $\partial_{t}s^{+}(t)=-(\nabla w_{1})^{+}(s^{+}(t),t)=-\nabla
v_{1}(0^{+},t)$, and that $\partial_{t}s^{-}(t)=-(\nabla
w_{2})^{-}(s^{-}(t),t)=\nabla v_{2}(0^{+},t)$, to derive the system of two
independent initial and boundary value problems

(2.11) $\begin{split}\partial_{t}v_{1}(y,t)=&\alpha\Delta
v_{1}(y,t)+\partial_{t}s^{+}(t)\nabla
v_{1}(y,t)+\sigma(y)\dot{W}(y,t)+\dot{\eta}_{1}(y,t)\\\ =&\alpha\Delta
v_{1}(y,t)-\nabla v_{1}(0^{+},t)\nabla
v_{1}(y,t)+\sigma(y)\dot{W}(y,t)+\dot{\eta}_{1}(y,t),\;\;y\in(0,\lambda)=:\mathcal{D},\;\;t>0,\\\
&v_{1}(0,t)=v_{1}(\lambda,t)=0,\;\;t>0,\;\;v_{1}(y,0)=w(y+s^{+}(0),0)\geq
0,\;\;y\in\mathcal{D},\\\ &{\rm and}\\\ \partial_{t}v_{2}(y,t)=&\alpha\Delta
v_{2}(y,t)-\partial_{t}s^{-}(t)\nabla
v_{2}(y,t)+\sigma(y)\dot{W}(y,t)+\dot{\eta}_{2}(y,t)\\\ =&\alpha\Delta
v_{2}(y,t)-\nabla v_{2}(0^{+},t)\nabla
v_{2}(y,t)+\sigma(y)\dot{W}(y,t)+\dot{\eta}_{2}(y,t),\;\;y\in(0,\lambda)=:\mathcal{D},\;\;t>0,\\\
&v_{2}(0,t)=v_{2}(\lambda,t)=0,\;\;t>0,\;\;v_{2}(y,0)=w(s^{-}(0)-y,0)\geq
0,\;\;y\in\mathcal{D}.\\\ \end{split}$

By using (2.5), the spread evolution is given by

(2.12) $\partial_{t}(s^{+}(t)-s^{-}(t))=-\nabla v_{1}(0^{+},t)-\nabla
v_{2}(0^{+},t).$

As we already mentioned, if the reflection measures exist and keep
$v_{1},v_{2}\geq 0$ and since $v_{1}=v_{2}=0$ at $y=0$, then the spread is
decreasing.

###  2.4. Case 3

As in Case 2, we consider for any $x\in\Omega$ the signed density $w$, and
define

$w_{1}(x,t):=w(x,t)\;\;x\geq s^{+},\;\;w_{2}(x,t)=w(x,t)\;\;x\leq s^{-}.$

We do not require here $w\geq 0$ and so we consider the unreflected equation
(1.1) posed in $\Omega_{\rm Liq},\;\;t>0,$ or equivalently for
$x\in\Omega_{\rm Liq}$

(2.13) $\begin{split}\partial_{t}w_{1}=&\alpha\Delta
w_{1}+\sigma(x-s^{+}(t))\dot{W}(x-s^{+}(t),t),\;\;\;\;\;\;\;\;\;\;\;x>s^{+}(t),\;\;t>0,\\\
\partial_{t}w_{2}=&\alpha\Delta
w_{2}+\sigma(-x+s^{-}(t))\dot{W}(-x+s^{-}(t),t),\;\;x<s^{-}(t),\;\;t>0,\end{split}$

while
$w(x,t)=w_{1}(x,t)=w_{2}(x,t)=0,\;\;\;\forall\;x\in[s^{-}(t),s^{+}(t)],\;\;\forall\;t>0,$
and $w_{1}(x,0)=w(x,0)$ for any $x\geq s^{+}(0)$, $w_{2}(x,0)=w(x,0)$ for any
$x\leq s^{-}(0)$. Using the Stefan condition (1.4), we obtain (2.9) again for
the velocity and the spread dynamics are given by (2.10). We apply the change
of variables $w_{1}(x,t)=v_{1}(y,t)$ for $y=x-s^{+}$, and
$w_{2}(x,t)=v_{2}(y,t)$ for $y=-x+s^{-}$ to obtain as in Case 2 the system of
two independent initial and boundary value problems

(2.14) $\begin{split}\partial_{t}v_{1}(y,t)=&\alpha\Delta v_{1}(y,t)-\nabla
v_{1}(0^{+},t)\nabla
v_{1}(y,t)+\sigma(y)\dot{W}(y,t),\;\;y\in(0,\lambda)=:\mathcal{D},\;\;t>0,\\\
&v_{1}(0,t)=v_{1}(\lambda,t)=0,\;\;t>0,\;\;v_{1}(y,0)=w(y+s^{+}(0),0),\;\;y\in\mathcal{D},\\\
&{\rm and}\\\ \partial_{t}v_{2}(y,t)=&\alpha\Delta v_{2}(y,t)-\nabla
v_{2}(0^{+},t)\nabla
v_{2}(y,t)+\sigma(y)\dot{W}(y,t),\;\;y\in(0,\lambda)=:\mathcal{D},\;\;t>0,\\\
&v_{2}(0,t)=v_{2}(\lambda,t)=0,\;\;t>0,\;\;v_{2}(y,0)=w(s^{-}(0)-y,0),\;\;y\in\mathcal{D}.\\\
\end{split}$

The spread evolution is given as in Case 2 by (2.12), but since $v_{1}$,
$v_{2}$ may change sign even if $v_{1}=v_{2}=0$ at $y=0$, the spread is not
monotone.

When reflection measures are considered, i.e., for the Cases 1,2, each
problem’s unknowns for $t\in[0,T]$ is a pair $(v,\eta)$ where the reflection
measure $\eta$ is defined to satisfy

(2.15) $\begin{split}&\mbox{for all}\mbox{ measurable functions
}\psi:\;\overline{\mathcal{D}}\times(0,T)\rightarrow[0,\infty)\\\
&\int_{0}^{t}\int_{\mathcal{D}}\psi(y,s)\eta(dy,ds)\;\;\mbox{is}\;\;\mathcal{F}_{t}-\mbox{measurable},\end{split}$

and the constraint

(2.16) $\int_{0}^{T}\int_{\mathcal{D}}v(y,s)\eta(dy,ds)=0.$

We shall assume that the noise diffusion $\sigma$ is a sufficiently smooth
function; its minimum regularity will be specified in the sequel. The random
measure $W(dy,ds)$ is defined as the $1$-dimensional space-time white noise
induced by the $2$-dimensional Wiener process
$W:=\\{W(y,t):\;t\in[0,T],\;y\in(0,\lambda)\\}$ which generates, for any
$t\geq 0$, the filtration $\mathcal{F}_{t}:=\sigma(W(y,s):\;s\leq
t,\;y\in(0,\lambda))$, where the notation $\sigma$ here denotes the
$\sigma$-algebra.

######  Remark 2.1.

In all the above cases, given the solutions $v_{1},v_{2}$ for $t\in[0,T]$,
$s^{+}(t),\;s^{-}(t)$ and the spread $s^{+}(t)-s^{-}(t)$ are derived by direct
formulae after integration of the Stefan condition in $[0,t]$.

######  Remark 2.2.

We observe that the transformed spdes of Cases 1,2,3 are of the general form
(2.2), i.e.,

$v_{t}(y,t)=\alpha\Delta v(y,t)\mp\nabla v(0^{+},t)\nabla
v(y,t)\pm\sigma(y)\dot{W}(y,t)+\dot{\eta}(y,t),$

posed on $\mathcal{D}:=(0,\lambda)$ for $t\in[0,T]$, with Dirichlet b.c.
$v(x,t)=0$ at $\partial\mathcal{D}$, and $v(y,0)$ given, for $v\geq 0$ when
$\eta$ not the zero measure, and signed $v$ when $\eta\equiv 0$.

######  Remark 2.3.

Given $v_{1,2}$ for any $y\in\mathcal{D}$ and any $t$ in $[0,T]$, then the
Stefan condition will determine after integration $s^{\pm}(t)$ in $(0,T]$. Let
$x\in\Omega_{\rm Liq}$ then for any given $t\in[0,T]$ and any $x\geq s^{+}(t)$
since $y=x-s^{+}(t)$, $w(x,t)$ will be defined by $v_{1}(x-s^{+}(t),t)$, while
for any $x\leq s^{-}(t)$ since $y=-x+s^{-}(t)$, $w(x,t)$ will be defined by
$-v_{2}(-x+s^{-}(t),t)$ for case (1) or by $v_{2}(-x+s^{-}(t),t)$ for Cases
2,3.

