The random process can easily be replaced by an other one and a different definition for the bounding area be chosen. Furthermore an extension of this approach to 3D is straight forward.

If you need to run this simulation multiple times, it is beneficial to put these steps together, e.g.

randomInTheBox = Prepend[
  Block[{randomMove = Transpose[Differences /@ 
         RandomFunction[WienerProcess[], {0, 100, .1}, 2]["States"]], 
         length, last, new},
    length = Length@randomMove;
    last = start;
    First@Last@Reap@Do[
      new = last + randomMove[[i]];
      If[new \[Element] box2D,
        last = new;
        Sow@new, Null],
      {i, Length@randomMove}]
  ],
  start];

The random process can easily be replaced by an other one and a different definition for the bounding area be chosen. Furthermore an extension of this approach to 3D is straight forward.

I chose the WienerProcess as the underlying random process, as this will simulate a Brownian motion.

Until Boundary Hit

Module[{rd = Transpose @ RandomFunction[WienerProcess[], {0, 1000, .01}, 2]["States"], length},
 length = LengthWhile[rd, # ∈ Rectangle[{-2, -2}, {+2, +2}] &];
 ListPlot[rd[[;; length]], Joined -> True, Mesh -> All, PlotRange -> {{-2.5, 2.5}, {-2.5, 2.5}}, 
  Epilog -> {EdgeForm[Thick], White, Opacity[0], Rectangle[{-2, -2}, {+2, +2}]}, ImageSize -> Large]
 ]

Other Direction Inside the Boundary

First the single moves as definded by a WienerProcess:

randomMove = Transpose[Differences /@ 
               RandomFunction[WienerProcess[], {0, 100, .1}, 2]["States"]];

These are

Length@randomMove

1000

steps.

We'll start at

start = {0., 0.};

and define the boundary as a square

box2D = Rectangle[{-2, -2}, {+2, +2}];

Now the random walk is inside this box is created with:

last = start;
walk = First@Last@Reap@Do[
  new = last + randomMove[[i]];
  If[new ∈ box2D,
   last = new;
   Sow@new, Null],
  {i, Length@randomMove}
  ];
randomInTheBox = Prepend[walk, start];

In my last run these where

Length@randomInTheBox

882

points.

A plot of the result:

ListPlot[randomInTheBox, Joined -> True, Mesh -> All, MeshStyle -> Black, AspectRatio -> 1, 
  Epilog -> {EdgeForm[{Thick, Red}], White, Opacity[0], 
             Rectangle[{-2, -2}, {+2, +2}]}, ImageSize -> Large]

The walk can be traced with

Manipulate[ListPlot[randomInTheBox, Joined -> True, Mesh -> All, 
  MeshStyle -> Black, AspectRatio -> 1, 
  Epilog -> {PointSize[Medium], Red, Point[randomInTheBox[[p]]], 
             EdgeForm[{Thick, Red}], Opacity[0], box2D}],
 {p, 1, Length@randomInTheBox, 1}]

The random process can easily replaced by an other one and a different definition for the bounding area be chosen. Furthermore an extension of this approach to 3D is straight forward.

I chose the WienerProcess as the underlying random process, as this will simulate a Brownian motion.

Until Boundary Hit

Module[{rd = Transpose @ RandomFunction[WienerProcess[], {0, 1000, .01}, 2]["States"], length},
 length = LengthWhile[rd, # ∈ Rectangle[{-2, -2}, {+2, +2}] &];
 ListPlot[rd[[;; length]], Joined -> True, Mesh -> All, PlotRange -> {{-2.5, 2.5}, {-2.5, 2.5}}, 
  Epilog -> {EdgeForm[Thick], White, Opacity[0], Rectangle[{-2, -2}, {+2, +2}]}, ImageSize -> Large]
 ]

Other Direction Inside the Boundary

First the single moves as definded by a WienerProcess:

randomMove = Transpose[Differences /@ 
               RandomFunction[WienerProcess[], {0, 100, .1}, 2]["States"]];

These are

Length@randomMove

1000

steps.

We'll start at

start = {0., 0.};

and define the boundary as a square

box2D = Rectangle[{-2, -2}, {+2, +2}];

Now the random walk inside this box is created with:

last = start;
walk = First@Last@Reap@Do[
  new = last + randomMove[[i]];
  If[new ∈ box2D,
   last = new;
   Sow@new, Null],
  {i, Length@randomMove}
  ];
randomInTheBox = Prepend[walk, start];

In my last run these where

Length@randomInTheBox

882

points.