######  Remark 2.4.

Evolution for $v_{1,2}$ will be observed as long as $a\leq s^{-}\leq s^{+}\leq
b$, while $a\leq x\leq b$. In particular, consider $y=\lambda=b-a$. Then if
$x\geq s^{+}$ then $y=b-a=x-s^{+}\leq b-s^{+}$ will yield $-a\leq-s^{+}$ i.e.,
$s^{+}\leq a$ and thus $s^{+}=a$ and $s^{-}=s^{+}=a$ and $x=b$ which is the
case when the the spread is zero and hits the boundary at $b$ and
$v_{1}(\lambda,t)=0$. If $x\leq s^{-}$ then $y=b-a=-x+s^{-}\leq-a+s^{-}$ will
yield $b\leq s^{-}$ and thus $s^{-}=b$ and $s^{+}=s^{-}=b$ and $x=a$ which is
the case when the the spread is zero and hits the boundary at $a$ and
$v_{2}(\lambda,t)=0$. When $y=0$ then either $x=s^{-}$ and $v_{2}(0,t)=0$ or
$x=s^{+}$ and $v_{1}(0,t)=0$. For all $x\in(s^{-},s^{+})$ the density $w(x,t)$
will be set to $0$. The initial values of $v_{1,2}$ are well defined through
the initial value $w(x,0)$ which is given for all $x\in\mathbb{R}$. We assume
that $w(x,0)$ is compactly supported in $\overline{\Omega}$ to obtain a
compatibility condition to $v_{1,2}(\lambda,t)=0$ at $t=0$.

We will analyze in detail in the sequel how the restrictions of a non-negative
spread and spread area in the domain $\Omega$, i.e., $a<s^{-}\leq s^{+}<b$,
reduce the stopping time up to which maximal solutions $w_{1,2}$ exist.

##  3\. Existence of maximal solutions with reflection

In what follows we shall present the analytical proof of existence of unique
maximal solutions $(v,\eta)$ for the initial and boundary value problem for

(3.1) $v_{t}(y,t)=\alpha\Delta v(y,t)-\nabla v(0^{+},t)\nabla
v(y,t)+\sigma(y)\dot{W}(y,t)+\dot{\eta}(y,t),$

posed for any $y$ in $\mathcal{D}=(0,\lambda)$ for $t\in[0,T]$ with Dirichlet
b.c., with $v(y,0)$ given, and $\eta$ a reflection measure satisfying (2.15)
and (2.16) keeping $v$ non-negative. As $\alpha>0$ the proof for the 2d i.b.v.
problem of (2.6) of Case 1 is completely analogous, while the results for Case
3 (unreflected problem) will be derived at a next section by setting
$\eta\equiv 0$. We will keep the absolute values on $\nabla v(0^{+},t)$
appearing in the following proofs (even if non-negative in (3.1)) so that the
results are applicable for these cases directly.

###  3.1. Weak formulation

Let us define an $L^{2}(\mathcal{D})$ basis of eigenfunctions
$w_{n}:=\sin\Big{(}\frac{n\pi}{\lambda}x\Big{)},\;\;n=0,1,2,\cdots,$
corresponding to the eigenvalues $\mu_{n}$, $n=0,1,\cdots$ of $-\Delta u=\mu
u,\;\;u(0)=0,\;\;u(\lambda)=0,$ where
$\mu_{n}:=\frac{n^{2}\pi^{2}}{\lambda^{2}},\;\;n=0,1,2,\cdots$ The associate
Green’s function for the negative of the Dirichlet Laplacian can then be given
by
$\frac{2}{\lambda}\displaystyle{\sum_{n=0}^{\infty}}e^{-\mu_{n}t}w_{n}(x)w_{n}(y),$
see [15], so that the Green’s function corresponding to $-\alpha\Delta$ with
Dirichlet b.c. is given by
$G(t,x,y)=\frac{2}{\lambda}\displaystyle{\sum_{n=0}^{\infty}}e^{-\alpha\mu_{n}t}w_{n}(x)w_{n}(y)$.

We say that $v$ is a weak (analytic) solution of (3.1) if it satisfies for all
$\phi=\phi(y)$ in $C^{2}(\overline{\mathcal{D}})$ with
$\phi(0)=\phi(\lambda)=0$, the following weak formulation

$\displaystyle\int_{\mathcal{D}}\Big{(}v(y,t)-v_{0}(y)\Big{)}\phi(y)dy=$
$\displaystyle\int_{0}^{t}\int_{\mathcal{D}}\Big{(}\alpha\Delta\phi(y)v(y,s)+\nabla\phi(y)\nabla
v(0^{+},s)v(y,s)\Big{)}dyds$ (3.2)
$\displaystyle+\int_{0}^{t}\int_{\mathcal{D}}\phi(y)\sigma(y)W(dy,ds)+\int_{0}^{t}\int_{\mathcal{D}}\phi(y)\eta(dy,ds).$

The solution of (3.1) admits for any $y\in\mathcal{D}$, $t\in[0,T]$, the next
integral representation

(3.3) $\begin{split}v(y,t)=&\int_{\mathcal{D}}v_{0}(z)G(y,z,t)dz\\\
&+\int_{0}^{t}\int_{\mathcal{D}}\nabla v(0^{+},s)\nabla
G(y,z,t-s)v(z,s)dzds\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\sigma(z)W(dz,ds)+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\eta(dz,ds),\end{split}$

and $\eta$ satisfies (2.15), (2.16).

###  3.2. Main Theorems

Let the Banach space $(\mathcal{B},\|\cdot\|_{\mathcal{B}})$

$\mathcal{B}:=\Big{\\{}f\in C(\overline{\mathcal{D}}):\;\exists
f^{\prime}(0),\;f(0)=f(\lambda)=0\Big{\\}},$

with the norm $\|\cdot\|_{\mathcal{B}}:\mathcal{B}\rightarrow\mathbb{R}^{+}$,
defined by

$\|f\|_{\mathcal{B}}:=\displaystyle{\sup_{y\in\mathcal{D}}}\Big{|}\frac{f(y)}{y}\Big{|}.$

Let $M>0$ fixed, we define in the Banach space $\mathcal{B}$, as in [20], the
operator $\mathcal{T}_{M}:\mathcal{B}\rightarrow\mathcal{B}$ given for any
$y\in\mathcal{\overline{D}}$ and $u\in\mathcal{B}$ by

(3.4)
$\mathcal{T}_{M}(u)(y,\cdot)=\begin{cases}y\min\Big{\\{}\frac{u(y,\cdot)}{y},M\Big{\\}}&y\neq
0,\\\ 0&y=0.\end{cases}$

We consider a truncated problem through the action of the operator
$\mathcal{T}_{M}$ on the gradient terms of the spde (3.1) for which we prove
the next existence-uniqueness theorem.

######  Theorem 3.1.

Let the noise diffusion $\sigma$ satisfy

(3.5) $\sigma\in
C(\overline{\mathcal{D}}),\;\;\sigma(0)=\sigma(\ell)=0,\;\;\exists\;\sigma^{\prime}(0).$

Let also $M>0$ fixed, and some $p\geq p_{0}>8$, and let $v_{0}(y)\in
L^{p}(\Omega,C[0,T];\mathcal{B})$ be the initial condition of (3.1). Then
there exists a unique week solution $(v^{M},\eta^{M})$ with $v^{M}\in
L^{p}(\Omega,C[0,T];\mathcal{B})$, depending on $M$, to the truncated problem

(3.6) $\begin{split}&v^{M}_{t}(y,t)=\alpha\Delta
v^{M}(y,t)-\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},t)\nabla(\mathcal{T}_{M}(v^{M}))(y,t)\\\
&\hskip
48.36958pt+\sigma(y)\dot{W}(y,t)+\dot{\eta}^{M}(y,t),\;\;t\in(0,T],\;\;y\in\mathcal{D},\\\
&v^{M}(y,0):=v_{0}(y),\;\;y\in\mathcal{D},\\\
&v^{M}(0,t)=v^{M}(\lambda,t)=0,\;\;t\in(0,T],\\\ \end{split}$

where $T:=T_{M}>0$ such that

(3.7)
$\displaystyle{\sup_{r\in(0,T)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},r)|^{p}\leq
C_{2}(M,p)<\infty\;\;a.s.$

More specifically, for any $t\in(0,T)$, $v^{M}$ satisfies the week formulation

(3.8) $\begin{split}v^{M}(y,t)=&\int_{\mathcal{D}}v_{0}(z)G(y,z,t)dz\\\
&+\int_{0}^{t}\int_{\mathcal{D}}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)\nabla
G(y,z,t-s)\mathcal{T}_{M}(v^{M})(z,s)dzds\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\sigma(z)W(dz,ds)\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\eta^{M}(dz,ds),\end{split}$

for $v^{M}(y,0):=v_{0}(y)$, and $\eta^{M}$, satisfies (2.15) and (2.16), i.e.,

(3.9) $\begin{split}&\mbox{for all}\mbox{ measurable functions
}\psi:\;\overline{\mathcal{D}}\times(0,T)\rightarrow[0,\infty)\\\
&\int_{0}^{t}\int_{\mathcal{D}}\psi(y,s)\eta^{M}(dy,ds)\;\;\mbox{is}\;\;\mathcal{F}_{t}-\mbox{measurable},\end{split}$

and the constraint

(3.10) $\int_{0}^{T}\int_{\mathcal{D}}v^{M}(y,s)\eta^{M}(dy,ds)=0.$

###### Proof.

The operator $\mathcal{T}_{M}:\mathcal{B}\rightarrow\mathcal{B}$ is well
defined, [20], and thus, for any $u$ in the space $\mathcal{B}$,
$\mathcal{T}_{M}(u)$ returns in $\mathcal{B}$, and so $\mathcal{T}_{M}(u)\in
C(\overline{\mathcal{D}})$ and vanishes at the boundary of $\mathcal{D}$,
while the gradient $\nabla(\mathcal{T}_{M}(u))$ at $x=0$ exists.

Motivated by the integral representation (3.8), we define through a Picard
iteration scheme the approximation $v_{n}^{M}$ of $v^{M}$ as the solution of
the approximate problem

(3.11) $\begin{split}v_{n}^{M}(y,t)=&\int_{\mathcal{D}}v_{0}(z)G(y,z,t)dz\\\
&+\int_{0}^{t}\int_{\mathcal{D}}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)\nabla
G(y,z,t-s)\mathcal{T}_{M}(v_{n-1}^{M})(z,s)dzds\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\sigma(z)W(dz,ds)\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\eta_{n}^{M}(dz,ds),\;\;n:=1,2\cdots\end{split}$

for $v_{0}^{M}(y,t):=v_{0}(y)$, and $\eta_{n}^{M}$, which approximates
$\eta^{M}$, satisfying (2.15) and (2.16), i.e.,

(3.12) $\begin{split}&\mbox{for all}\mbox{ measurable functions
}\psi:\;\overline{\mathcal{D}}\times(0,T)\rightarrow[0,\infty)\\\
&\int_{0}^{t}\int_{\mathcal{D}}\psi(y,s)\eta_{n}^{M}(dy,ds)\;\;\mbox{is}\;\;\mathcal{F}_{t}-\mbox{measurable},\end{split}$

and the constraint

(3.13) $\int_{0}^{T}\int_{\mathcal{D}}v_{n}^{M}(y,s)\eta_{n}^{M}(dy,ds)=0.$

In order to keep $v_{n}^{M}$ non-negative, and having in mind the integral
property (3.13), we will absorb the reflection term $\eta_{n}^{M}$ in the
Picard scheme (3.11), by spliting $v_{n}^{M}$ as follows

(3.14) $v_{n}^{M}(y,t)=u_{n}(y,t)+\mathbb{O}_{n}(y,t),$

where $\mathbb{O}_{n}(y,t)$ solves in the weak sense the Heat Equation
Obstacle problem for any $y\in\mathcal{D}$, $t\in[0,T]$

(3.15)
$\begin{split}&\partial_{t}\mathbb{O}_{n}(y,t)=\alpha\Delta\mathbb{O}_{n}(y,t)+\tilde{\eta}_{n}(dy,dt),\;\;u_{n}+\mathbb{O}_{n}\geq
0(\Leftrightarrow\mathbb{O}_{n}\geq-u_{n}),\\\
&\mathbb{O}_{n}(0,t)=\mathbb{O}_{n}(\lambda,t)=0,\;\;\mathbb{O}_{n}(y,0)=0,\\\
&\int_{0}^{T}\int_{\mathcal{D}}(u_{n}(y,s)+\mathbb{O}_{n}(y,s))\tilde{\eta}_{n}(dy,ds)=0.\end{split}$

Note that the above problem has a unique weak solution
$(\mathbb{O}_{n},\tilde{\eta}_{n})$ as long as $u_{n}$ exists and is smooth.
We observe that $\mathbb{O}_{n}(y,0)=0$ yields that
$u_{n}(y,0)=v_{n}^{M}(y,0)=v_{0}(y)$.

We define $\eta_{n}^{M}:=\tilde{\eta}_{n}$, and as we shall see it satisfies
(3.11) when $v_{n}^{M}$ satisfies (3.14). Indeed, we replace
$v_{n}^{M}=u_{n}(y,t)+\mathbb{O}_{n}(y,t)$ at the left-hand side of (3.11) and
obtain for $\eta_{n}^{M}:=\tilde{\eta}_{n}$

(3.16)
$\begin{split}u_{n}&(y,t)+\mathbb{O}_{n}(y,t)=\int_{\mathcal{D}}v_{0}(z)G(y,z,t)dz\\\
&+\int_{0}^{t}\int_{\mathcal{D}}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)\nabla
G(y,z,t-s)\mathcal{T}_{M}(v_{n-1}^{M})(z,s)dzds\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\sigma(z)W(dz,ds)\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\eta_{n}^{M}(dz,ds),\\\
=&\int_{\mathcal{D}}v_{0}(z)G(y,z,t)dz\\\
&+\int_{0}^{t}\int_{\mathcal{D}}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)\nabla
G(y,z,t-s)\mathcal{T}_{M}(v_{n-1}^{M})(z,s)dzds\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\sigma(z)W(dz,ds)\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\tilde{\eta}_{n}(dz,ds)\;\;n:=1,2\cdots\end{split}$

Since $\mathbb{O}_{n}$ solves in the weak sense (3.15), and
$\mathbb{O}_{n}(y,0)=0$, then using the same Green’s function $G$ for the
integral representation of $\mathbb{O}_{n}$, we see that the last term of
(3.16) coincides with $\mathbb{O}_{n}(y,t)$, so we obtain

(3.17) $\begin{split}u_{n}(y,t)=&\int_{\mathcal{D}}v_{0}(z)G(y,z,t)dz\\\
&+\int_{0}^{t}\int_{\mathcal{D}}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)\nabla
G(y,z,t-s)\mathcal{T}_{M}(v_{n-1}^{M})(z,s)dzds\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\sigma(z)W(dz,ds),\;\;n:=1,2\cdots\end{split}$

We split now $v^{M}$ by

(3.18) $v^{M}(y,t)=u(y,t)+\mathbb{O}(y,t),$

and set $\eta^{M}:=\tilde{\eta}$, where $(\mathbb{O}(y,t),\tilde{\eta}(y,t))$
solves in the weak sense the Heat Equation Obstacle problem for any
$y\in\mathcal{D}$, $t\in[0,T]$

(3.19)
$\begin{split}&\partial_{t}\mathbb{O}(y,t)=\alpha\Delta\mathbb{O}(y,t)+\tilde{\eta}(dy,dt),\;\;u+\mathbb{O}\geq
0(\Leftrightarrow\mathbb{O}\geq-u),\\\
&\mathbb{O}(0,t)=\mathbb{O}(\lambda,t)=0,\;\;\mathbb{O}(y,0)=0,\\\
&\int_{0}^{T}\int_{\mathcal{D}}(u(y,s)+\mathbb{O}(y,s))\tilde{\eta}(dy,ds)=0.\end{split}$

We observe that $\mathbb{O}(y,0)=0$ yields that $u(y,0)=v^{M}(y,0)=v_{0}(y)$,
and as we argued for the derivation of (3.17), we obtain that $u$ satisfies

(3.20) $\begin{split}u(y,t)=&\int_{\mathcal{D}}v_{0}(z)G(y,z,t)dz\\\
&+\int_{0}^{t}\int_{\mathcal{D}}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)\nabla
G(y,z,t-s)\mathcal{T}_{M}(v^{M})(z,s)dzds\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\sigma(z)W(dz,ds).\end{split}$

Using (3.17) for $u_{n}$, $u_{n-1}$, by substraction, we get for
$n=2,3,\cdots$

(3.21)
$\begin{split}u_{n}&(y,t)-u_{n-1}(y,t)=\int_{0}^{t}\int_{\mathcal{D}}\Big{[}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)\mathcal{T}_{M}(v_{n-1}^{M})(z,s)\\\
&-\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)\mathcal{T}_{M}(v_{n-2}^{M})(z,s)\Big{]}\nabla
G(y,z,t-s)dzds\\\
=&\int_{0}^{t}\int_{\mathcal{D}}\Big{[}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)z\min\Big{\\{}\frac{v_{n-1}^{M}(z,s)}{z},M\Big{\\}}\\\
&-\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)z\min\Big{\\{}\frac{v_{n-2}^{M}(z,s)}{z},M\Big{\\}}\Big{]}\nabla
G(y,z,t-s)dzds.\end{split}$

In the above, we apply $\|\cdot\|_{\mathcal{B}}$-norm at both sides and then
take $p$-powers for some $p>0$, and then $\displaystyle{\sup_{t\in(0,T)}}$,
and then expectation, to obtain for $n=2,3,\cdots$

(3.22)
$\begin{split}&E\Big{(}\displaystyle{\sup_{t\in(0,T)}}\|u_{n}(\cdot,t)-u_{n-1}(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\\\
=&E\Big{(}\displaystyle{\sup_{t\in(0,T)}}\Big{\|}\int_{0}^{t}\int_{\mathcal{D}}\Big{[}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)z\min\Big{\\{}\frac{v_{n-1}^{M}(z,s)}{z},M\Big{\\}}\\\
&-\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)z\min\Big{\\{}\frac{v_{n-2}^{M}(z,s)}{z},M\Big{\\}}\Big{]}\nabla
G(y,z,t-s)dzds\Big{\|}^{p}_{\mathcal{B}}\Big{)}.\end{split}$

In [20], various useful bounds were proven in the norm
$\|\cdot\|_{\mathcal{B}}$ for the heat kernel $G$ defined explicitly by a
different series representation than the standard trigonometric series (bounds
holding obviously true for $\alpha t$ in place of the time variable $t$, and
$\mathcal{D}=(0,\lambda)$ in place of $(0,1)$ there). In particular, we use
the estimate of Proposition 4.4. therein, to derive directly for some constant
$c=c(T,p)>0$, that