A plot of the result:

ListPlot[randomInTheBox, Joined -> True, Mesh -> All, MeshStyle -> Black, AspectRatio -> 1, 
  Epilog -> {EdgeForm[{Thick, Red}], White, Opacity[0], 
             Rectangle[{-2, -2}, {+2, +2}]}, ImageSize -> Large]

The walk can be traced with

Manipulate[ListPlot[randomInTheBox, Joined -> True, Mesh -> All, 
  MeshStyle -> Black, AspectRatio -> 1, 
  Epilog -> {PointSize[Medium], Red, Point[randomInTheBox[[p]]], 
             EdgeForm[{Thick, Red}], Opacity[0], box2D}],
 {p, 1, Length@randomInTheBox, 1}]

The random process can easily be replaced by an other one and a different definition for the bounding area be chosen. Furthermore an extension of this approach to 3D is straight forward.

I chose the WienerProcess as the underlying random process, as this will simulate a Brownian motion.

Until Boundary Hit

Module[{rd = Transpose @ RandomFunction[WienerProcess[], {0, 1000, .01}, 2]["States"], length},
 length = LengthWhile[rd, # \[Element] Rectangle[{-2, -2}, {+2, +2}] &];
 ListPlot[rd[[;; length]], Joined -> True, Mesh -> All, PlotRange -> {{-2.5, 2.5}, {-2.5, 2.5}}, 
  Epilog -> {EdgeForm[Thick], White, Opacity[0], Rectangle[{-2, -2}, {+2, +2}]}, ImageSize -> Large]
 ]

Other Direction Inside the Boundary

First the single moves as definded by a WienerProcess:

randomMove = Transpose[Differences /@ 
               RandomFunction[WienerProcess[], {0, 100, .1}, 2]["States"]];

These are

Length@randomMove

1000

steps.

We'll start at

start = {0., 0.};

and define the boundary as a square

box2D = Rectangle[{-2, -2}, {+2, +2}];

Now the random walk is inside this box is created with:

last = start;
walk = First@Last@Reap@Do[
  new = last + randomMove[[i]];
  If[new \[Element] box2D,
   last = new;
   Sow@new, Null],
  {i, Length@randomMove}
  ];
randomInTheBox = Prepend[walk, start];

In my last run these where

Length@randomInTheBox

882

points.

A plot of the result:

ListPlot[randomInTheBox, Joined -> True, Mesh -> All, MeshStyle -> Black, AspectRatio -> 1, 
  Epilog -> {EdgeForm[{Thick, Red}], White, Opacity[0], 
             Rectangle[{-2, -2}, {+2, +2}]}, ImageSize -> Large]

The walk can be traced with

Manipulate[ListPlot[randomInTheBox, Joined -> True, Mesh -> All, 
  MeshStyle -> Black, AspectRatio -> 1, 
  Epilog -> {PointSize[Medium], Red, Point[randomInTheBox[[p]]], 
             EdgeForm[{Thick, Red}], Opacity[0], box2D}],
 {p, 1, Length@randomInTheBox, 1}]

I chose the WienerProcess as the underlying random process, as this will simulate a Brownian motion.

Until Boundary Hit

Module[{rd = Transpose @ RandomFunction[WienerProcess[], {0, 1000, .01}, 2]["States"], length},
 length = LengthWhile[rd, # ∈ Rectangle[{-2, -2}, {+2, +2}] &];
 ListPlot[rd[[;; length]], Joined -> True, Mesh -> All, PlotRange -> {{-2.5, 2.5}, {-2.5, 2.5}}, 
  Epilog -> {EdgeForm[Thick], White, Opacity[0], Rectangle[{-2, -2}, {+2, +2}]}, ImageSize -> Large]
 ]

Other Direction Inside the Boundary

First the single moves as definded by a WienerProcess:

randomMove = Transpose[Differences /@ 
               RandomFunction[WienerProcess[], {0, 100, .1}, 2]["States"]];

These are

Length@randomMove

1000

steps.

We'll start at

start = {0., 0.};

and define the boundary as a square

box2D = Rectangle[{-2, -2}, {+2, +2}];

Now the random walk is inside this box is created with:

last = start;
walk = First@Last@Reap@Do[
  new = last + randomMove[[i]];
  If[new ∈ box2D,
   last = new;
   Sow@new, Null],
  {i, Length@randomMove}
  ];
randomInTheBox = Prepend[walk, start];

In my last run these where

Length@randomInTheBox

882

points.

A plot of the result:

ListPlot[randomInTheBox, Joined -> True, Mesh -> All, MeshStyle -> Black, AspectRatio -> 1, 
  Epilog -> {EdgeForm[{Thick, Red}], White, Opacity[0], 
             Rectangle[{-2, -2}, {+2, +2}]}, ImageSize -> Large]

The walk can be traced with

Manipulate[ListPlot[randomInTheBox, Joined -> True, Mesh -> All, 
  MeshStyle -> Black, AspectRatio -> 1, 
  Epilog -> {PointSize[Medium], Red, Point[randomInTheBox[[p]]], 
             EdgeForm[{Thick, Red}], Opacity[0], box2D}],
 {p, 1, Length@randomInTheBox, 1}]

The random process can easily replaced by an other one and a different definition for the bounding area be chosen. Furthermore an extension of this approach to 3D is straight forward.