$E(\displaystyle{\sup_{t\in(0,T)}}\|J\|^{p}_{\mathcal{B}})\leq
c(T,p)\int_{0}^{T}E(\displaystyle{\sup_{\tau\in(0,s)}}\|f(\cdot,\tau)\|^{p}_{\mathcal{B}})ds,$

for

$J(y,t):=\int_{0}^{t}\int_{\mathcal{D}}f(z,s)\nabla G(y,z,t-s)dzds,$

and

$\begin{split}f(z,s):=&\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)z\min\Big{\\{}\frac{v_{n-1}^{M}(z,s)}{z},M\Big{\\}}\\\
&-\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)z\min\Big{\\{}\frac{v_{n-2}^{M}(z,s)}{z},M\Big{\\}}.\end{split}$

Using the above in (3.22) yields for $n=2,3,\cdots$

(3.23)
$\begin{split}&E\Big{(}\displaystyle{\sup_{t\in(0,T)}}\|u_{n}(\cdot,t)-u_{n-1}(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\\\
&\leq
cC(T,p)\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)\min\Big{\\{}\frac{v_{n-1}^{M}(z,\tau)}{z},M\Big{\\}}\\\
&\hskip
128.0374pt-\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)\min\Big{\\{}\frac{v_{n-2}^{M}(z,\tau)}{z},M\Big{\\}}\Big{|}^{p}\Big{)}ds\\\
=&cC(T,p)\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)\Big{|}^{p}\\\
&\hskip
128.0374pt\Big{|}\min\Big{\\{}\frac{v_{n-1}^{M}(z,\tau)}{z},M\Big{\\}}-\min\Big{\\{}\frac{v_{n-2}^{M}(z,\tau)}{z},M\Big{\\}}\Big{|}^{p}\Big{)}ds.\end{split}$

Observing that $a\leq M$ and $b\leq M$ yields
$\min\\{a,M\\}-\min\\{b,M\\}=a-b$, while $a\geq M$ and $b\geq M$ yields
$\min\\{a,M\\}-\min\\{b,M\\}=M-M=0\leq|a-b|$, while $a\leq M$ and $b\geq M$
yields $\min\\{a,M\\}-\min\\{b,M\\}=a-M\leq 0\leq|a-b|$, we have

$|\min\\{a,M\\}-\min\\{b,M\\}|\leq|a-b|,$

and so, we obtain by (3.23) for $n=2,3,\cdots$

(3.24)
$\begin{split}&E\Big{(}\displaystyle{\sup_{t\in(0,T)}}\|u_{n}(\cdot,t)-u_{n-1}(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\\\
&\leq
cC(T,p)\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)\Big{|}^{p}\\\
&\hskip
128.0374pt\Big{|}\min\Big{\\{}\frac{v_{n-1}^{M}(z,\tau)}{z},M\Big{\\}}-\min\Big{\\{}\frac{v_{n-2}^{M}(z,\tau)}{z},M\Big{\\}}\Big{|}^{p}\Big{)}ds\\\
&\leq
cC(T,p)\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)|^{p}\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\frac{v_{n-1}^{M}(z,\tau)}{z}-\frac{v_{n-2}^{M}(z,\tau)}{z}\Big{|}^{p}\Big{)}ds\\\
&=cC(T,p)\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}|(\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)|^{p}\displaystyle{\sup_{\tau\in(0,s)}}\|v_{n-1}^{M}(\cdot,\tau)-v_{n-2}^{M}(\cdot,\tau)\|^{p}_{\mathcal{B}}\Big{)}ds.\end{split}$

But by (3.14), it holds that

(3.25)
$\begin{split}\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\frac{v_{n-1}^{M}(z,\tau)}{z}-\frac{v_{n-2}^{M}(z,\tau)}{z}\Big{|}^{p}\leq&c(p)\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\frac{u_{n-1}(z,\tau)}{z}-\frac{u_{n-2}(z,\tau)}{z}\Big{|}^{p}\\\
&+c(p)\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\frac{\mathbb{O}_{n-1}(z,\tau)}{z}-\frac{\mathbb{O}_{n-2}(z,\tau)}{z}\Big{|}^{p}\\\
\leq&c(p)\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\frac{u_{n-1}(z,\tau)}{z}-\frac{u_{n-2}(z,\tau)}{z}\Big{|}^{p},\end{split}$

where for the last inequality we used the stability bound in
$\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}$ of
the obstacle problem solutions by the obstacle, cf. [20] in the proof of
Theorem 3.2. So, for $s=T$, we obtain by (3.24), (3.25), and for
$n=2,3,\cdots$

(3.26)
$\begin{split}c(p)E\Big{(}&\displaystyle{\sup_{t\in(0,T)}}\|v_{n}^{M}(\cdot,t)-v_{n-1}^{M}(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\leq
E\Big{(}\displaystyle{\sup_{t\in(0,T)}}\|u_{n}(\cdot,t)-u_{n-1}(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\\\
&\leq
c(p)2^{p}C(T,p)\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)|^{p}\displaystyle{\sup_{\tau\in(0,s)}}\|v_{n-1}^{M}(\cdot,\tau)-v_{n-2}^{M}(\cdot,\tau)\|^{p}_{\mathcal{B}}\Big{)}ds\\\
&\leq
C(T,p)\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)|^{p}\displaystyle{\sup_{\tau\in(0,s)}}\|u_{n-1}(\cdot,\tau)-u_{n-2}(\cdot,\tau)\|^{p}_{\mathcal{B}}\Big{)}ds.\end{split}$

We apply the same argumentation as for deriving the above inequality, on
(3.17) now. By using that $v_{0}\in L^{p}(\Omega,C[0,T];\mathcal{B})$, and the
estimate of Proposition 4.3 from [20], as $p>8>2$, we obtain for
$n=1,2,\cdots$

(3.27)
$\begin{split}c(p)E\Big{(}&\displaystyle{\sup_{t\in(0,T)}}\|v_{n}^{M}(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\leq
E\Big{(}\displaystyle{\sup_{t\in(0,T)}}\|u_{n}(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\\\
\leq&C(T,p)\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)|^{p}\displaystyle{\sup_{\tau\in(0,s)}}\|v_{n-1}^{M}(\cdot,\tau)\|^{p}_{\mathcal{B}}\Big{)}ds\\\
&+CT\displaystyle{\sup_{\tau\in(0,T)}}\|\sigma(\cdot,\tau)\|^{p}_{\mathcal{B}}+C\\\
\leq&C(T,p)\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)|^{p}\displaystyle{\sup_{\tau\in(0,s)}}\|u_{n-1}(\cdot,\tau)\|^{p}_{\mathcal{B}}\Big{)}ds\\\
&+CT\displaystyle{\sup_{\tau\in(0,T)}}\|\sigma(\cdot,\tau)\|^{p}_{\mathcal{B}}+C.\end{split}$

Here, we used (3.14) and the bound of the solution $\mathbb{O}_{n}$ by the
obstacle (comparing with the zero solution) for the first and the third
inequality. Moreover, since $p>8$ we applied the estimate of Proposition 4.5
in [20] to bound the noise term. In the above, note that since $\sigma$
satisfies (3.5), i.e., $\sigma\in C(\overline{\mathcal{D}})$,
$\sigma(0)=\sigma(\ell)=0$, and that it exists $\sigma^{\prime}(0)$, then
$\|\sigma\|_{\mathcal{B}}$ is well defined. This assumption models a zero
volatility at the boundary of $\mathcal{D}$ in accordance to the Dirichlet
b.c. for the density of the transactions $v$, i.e., the solution of (3.1),
that vanishes on $\partial\mathcal{D}$.

Thus, by (3.27), since $\sigma$ satisfies (3.5), $v_{n}^{M},u_{n}$ stay in
$L^{p}(\Omega,C[0,T];\mathcal{B})$ if $v_{0}^{M}(y,t):=v_{0}(y)\in
L^{p}(\Omega,C[0,T];\mathcal{B})$ and $T=T_{M}$ such that (4.3) holds true,
i.e.,

$\displaystyle{\sup_{r\in(0,T)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},r)|^{p}\leq
C_{2}(M,p)<\infty\;\;a.s.$

Furthermore, by (3.26), we get that $v_{n}^{M}$, $u_{n}$ are Cauchy in
$L^{p}(\Omega,C[0,T];\mathcal{B})$ while they also stay in
$L^{p}(\Omega,C[0,T];\mathcal{B})$, for $T=T_{M}$. So, by the completeness of
the Banach space $\mathcal{B}$ in this norm, both they converge in
$L^{p}(\Omega,C[0,T];\mathcal{B})$ to some unique $\hat{v}^{M},\hat{u}\in
L^{p}(\Omega,C[0,T];\mathcal{B})$ as $n\rightarrow\infty$. In details, by
using (3.26), we obtain

(3.28)
$\begin{split}E\Big{(}&\displaystyle{\sup_{t\in(0,T)}}\|v_{n}^{M}(\cdot,t)-v_{n-1}^{M}(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\leq
CE\Big{(}\displaystyle{\sup_{t\in(0,T)}}\|u_{n}(\cdot,t)-u_{n-1}(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\\\
&\leq\tilde{c}\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)|^{p}\displaystyle{\sup_{\tau\in(0,s)}}\|v_{n-1}^{M}(\cdot,\tau)-v_{n-2}^{M}(\cdot,\tau)\|^{p}_{\mathcal{B}}\Big{)}ds\\\
&\leq\tilde{c}\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}\|v_{n-1}^{M}(\cdot,\tau)-v_{n-2}^{M}(\cdot,\tau)\|^{p}_{\mathcal{B}}\Big{)}ds\\\
&\leq
c^{n-1}\int_{0}^{T}\int_{0}^{s_{n-1}}\int_{0}^{s_{n-2}}\cdots\int_{0}^{s_{2}}1d_{s_{1}}\cdots
d_{s_{n-2}}d_{s_{n-1}}E\Big{(}\displaystyle{\sup_{\tau\in(0,T)}}\|v_{1}^{M}(\cdot,\tau)-v_{0}^{M}(\cdot,\tau)\|^{p}_{\mathcal{B}}\Big{)}\\\
&\leq C\frac{c^{n-1}}{(n-1)!}\rightarrow
0\;\;\;as\;\;n\rightarrow\infty,\end{split}$

where we used that $v_{1}^{M},\;v_{0}^{M}\in L^{p}(\Omega,C[0,T];\mathcal{B})$
since $v_{n}^{M}$ stays in $L^{p}(\Omega,C[0,T];\mathcal{B})$ for all $n$ if
$v_{0}^{M}(y,t):=v_{0}(y)\in L^{p}(\Omega,C[0,T];\mathcal{B})$. Therefore,
$v_{n}^{M}$, $u_{n}$ are Cauchy in $L^{p}(\Omega,C[0,T];\mathcal{B})$.

Moreover, we also obtain, as in (3.27)

(3.29)
$\begin{split}E\Big{(}&\displaystyle{\sup_{t\in(0,T)}}\|v^{M}(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\leq
CE\Big{(}\displaystyle{\sup_{t\in(0,T)}}\|u(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\\\
\leq&C\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)|^{p}\displaystyle{\sup_{\tau\in(0,s)}}\|v^{M}(\cdot,\tau)\|^{p}_{\mathcal{B}}\Big{)}ds\\\
&+CT\displaystyle{\sup_{\tau\in(0,T)}}\|\sigma(\cdot,\tau)\|^{p}_{\mathcal{B}}+C\\\
\leq&C\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)|^{p}\displaystyle{\sup_{\tau\in(0,s)}}\|u(\cdot,\tau)\|^{p}_{\mathcal{B}}\Big{)}ds\\\
&+CT\displaystyle{\sup_{\tau\in(0,T)}}\|\sigma(\cdot,\tau)\|^{p}_{\mathcal{B}}+C.\end{split}$

Therefore, by Gronwall’s inequality on (3.29), and using (4.3), (3.5), we
arrive at

(3.30)
$\begin{split}E\Big{(}&\displaystyle{\sup_{t\in(0,T)}}\|v^{M}(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}+E\Big{(}\displaystyle{\sup_{t\in(0,T)}}\|u(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\leq
C.\end{split}$

Thus, we get that $v^{M},u\in L^{p}(\Omega,C[0,T];\mathcal{B})$. By
substraction of the integral representations (3.17), (3.20), we obtain (as
previously when deriving the 4th inequality of (3.26)), that $u_{n},u$ satisfy
for $n=1,2,\cdots$

(3.31)
$\begin{split}E\Big{(}&\displaystyle{\sup_{t\in(0,T)}}\|u_{n}(\cdot,t)-u(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\\\
&\leq
C\int_{0}^{T}E\Big{(}\displaystyle{\sup_{\tau\in(0,s)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},\tau)|^{p}\displaystyle{\sup_{\tau\in(0,s)}}\|u_{n-1}(\cdot,\tau)-u(\cdot,\tau)\|^{p}_{\mathcal{B}}\Big{)}ds\\\
&\leq\frac{C^{n-1}}{(n-1)!}E\Big{(}\displaystyle{\sup_{t\in(0,T)}}\|u_{1}(\cdot,t)-u(\cdot,t)\|^{p}_{\mathcal{B}}\Big{)}\rightarrow
0\;\;as\;\;n\rightarrow\infty.\end{split}$

Here, we used once again (3.5), (4.3), the argument for the last bound of
(3.28), together with the fact that, as proven, $u_{1},u\in
L^{p}(\Omega,C[0,T];\mathcal{B})$. Therefore, for $T=T_{M}$, $u_{n}\rightarrow
u$ as $n\rightarrow\infty$ in $L^{p}(\Omega,C[0,T];\mathcal{B})$. By
uniqueness of the limits, we get that $\hat{u}=u$ a.s.

Since $u$ exists uniquely, then the solution $(\mathbb{O},\tilde{\eta})$ of
(3.19) exists uniquely. But it holds that

$\begin{split}\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\frac{v_{n}^{M}(z,\tau)}{z}-\frac{v^{M}(z,\tau)}{z}\Big{|}^{p}\leq&c\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\frac{u_{n}(z,\tau)}{z}-\frac{u(z,\tau)}{z}\Big{|}^{p}\\\
&+c\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\frac{\mathbb{O}_{n}(z,\tau)}{z}-\frac{\mathbb{O}(z,\tau)}{z}\Big{|}^{p}\\\
\leq&c\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}\Big{|}\frac{u_{n}(z,\tau)}{z}-\frac{u(z,\tau)}{z}\Big{|}^{p},\end{split}$

where again we used the stability bound in
$\displaystyle{\sup_{\tau\in(0,s)}}\displaystyle{\sup_{z\in\mathcal{D}}}$ of
the obstacle problem solutions by the obstacle, see in [20] in the proof of
Theorem 3.2. So, for $T=T_{M}$, by taking expectation, since, as we have
shown, $u_{n}\rightarrow u$ in $L^{p}(\Omega,C[0,T];\mathcal{B})$, we obtain
that $v_{n}^{M}\rightarrow v^{M}$ as $n\rightarrow\infty$ in
$L^{p}(\Omega,C[0,T];\mathcal{B})$. By uniqueness of the limits, we get that
$\hat{v}^{M}=v^{M}$ a.s.

So, for all $T=T_{M}>0$ such that (4.3) holds true, we derive the following:

  1. (1)

$v^{M},u$ exist and belong in $L^{p}(\Omega,C[0,T];\mathcal{B})$.

  2. (2)

Since $u$ exists, we may define $\eta^{M}:=\tilde{\eta}$ a.s. for
$\tilde{\eta}$ the second term of the solution $(\mathbb{O},\tilde{\eta})$ of
(3.19).

  3. (3)

The pair $(v^{M},\eta^{M})$ exists, $v^{M}\in
L^{p}(\Omega,C[0,T];\mathcal{B})$, and is the week solution of the truncated
problem (3.6) in $(0,T)$, where $v^{M}$, $\eta^{M}$ satisfy the week
formulation (3.8) for $v^{M}(y,0):=v_{0}(y)$, where $\eta^{M}$ satisfies (3.9)
and (3.10).

  4. (4)

The pair $(v^{M},\eta^{M})$ is unique. Indeed, uniqueness of the limit of
$v_{n}^{M}$ showed that $v^{M}$ is unique. Uniqueness of $\eta^{M}$ follows by
the uniqueness of $\tilde{\eta}$ of the obstacle problem (3.19) since as we
have shown $u$ exists uniquely as the limit of $u_{n}$ in
$L^{p}(\Omega,C[0,T];\mathcal{B})$.

Therefore, there exists a unique solution $(v^{M},\eta^{N})$ of the week
formulation (3.8) with $v^{M}\in L^{p}(\Omega,C[0,T];\mathcal{B})$ and with
$\eta^{M}$ satisfying (3.9) and (3.10), which completes the proof. ∎

We return to the $M$-independent problem (3.1), and we shall prove that it
admits a unique maximal solution by concatenation of the solution of the
$M$-truncated problem (3.6)-(3.9)-(3.10). This is established by the next Main
Theorem.

######  Theorem 3.2.

Let the noise diffusion $\sigma$ satisfy (3.5), and $v_{0}(y)\in
L^{p}(\Omega,C[0,T];\mathcal{B})$ for $p\geq p_{0}>8$. Then, there exists a
unique weak maximal solution $(v,\eta)$ to the problem (3.1)-(2.15)-(2.16) in
the maximal interval $[0,\displaystyle{\sup_{M>0}}\tau_{M})$, where

(3.32) $\begin{split}\tau_{M}:=&\inf\Big{\\{}T\geq
0:\;\displaystyle{\sup_{r\in(0,T)}}|\nabla v(0^{+},r)|\geq M\Big{\\}}\\\
=&\inf\Big{\\{}T\geq 0:\;\displaystyle{\sup_{r\in(0,T)}}\nabla v(0^{+},r)\geq
M\Big{\\}}.\end{split}$

###### Proof.

We note that for the reflected problem since $v\geq 0$ a.s. and $v(0,t)=0$,
then $\nabla v(0^{+},t)\geq 0$ a.s. for any $t$. As we mentioned, we continue
to keep the absolute value on $\nabla v(0^{+},t)$ in this proof also in order
to present a more general result applicable to the 2d i.b.v. problem of (2.6),
and to the problem without reflection of next section.

Let $v^{M}$ as in Theorem 3.1. We observe first that by the operator
definition (3.4), and since $v^{M}(0,t)=0$, we have

(3.33)
$\begin{split}\displaystyle{\sup_{r\in(0,T)}}&|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},r)|^{p}\leq\min\Big{\\{}\displaystyle{\sup_{r\in(0,T)}}|\nabla
v^{M}(0^{+},r)|^{p},M^{p}\Big{\\}}\leq M^{p},\end{split}$

and so by (4.3), Theorem 3.1 holds also for any $T=T(M)$ such that

(3.34) $\min\Big{\\{}\displaystyle{\sup_{r\in(0,T)}}|\nabla
v^{M}(0^{+},r)|,M\Big{\\}}\leq M<C_{2}(M,p)^{1/p}<\infty\;\;a.s.$

We fix $M>0$, and consider arbitrary $\tilde{M}>0$ such that $\tilde{M}\leq
M$. Thus, the weak solution $(v^{M},\eta^{M})$ of (3.6)-(3.9)-(3.10) solves
weakly the $M$-independent problem (3.1)-(2.15)-(2.16) (since they share the
same initial condition), until the (random) stopping time $\tau$ defined as
follows

(3.35) $\begin{split}\tau:=&\inf\Big{\\{}T\geq
0:\;\min\Big{\\{}\displaystyle{\sup_{r\in(0,T)}}|\nabla
v^{M}(0^{+},r)|,M\Big{\\}}\geq\tilde{M}\Big{\\}}\\\ =&\inf\Big{\\{}T\geq
0:\;\displaystyle{\sup_{r\in(0,T)}}|\nabla
v^{M}(0^{+},r)|\geq\tilde{M}\Big{\\}}.\end{split}$

Let now an arbitrary deterministic $t>0$, then we have as in (3.26)

(3.36)
$\begin{split}&cE\Big{(}\displaystyle{\sup_{s\in(0,\min\\{t,\tau\\})}}\|v^{M}(\cdot,s)-v^{\tilde{M}}(\cdot,s)\|^{p}_{\mathcal{B}}\Big{)}\leq
E\Big{(}\displaystyle{\sup_{s\in(0,\min\\{t,\tau\\})}}\|u(\cdot,s)-\tilde{u}(\cdot,s)\|^{p}_{\mathcal{B}}\Big{)}\\\
&\leq
cC(t,p)\int_{0}^{\min\\{t,\tau\\}}E\Big{(}\displaystyle{\sup_{r\in(0,s)}}|\nabla(\mathcal{T}_{\tilde{M}}(v^{\tilde{M}}))(0^{+},r)|^{p}\displaystyle{\sup_{r\in(0,s)}}\|v^{M}(\cdot,r)-v^{\tilde{M}}(\cdot,r)\|^{p}_{\mathcal{B}}\Big{)}ds\\\
&\leq
cC(t,p)\int_{0}^{\min\\{t,\tau\\}}E\Big{(}\displaystyle{\sup_{r\in(0,s)}}|\nabla(\mathcal{T}_{\tilde{M}}(v^{\tilde{M}}))(0^{+},r)|^{p}\displaystyle{\sup_{r\in(0,s)}}\|u(\cdot,r)-\tilde{u}(\cdot,r)\|^{p}_{\mathcal{B}}\Big{)}ds\\\
&\leq
C(t,p)\tilde{M}^{p}\int_{0}^{\min\\{t,\tau\\}}E\Big{(}\displaystyle{\sup_{r\in(0,s)}}\|u(\cdot,r)-\tilde{u}(\cdot,r)\|^{p}_{\mathcal{B}}\Big{)}ds,\end{split}$

for $\tilde{u}$ given by

(3.37) $\begin{split}\tilde{u}(y,t)=&\int_{\mathcal{D}}v_{0}(z)G(y,z,t)dz\\\
&+\int_{0}^{t}\int_{\mathcal{D}}\nabla(\mathcal{T}_{\tilde{M}}(v^{\tilde{M}}))(0^{+},s)\nabla
G(y,z,t-s)\mathcal{T}_{\tilde{M}}(v^{\tilde{M}})(z,s)dzds\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\sigma(z)W(dz,ds),\end{split}$

while

$\begin{split}&cC(t,p)\int_{0}^{\min\\{t,\tau\\}}E\Big{(}\displaystyle{\sup_{r\in(0,s)}}|\nabla(\mathcal{T}_{\tilde{M}}(v^{\tilde{M}}))(0^{+},r)|^{p}\displaystyle{\sup_{r\in(0,s)}}\|v^{M}(\cdot,r)-v^{\tilde{M}}(\cdot,r)\|^{p}_{\mathcal{B}}\Big{)}ds\\\
&\leq
C(t,p)\tilde{M}^{p}\int_{0}^{\min\\{t,\tau\\}}E\Big{(}\displaystyle{\sup_{r\in(0,s)}}\|v^{M}(\cdot,r)-v^{\tilde{M}}(\cdot,r)\|^{p}_{\mathcal{B}}\Big{)}ds.\end{split}$

Therefore, by Gronwall’s inequality, we get

$u(\cdot,s)=\tilde{u}(\cdot,s),\;\;\;\;v^{M}(\cdot,s)=v^{\tilde{M}}(\cdot,s)\;\;\;\forall\;s\leq\min\\{t,\tau\\},$

and thus, by the uniqueness of the obstacle problem solution
$\eta^{\tilde{M}}$ when $\tilde{u}$ exists, we arrive at

$v^{M}(\cdot,s)=v^{\tilde{M}}(\cdot,s),\;\;\;\eta^{M}(\cdot,s)=\eta^{\tilde{M}}(\cdot,s)\;\;\;\forall\;s\leq\min\\{t,\tau\\}.$

Since $t$ is a deterministic arbitrary constant, the above yields that

$v^{M}(\cdot,s)=v^{\tilde{M}}(\cdot,s),\;\;\;\eta^{M}(\cdot,s)=\eta^{\tilde{M}}(\cdot,s)\;\;\forall\;s\leq\tau\;\;a.s.$

So, the weak solutions of the $M$-truncated problem (3.6)-(3.9)-(3.10) are
consistent, and we can proceed to concatenation.

Let us define the stochastic process $(v,\eta)$ such that for all $M>0$ it
coincides with the weak solution $(v^{M},\eta^{M})$ of the $M$-truncated
problem (3.6)-(3.9)-(3.10) until the stopping time

$\begin{split}\tau_{M}=&\inf\Big{\\{}T\geq
0:\;\displaystyle{\sup_{r\in(0,T)}}|\nabla v^{M}(0^{+},r)|\geq M\Big{\\}}\\\
=&\inf\Big{\\{}T\geq 0:\;\displaystyle{\sup_{r\in(0,T)}}|\nabla
v(0^{+},r)|\geq M\Big{\\}}.\end{split}$

By its definition, $(v(\cdot,s),\eta(\cdot,s))$ is a weak solution of the
$M$-independent problem (3.1)-(2.15)-(2.16), for any
$s\in[0,\displaystyle{\sup_{M>0}}\tau_{M})$, and $\tau_{M}$ is a localising
sequence. Then, $(v(\cdot,s),\eta(\cdot,s))$ is a maximal weak solution of
(3.1)-(2.15)-(2.16), since

$\displaystyle{\lim_{t\rightarrow\Big{(}\displaystyle{\sup_{M>0}}\tau_{M}\Big{)}^{-}}}\displaystyle{\sup_{r\in(0,t)}}|\nabla
v^{M}(0^{+},r)|=\infty\;\;a.s.$

Uniqueness of the maximal weak solution $(v(\cdot,s),\eta(\cdot,s))$ for
$s\in[0,\displaystyle{\sup_{M>0}}\tau_{M})$, follows from the consistency of
the solution of the $M$-truncated problem with which by its definition
coincides. ∎

Let us now consider Case 1. The above analysis is valid for both i.b.v.
problems of (2.6), and due to Theorem 3.2, and under its assumptions there
exist unique weak maximal solutions $(v_{1},\eta_{1})$, $(v_{2},\eta_{2})$
satisfying (2.15)-(2.16) in the maximal interval
$[0,\displaystyle{\sup_{M>0}}\tau_{1M})$, where

(3.38) $\begin{split}\tau_{1M}:=&\inf\Big{\\{}T\geq
0:\;\displaystyle{\sup_{r\in(0,T)}}(|\nabla v_{1}(0^{+},r)|+|\nabla
v_{2}(0^{+},r)|)\geq M\Big{\\}}\\\ =&\inf\Big{\\{}T\geq
0:\;\displaystyle{\sup_{r\in(0,T)}}(\nabla v_{1}(0^{+},r)+\nabla
v_{2}(0^{+},r))\geq M\Big{\\}}.\end{split}$

Recall that $\Omega=(a,b)$, $\lambda=b-a$, and $a\leq s^{-}(0)\leq
s^{+}(0)\leq b$. We need $a\leq s^{-}(t)\leq s^{+}(t)\leq b$ in order to
return to the initial variables. This will restrict the stopping time. By
using the Stefan condition (2.4) we obtain

$\partial_{t}s^{-}(t)=-\nabla v_{2}(0^{+},t)\leq
0,\;\;\partial_{t}s^{+}(t)=-\nabla v_{1}(0^{+},t)\leq 0,$

and so

$s^{-}(t)\leq s^{-}(0)\leq b,\;\;s^{+}(t)\leq s^{+}(0)\leq b,$

so we need $a\leq s^{-}(t)\leq s^{+}(t)$ which yields

$a\leq s^{-}(0)-\int_{0}^{t}\nabla v_{2}(0^{+},s)ds\leq
s^{+}(0)-\int_{0}^{t}\nabla v_{1}(0^{+},s)ds.$

We define the stopping time

(3.39) $\tau_{1s}:=\inf\Big{\\{}T>0:\;\sup_{r\in(0,T)}|\nabla
v_{1}(0^{+},r)-\nabla
v_{2}(0^{+},r)|\geq(T)^{-1}(s^{+}(0)-s^{-}(0))\Big{\\}},$

to keep the spread non-negative and

(3.40) $\tau_{1}^{*}:=\inf\Big{\\{}T>0:\;\sup_{r\in(0,T)}\nabla
v_{2}(0^{+},r)\geq(T)^{-1}(s^{-}(0)-a)\Big{\\}},$

to keep the spread area in $\mathcal{D}$.

So, the next theorem holds.

######  Theorem 3.3.

Under the assumptions of Theorem 3.2, and if the initial spread satisfies
$\lambda>s^{+}(0)-s^{-}(0)\geq 0$, then there exist unique weak maximal
solutions $(w_{1},\eta_{1})$, $(w_{2},\eta_{2})$ to the reflected Stefan
problem (2.3)-(2.15)-(2.16), and $w|_{x\geq s^{+}}=w_{1}$, $w|_{x\leq
s^{-}}=-w_{2}$, in the maximal interval $\mathcal{I}_{1}:=[0,\hat{\tau})$ for
$\hat{\tau}:=\min\\{\displaystyle{\sup_{M>0}}\tau_{1M},\tau_{1s},\tau_{1}^{*}\\}$,
with $\tau_{1M},\tau_{1s},\tau_{1}^{*}$ given by (3.38), (3.39), (3.40) for
which the spread $s^{+}(t)-s^{-}(t)$ defined by the Stefan condition (2.4)
exists and stays a.s. non-negative for any $t\in\mathcal{I}_{1}$.

We consider now Case 2. Due to Theorem 3.2, and under its assumptions there
exist unique weak maximal solutions $(v_{1},\eta_{1})$, $(v_{2},\eta_{2})$
satisfying (2.15)-(2.16) in the maximal interval
$[0,\displaystyle{\sup_{M>0}}\tau_{1M})$ for $\tau_{1M}$ given by (3.38). We
need $a\leq s^{-}(t)\leq s^{+}(t)\leq b$ in order to return to the initial
variables. By using the Stefan condition (2.9) we obtain

$\partial_{t}s^{-}(t)=\nabla v_{2}(0^{+},t)\geq
0,\;\;\partial_{t}s^{+}(t)=-\nabla v_{1}(0^{+},t)\leq 0,$

and so

$a\leq s^{-}(0)\leq s^{-}(t),\;\;s^{+}(t)\leq s^{+}(0)\leq b,$

so we need $s^{-}(t)\leq s^{+}(t)$ which yields

$s^{-}(0)+\int_{0}^{t}\nabla v_{2}(0^{+},s)ds\leq s^{+}(0)-\int_{0}^{t}\nabla
v_{1}(0^{+},s)ds.$

We define the stopping time

(3.41) $\tau_{2s}:=\inf\Big{\\{}T>0:\;\sup_{r\in(0,T)}(\nabla
v_{1}(0^{+},r)+\nabla
v_{2}(0^{+},r))\geq(T)^{-1}(s^{+}(0)-s^{-}(0))\Big{\\}},$

to keep the spread non-negative, while the spread area stays in $\mathcal{D}$
as the spread is decreasing.

So, the next theorem holds.

######  Theorem 3.4.

Under the assumptions of Theorem 3.2, and if the initial spread satisfies
$\lambda>s^{+}(0)-s^{-}(0)\geq 0$, then there exist unique weak maximal
solutions $(w_{1},\eta_{1})$, $(w_{2},\eta_{2})$ to the reflected Stefan
problem (2.8)-(2.15)-(2.16), and $w|_{x\geq s^{+}}=w_{1}$, $w|_{x\leq
s^{-}}=w_{2}$, in the maximal interval $\mathcal{I}_{2}:=[0,\hat{\tau})$ for
$\hat{\tau}:=\min\\{\displaystyle{\sup_{M>0}}\tau_{1M},\tau_{2s}\\}$, with
$\tau_{1M},\tau_{2s}$ given by (3.38), (3.41), for which the spread
$s^{+}(t)-s^{-}(t)$ defined by the Stefan condition (2.9) exists and stays
a.s. non-negative for any $t\in\mathcal{I}_{2}$.

##  4\. The problem without reflection

###  4.1. Existence of maximal solutions

We shall consider the unreflected initial and boundary value problem for

(4.1) $v_{t}(y,t)=\alpha\Delta v(y,t)-\nabla v(0^{+},t)\nabla
v(y,t)+\sigma(y)\dot{W}(y,t),$

posed for any $y$ in $\mathcal{D}=(0,\lambda)$ for $t\in[0,T]$ with Dirichlet
b.c., with $v(y,0)$ given.

In the proofs of the previous section we replace the reflection measure by $0$
and keep as presented the absolute value on the changing in general sign
$\nabla v(0^{+},t)$ (as $v$ may take negative values), and we derive the next
results.

######  Theorem 4.1.

Let the noise diffusion $\sigma$ satisfy the condition (3.5), $M>0$ fixed,
$p\geq p_{0}>8$, and let $v_{0}(y)\in L^{p}(\Omega,C[0,T];\mathcal{B})$ be the
initial condition of (4.1). Then there exists a unique week solution $v^{M}\in
L^{p}(\Omega,C[0,T];\mathcal{B})$ to the truncated problem

(4.2) $\begin{split}&v^{M}_{t}(y,t)=\alpha\Delta
v^{M}(y,t)-\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},t)\nabla(\mathcal{T}_{M}(v^{M}))(y,t)\\\
&\hskip 71.13188pt+\sigma(y)\dot{W}(y,t),\;\;t\in(0,T],\;\;y\in\mathcal{D},\\\
&v^{M}(y,0):=v_{0}(y),\;\;y\in\mathcal{D},\\\
&v^{M}(0,t)=v^{M}(\lambda,t)=0,\;\;t\in(0,T],\\\ \end{split}$

where $T:=T_{M}>0$ such that

(4.3)
$\displaystyle{\sup_{r\in(0,T)}}|\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},r)|^{p}<\infty\;\;a.s.,$

where for any $t\in(0,T)$, $v^{M}$ satisfies the week formulation

(4.4) $\begin{split}v^{M}(y,t)=&\int_{\mathcal{D}}v_{0}(z)G(y,z,t)dz\\\
&+\int_{0}^{t}\int_{\mathcal{D}}\nabla(\mathcal{T}_{M}(v^{M}))(0^{+},s)\nabla
G(y,z,t-s)\mathcal{T}_{M}(v^{M})(z,s)dzds\\\
&+\int_{0}^{t}\int_{\mathcal{D}}G(y,z,t-s)\sigma(z)W(dz,ds),\end{split}$

for $v^{M}(y,0):=v_{0}(y)$.

######  Theorem 4.2.

Let the noise diffusion $\sigma$ satisfy (3.5), and $v_{0}(y)\in
L^{p}(\Omega,C[0,T];\mathcal{B})$ for $p\geq p_{0}>8$. Then, there exists a
unique weak maximal solution $v$ to the problem (4.1) in the maximal interval
$[0,\displaystyle{\sup_{M>0}}\tilde{\tau}_{M})$, where

(4.5) $\tilde{\tau}_{M}:=\inf\Big{\\{}T\geq
0:\;\displaystyle{\sup_{r\in(0,T)}}|\nabla v(0^{+},r)|\geq M\Big{\\}}.$

We consider now Case 3. Due to Theorem 4.2, and under its assumptions there
exist unique weak maximal solutions $(v_{1},\eta_{1})$, $(v_{2},\eta_{2})$
satisfying (2.15)-(2.16) in the maximal interval
$[0,\displaystyle{\sup_{M>0}}\tau_{3M})$ for $\tau_{3M}$ given by

(4.6) $\begin{split}\tau_{3M}:=&\inf\Big{\\{}T\geq
0:\;\displaystyle{\sup_{r\in(0,T)}}(|\nabla v_{1}(0^{+},r)|+|\nabla
v_{2}(0^{+},r)|)\geq M\Big{\\}}.\end{split}$

We need $a\leq s^{-}(t)\leq s^{+}(t)\leq b$ in order to return to the initial
variables. By using the Stefan condition (2.9) we obtain

$\partial_{t}s^{-}(t)=\nabla v_{2}(0^{+},t),\;\;\partial_{t}s^{+}(t)=-\nabla
v_{1}(0^{+},t),$

and we need

$a\leq s^{-}(0)+\int_{0}^{t}\nabla v_{2}(0^{+},s)ds\leq
s^{+}(0)-\int_{0}^{t}\nabla v_{1}(0^{+},s)ds\leq b.$

We define the stopping time

(4.7) $\tau_{3s}:=\inf\Big{\\{}T>0:\;\sup_{r\in(0,T)}|\nabla
v_{1}(0^{+},r)+\nabla
v_{2}(0^{+},r))|\geq(T)^{-1}(s^{+}(0)-s^{-}(0))\Big{\\}},$

to keep the spread non-negative, and

(4.8) $\tau_{3}^{*}:=\inf\Big{\\{}T>0:\;\sup_{r\in(0,T)}|\nabla
v_{2}(0^{+},r)|\geq(T)^{-1}(s^{-}(0)-a)\Big{\\}},$ (4.9)
$\tau_{3}^{**}:=\inf\Big{\\{}T>0:\;\sup_{r\in(0,T)}|\nabla
v_{1}(0^{+},r)|\geq(T)^{-1}(b-s^{+}(0))\Big{\\}}\Big{\\}},$

to keep the spread area in $\mathcal{D}$.

So, the next theorem holds.

######  Theorem 4.3.

Under the assumptions of Theorem 4.2, and if the initial spread satisfies
$\lambda>s^{+}(0)-s^{-}(0)\geq 0$, then there exist unique weak maximal
solutions $w_{1}$, $w_{2}$ to the unreflected Stefan problem (2.13), and
$w|_{x\geq s^{+}}=w_{1}$, $w|_{x\leq s^{-}}=w_{2}$, in the maximal interval
$\mathcal{I}_{3}:=[0,\hat{\tau})$ for
$\hat{\tau}:=\min\\{\displaystyle{\sup_{M>0}}\tau_{3M},\tau_{3s},\tau_{3}^{*},\tau_{3}^{**}\\}$,
with $\tau_{3M},\tau_{3s},\tau_{3}^{*},\tau_{3}^{**}$ given by (4.6), (4.7),
(4.8), (4.9) for which the spread $s^{+}(t)-s^{-}(t)$ defined by the Stefan
condition (2.9) exists and stays a.s. non-negative for any
$t\in\mathcal{I}_{3}$.

###  4.2. Formal asymptotics

Let us consider the Stefan problem (1.1) without reflection, and the
multiplicative noise defined as $\sigma(y)\dot{W}(y,t)\equiv\dot{W}(t)$. We
conjecture that the spread $s^{+}-s^{-}$ satisfies $\forall\;t>0$

(4.10)
$\begin{split}&\partial_{t}(s^{+}-s^{-})(t)=-\frac{2}{\lambda}w_{\infty},\\\
&s^{+}(0)-s^{-}(0)=\mbox{given},\end{split}$

for $w_{\infty}(t)$ the solution of the initial value stochastic equation
problem

(4.11)
$\partial_{t}w_{\infty}(t)=\frac{2\alpha}{\lambda^{2}}w_{\infty}(t)+\dot{W}(t),\;\;t>0,\;\;\;\;w_{\infty}(0)=\mbox{given}.$

We shall present some formal arguments for the derivation of the above system.
Let $w_{\infty}(t)$ be the mean field profile of the diffusion when the domain
is infinite. We define $\Omega=\Omega_{\rm Liq}(t)\cup[s^{-}(t),s^{+}(t)]$ of
diameter $\lambda>>0$. Moreover, we consider $\alpha>>0$ so that $\Delta w=0$
on $\Omega_{\rm Liq}$ approximates the spde of the Stefan problem (1.1). Then
$w$ such that

$w(x,t)=\frac{w_{\infty}(t)}{\lambda}(x-s^{+}(t))\;\;\;x\geq
s^{+}(t),\;\;\;w(x,t)=\frac{w_{\infty}(t)}{\lambda}(-x+s^{-}(t))\;\;\;x\leq
s^{-}(t)$

satisfies exactly

(4.12) $\begin{split}&\Delta w(x,t)=0,\;\;t>0,\\\ &w(s^{\pm}(t),t)=0,\\\
&w(x,t)=w_{\infty}(t)\;\;\mbox{at}\;\;x=\lambda+s^{+}(t)\;\;\mbox{where}\;\;x-s^{+}(t)=\lambda,\\\
&w(x,t)=w_{\infty}(t)\;\;\mbox{at}\;\;x=s^{-}(t)-\lambda\;\;\mbox{where}\;\;-x+s^{-}(t)=\lambda,\\\
&\partial_{t}(s^{+}-s^{-})(t)=-(\nabla w)^{+}(s^{+}(t),t)+(\nabla
w)^{-}(s^{-}(t),t)=-\frac{2}{\lambda}w_{\infty}(t)\;\;t>0.\end{split}$

Here, the value of $w=w_{\infty}$ at $x=s^{\mp}\mp\lambda$, for $\lambda>>0$,
approximates the condition $w\rightarrow w_{\infty}$ at infinite distance from
the solid phase of [29, 30, 7], i.e., convergence to the mean-field solution.

From the above approximate problem (4.12) we keep the spread evolution through
$w_{\infty}$, i.e.,
$\partial_{t}(s^{+}-s^{-})(t)=-\frac{2}{\lambda}w_{\infty}(t)$, and proceed by
matching it to the asymptotics of the solution $w$ of the stochastic parabolic
equation (1.1) on $\Omega_{\rm Liq}$. This will yield the sde for
$w_{\infty}(t)$ that is missing. We observe that
$\lambda=|\Omega|\simeq|\Omega_{\rm Liq}|$ and so

$w_{\infty}(t)\simeq\frac{1}{|\Omega|}\int_{\Omega_{\rm Liq}}w(x,t)dx.$

By differentiating in time we have

$\begin{split}|\Omega|\partial_{t}w_{\infty}\simeq&\int_{\Omega_{\rm
Liq}}w_{t}(x,t)dx-\int_{\partial\Omega_{\rm
Liq}}\partial_{t}(s^{+}(t)-s^{-}(t))wdx=\alpha\int_{\Omega_{\rm Liq}}\Delta
w(x,t)dx\\\ &+\int_{\Omega_{\rm
Liq}}\sigma\dot{W}_{s}(x,t)-\int_{\partial\Omega_{\rm
Liq}}\partial_{t}(s^{+}(t)-s^{-}(t))wdx\simeq-\alpha((\nabla
w)^{+}(s^{+}(t),t)\\\ &-(\nabla w)^{-}(s^{-}(t),t))+\int_{\Omega_{\rm
Liq}}\sigma\dot{W}_{s}(x,t)-\int_{\partial\Omega_{\rm
Liq}}\partial_{t}(s^{+}(t)-s^{-}(t))wdx\\\
=&-\alpha\partial_{t}(s^{+}(t)-s^{-}(t))+\int_{\Omega_{\rm
Liq}}\sigma\dot{W}_{s}(x,t)-\int_{\partial\Omega_{\rm
Liq}}\partial_{t}(s^{+}(t)-s^{-}(t))wdx,\end{split}$

where we assumed that as $x\rightarrow\pm\infty$ $w\rightarrow w_{\infty}(t)$
and so $\nabla w\rightarrow 0$ (this can be modeled by a Neumann condition for
$w$ at $\partial\Omega$ when the diameter $\lambda$ of $\Omega$ is very
large). Since $\alpha>>0$ the last term can be ignored, and therefore

$\begin{split}|\Omega|\partial_{t}w_{\infty}\simeq-\alpha\partial_{t}(s^{+}(t)-s^{-}(t))+\int_{\Omega_{\rm
Liq}}\sigma\dot{W}_{s}(x,t).\end{split}$

Replacing $\lambda=|\Omega|$, and
$\partial_{t}(s^{+}(t)-s^{-}(t))=-\frac{2}{\lambda}w_{\infty}(t)$, we get for
$\sigma=1$ and $\dot{W}_{s}(x,t)=\dot{W}(t)$,

$\partial_{t}w_{\infty}\simeq\frac{2\alpha}{\lambda^{2}}w_{\infty}+\dot{W}(t),\;\;\;\partial_{t}(s^{+}-s^{-})(t)=-\frac{2}{\lambda}w_{\infty}(t),$

with initial values $w_{\infty}(0)$ and $s^{+}(0)-s^{-}(0)$ (the initial
spread).

######  Remark 4.4.

Setting $w_{\infty}:=\beta+W$ we derive the following equivalent problem to
(4.10), (4.11)

$\begin{split}&\partial_{t}(s^{+}-s^{-})(t)=-\frac{2}{\lambda}(\beta(t)+W(t)),\\\
&s^{+}(0)-s^{-}(0)=\mbox{given},\end{split}$

for $\beta(t)$ the solution of

$\partial_{t}\beta(t)=\frac{2\alpha}{\lambda^{2}}(\beta(t)+W(t)),\;\;t>0,\;\;\;\;\beta(0)=\mbox{given}.$

## Acknowledgment

The research work was supported by the Hellenic Foundation for Research and
Innovation (H.F.R.I.) under the First Call for H.F.R.I. Research Projects to
support Faculty members and Researchers. (Project Number: HFRI-FM17-45).

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